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We've got more than one million unique sudoku puzzles available for play, both online and the old fashioned way - with pencil and paper. Feel free to solve online just for fun, or, for an added challenge, register a free account and compete against thousands of other solvers to make it into our Hall of Fame!

Solve a Sudoku "

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We've got more than one million unique sudoku puzzles available for play, both online and the old fashioned way - with pencil and paper. Feel free to solve online just for fun, or, for an added challenge, register a free account and compete against thousands of other solvers to make it into our Hall of Fame!

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Computation Castle is a fun game that requires students to use their fractions, measurement, place value, and exponents skills to reverse a spell placed that caused the royal family to be turned into various animals.

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Computation Castle is a fun game that requires students to use their fractions, measurement, place value, and exponents skills to reverse a spell placed that caused the royal family to be turned into various animals.

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Reading time: ~10 min Reveal all steps

Welcome to the most spectacular game show on the planet! You now have a once-in-a-lifetime chance of winning a fantastic sports car which is hidden behind one of these three doors. Unfortunately, there are only goats behind the other two doors. Select one to make your choice!

Are you sure about that? You can still change your mind and select a different door...

I'm sure!

A great choice, but let me make life a little easier for you. I'll open one of the other doors with a goat, so that there are only two doors left for you to pick from. Do you want to stick with your choice, or do you want to swap?

I want to stay! I want to swap!

Ok - let's see how you did...

Open the doors...

Looks like you made the right choice. Congratulations, you just won a beautiful new sports car!

Replay game

If you play this game many times, you'll notice that you're more likely to win if you swap don't swap after the first door is opened, rather than sticking with your initial choice.

But how can this be - surely the car is equally likely to be behind each of the two remaining doors?

The explanation is very subtle. When you pick the initial door, the probability of being correct is 1 3 and the probability of being wrong is 2 3.

Artboard 20

After the game master opens one of the other doors, the probability of being wrong is This means that swapping doors still doubles triples halves your chance of winning. 2 3, except now all this probability is on just one door.

Artboard 21

Even if this doesn't seem very intuitive, we can prove that it is correct - simply by listing all different possibilities:

Out of the 9 possibilities need you to switch doors, to win. This gives a chance of 6 9=2 3 like before.

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The Monty Hall Problem - Probability - Mathigon

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Reading time: ~10 min Reveal all steps Welcome to the most spectacular game show on the planet! You now have a once-in-a-lifetime chance of winning a fantastic sports car which is hidden behind one of these three doors. Unfortunately, there are only goats behind the other two doors.

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Fruit Splat - Decimal Place Value Expanded Form

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Learn place value decimal expanded form with this fun arcade math game.

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Horrendous Soup: Capacity, Mass, & Distance 

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Horrendous Soup: Capacity, Mass, & Distance 

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Horrendous Soup is a fun game in which students have to make a recipe for the most disgusting soup they can imagine. Students progress through the game by adding ingredients to the soup. Some of the ingredients are measured in capacity, some are measured in mass, and some are measured in distance (for example, the recipe might call for a certain amount of millimeters of toothpaste or tree branch). Given this is a bit far-fetched, but still a perfect way to reinforce metric conversion skills. To successfully print out the recipe, students must demonstrate the ability to convert the amounts of ingredients to other metric units. For example, liters to milliliters and vice versa, grams to milligrams and vice versa, and millimeters to centimeters to meters. See demonstration video for more information.

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2023 MrNussbaum.com is a copyright of the Nussbaum Education Network,LLC.

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Mr. Nussbaum - Horrendous Soup - Online Game

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Horrendous Soup is a fun game in which students have to make a recipe for the most disgusting soup they can imagine. Students progress through the game by

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This site has been developed to assist Grades 4, 5, and 6 teachers to integrate technology into their mathematics classrooms, through the use of online math games

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Whether you know them as numbergrids, nonograms, nonogrids, paint-by-number or any other name (there are dozens!), we've got hundreds of thousands of unique puzzles here for you to solve. You can play for fun, or register a free account and compete against other players for the Numbergrid Hall of Fame!

How are our puzzles different? It may sound silly to say... but each of our puzzles is actually solvable! There are many online sites offering numbergrid-type puzzles, but far too many of these are computer-generated and either have multiple solutions, or they require the user to guess or use trail-and-error to complete them. Not ours - we've engineered every single puzzle on this site so that each has one unique solution, and is solvable using pure logical deduction. Guessing and trial-and-error are never required for any puzzle on this site.

If you are new to numbergrid puzzles, check out this Wikipedia page for helpful background information and many helpful solving tips. Then try a small 5x5 puzzle to get your feet wet.

Start a New Puzzle "

You'll never have to register an account to play on this site, but there are several reasons you may want to play as a registered user:

IT'S FREE! :-)

Monthly Competitions - registered players are automatically entered into our monthly scoring competitions, and are eligible for our Hall of Fame listings. Record solve times are also kept exclusively for registered members.

Resume Puzzles - save your progress on any puzzle and return to finish it later anytime within 24 hours.

Premium Upgrades - registered users are eligible for premium upgrades which remove all advertising from the site and result in a noticeably larger playing area on the puzzle solving page.

If any or all of the above features sound good to you, click on the button below to register for a free account:

Register for Free "

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Whether you know them as numbergrids, nonograms, nonogrids, paint-by-number or any other name (there are dozens!), we've got hundreds of thousands of unique puzzles here for you to solve. You can play for fun, or register a free account and compete against other players for the Numbergrid Hall of Fame!

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Fun Games for Math Practice
This exercise will help kids practice Math in a fun way. Kids related to games very well. From preschool / kindergarten to grade 6 levels of math games. There are games for the following topics:

Games to practice Addition, subtraction, Geometry, Comparison, Algebra, Shapes, Time, Fractions, Decimals, Sequence, Division, Metric system, Logarithms, ratios, probability, multiplication and more>>

The games include among other : memory games, Walk the plank, Fling the Teacher, En Garde Duel, Basketball Game, Penalty Shoot and more.

Math Worksheets & Printables

These worksheets are printable PDF exercises of the highest quality. Writing reinforces Maths learnt. These worksheets are from preschool, kindergarten to grade 6 levels of maths. The following topics are covered among others:

Worksheets to practice Addition, subtraction, Geometry, Comparison, Algebra, Shapes, Time, Fractions, Decimals, Sequence, Division, Metric system, Logarithms, ratios, probability, multiplication and more>>>

Ever wondered how to teach maths to kids as a teacher or parent? Simply let them watch our vividly presented videos. These videos cover a number of math topics and simply teach the lesson. The good thing is that they are videos which means they can be watched over and over again. We even have ipod compatible videos along these topics : Lessons to teach Addition, subtraction, Geometry, Comparison, Algebra, Shapes, Time, Fractions, Decimals, Sequence, Division, Metric system, Logarithms, ratios, probability, multiplication and more>>>

Math Quizzes and Online Tests

Math Quizzes Online - Addition, subtraction, Geometry, Comparison, Algebra, Shapes, Time, Fractions, Decimals, Sequence, Division, Metric system, Logarithms, ratios, probability, multiplication

These quizzes range from multiple choice math quizzes, gap fill quizzes, matching exercises, hotspot quizzes with graphics and more for interactive math practice.

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Lowest Common Multiple practice game for 5th grade

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Fun Games for Math Practice This exercise will help kids practice Math in a fun way. Kids related to games very well. From preschool / kindergarten to grade 6 levels of math games.

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Reading time: ~35 min Reveal all steps

For as long as humans have existed, we have looked to the sky and tried to explain life on Earth using the motion of stars, planets and the moon.

Ancient Greek astronomers were the first to discover that all celestial objects move on regular paths, called orbits. They believed that these orbits are always circular. After all, circles are the "most perfect" of all shapes: symmetric in every direction, and thus a fitting choice for the underlying order of our universe.

Every point on a circle has the same distance from its center. This means that they can be drawn using a

There are three important measurements related to circles that you need to know:

The radius is the distance from the center of a circle to its outer rim.

The diameter is the distance between two opposite points on a circle. It goes through its center, and its length is the radius.

The circumference (or perimeter) is the distance around a circle.

One important property of circles is that all circles are You can prove that by showing how all circles can be matched up using simply translations and

You might remember that, for similar polygons, the ratio between corresponding sides is always constant. Something similar works for circles: the ratio between the It is always 3.14159... - a mysterious number called which is often written as the Greek letter circumference and the diameter is equal for π for "p". all circles. Pi has infinitely many decimal digits that go on forever without any specific pattern:

Circles are perfectly symmetric, and they don't have any "weak points" like the corners of a polygon. This is one of the reasons why they can be found everywhere in nature:

And there are so many other examples: from rainbows to water ripples. Can you think of anything else?

It also turns out that a circle is the shape with the largest area for a given circumference. For example, if you have a rope of length 100 m, you can use it to enclose the largest space if you form a circle (rather than other shapes like a rectangle or triangle).

In nature, objects like water drops or air bubbles can save energy by becoming circular or spherical, and reducing their surface area.

The Area of a Circle

But how do we actually calculate the area of a circle? Let's try the same technique we used for finding the area of quadrilaterals: we cut the shape into multiple different parts, and then rearrange them into a different shape we already know the area of (e.g. a rectangle or a triangle).

The only difference is that, because circles are curved, we have to use some approximations:

Here you can see a circle divided into ${toWord(n1)} wedges. Move the slider, to line up the wedges in one row.

If we increase the number of wedges to this shape starts to look more and more like a

The height of the rectangle is equal to the of the circle. The width of the rectangle is equal to half the circumference the circumference twice the radius of the circle. (Notice how half of the wedges face down and half of them face up.)

If we could use infinitely many rings or wedges, the approximations above would be perfect - and they both give us the same formula for the area of a circle:

Calculating Pi

As you saw above, is not a simple integer, and its decimal digits go on forever, without any repeating pattern. Numbers with this property are called and it means that cannot be expressed as a simple fraction

It also means that we can never write down all the digits of Pi - after all, there are infinitely many. Ancient Greek and Chinese mathematicians calculated the first four decimal digits of Pi by approximating circles using regular polygons. Notice how, as you add more sides, the polygon starts to look like a circle:

In 1665, Isaac Newton managed to calculate 15 digits. Today, we can use powerful computers to calculate the value of Pi to much higher accuracy.

The current record is 31.4 trillion digits. A printed book containing all these digits would be approximately 400 km thick - that's the height at which the International Space Station orbits Earth!

Of course, you don't need to remember that many digits of Pi.

Many mathematicians believe that Pi has an even more curious property: that it is a normal number. This means that the digits from 0 to 9 appear completely at random, as if nature had rolled a 10-sided dice infinitely many times, to determine the value of Pi.

Here you can see the first 100 digits of Pi. Move over some of the cells, to see how the digits are distributed.

If Pi is normal, it means that you can think of any string of digits, and it will appear somewhere in its digits. Here you can search the first one million digits of Pi - do they contain your birthday?

We could even convert an entire book, like Harry Potter, into a very long string of digits (a = 01, b = 02, and so on). If Pi is normal, this string will appear somewhere in its digits - but it would take millions of years to calculate enough digits to find it.

Pi is easy to understand, but of fundamental importance in science and mathematics. That might be a reason why Pi has become unusually popular in our culture (at least, compared to other topics of mathematics):

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Reading time: ~35 min Reveal all steps For as long as humans have existed, we have looked to the sky and tried to explain life on Earth using the motion of stars, planets and the moon. Ancient Greek astronomers were the first to discover that all celestial objects move on regular paths, called orbits.

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Reading time: ~25 min Reveal all steps

Polygons appear everywhere in nature. They are especially useful if you want to tile a large area, because you can fit polygons together without any gaps or overlaps. Patterns like that are called tessellations.

Humans have copied many of these natural patterns in art, architecture and technology - from ancient Rome to the present. Here are a few examples:

Here you can create your own tessellations using regular polygons. Simply drag new shapes from the sidebar onto the canvas. Which shapes tessellate well? Are there any shapes that don't tessellate at all? Try to create interesting patterns!

Examples of other students' tessellations

Tessellations from regular polygons

You might have noticed that some regular polygons (like squares pentagons) tessellate very easily, while others (like pentagons triangles hexagons) don't seem to tessellate at all.

This has to do with the size of their At every internal angles, which we learned to calculate before. vertex in the tessellation, the internal angles of multiple different polygons meet. We need all of these angles to add up to °, otherwise there will either be a gap or an overlap.

You can similarly check that, just like pentagons, any regular polygon with 7 or more sides doesn't tessellate. This means that the only regular polygons that tessellate are triangles, squares and hexagons!

Of course you could combine different kinds of regular polygons in a tessellation, provided that their internal angles can add up to 360°:

Tessellations from irregular polygons

We can also try making tessellations out of irregular polygons - as long as we are careful when rotating and arranging them.

It turns out that you can tessellate not just equilateral triangles, but any triangle! Try moving the vertices in this diagram.

The sum of the internal angles in a triangle is °. If we use each angle twice once three times at every vertex in the tessellation, we get 360°:

More surprisingly, Their internal angle sum is °, so if we use each angle any quadrilateral also tessellates! once twice three times at every vertex in the tessellation, we we get 360°.

Pentagons are a bit trickier. We already saw that regular pentagons don't tessellate tessellate, but what about non-regular ones?

Here are three different examples of tessellations with pentagons. They are not regular, but they are perfectly valid 5-sided polygons.

So far, mathematicians have only found 15 different kinds of tessellations with (convex) pentagons - the most recent of which was discovered in 2015. No one knows if there are any others, or if these 15 are the only ones...

Tessellations in Art

Many artists, architects and designers use tessellations in their work. One of the most famous examples is the Dutch artist M. C. Escher. His works contain strange, mutating creatures, patterns and landscapes:

These artworks often look fun and effortless, but the underlying mathematical principles are the same as before: angles, rotations, translations and polygons. If the maths isn't right, the tessellation is not going to work!

Penrose Tilings

All the tessellations we saw so far have one thing in common: they are periodic. That means they consist of a regular pattern that is repeated again and again. They can continue forever in all directions and they will look the same everywhere.

In the 1970s, the British mathematician and physicist These are called Roger Penrose discovered Penrose tilings, and you only need a few different kinds of polygons to create one: non-periodic tessellations - they still continue infinitely in all directions, but never look exactly the same.

Penrose was exploring tessellations purely for fun, but it turns out that the internal structure of some real materials (like aluminium) follow a similar pattern. The pattern was even used on toilet paper, because the manufacturers noticed that a non-periodic pattern can be rolled up without any bulges.

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Tessellations - Polygons and Polyhedra - Mathigon

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Reading time: ~25 min Reveal all steps Polygons appear everywhere in nature. They are especially useful if you want to tile a large area, because you can fit polygons together without any gaps or overlaps. Patterns like that are called tessellations.

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Math Games: Fruit Splat Least Common Multiple

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Practice finding the least common multiple in this fun free arcade flash math game.

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Welcome to Whimsical Windows. It is here, behind one of the 15 windows, where you can find the long lost black unicorn stallion. Each window, however, is locked and only a special equation can open it. To find the equation, you'll need to look at the relationship between the ordered pairs etched into each window. One equation governs each of the ordered pairs. Such a relationship is referred to as a function. Click on a window and find the equation. If you are correct, the window will open and you'll discover one of forty possible animated animals. You may find a tiger, giraffe, giant panda, koala, or others such as sabertooth tiger, wholly mammoth, the kraken, white unicorn, and of course the mythical and mysterious black unicorn stallion. The game ends, and all of the windows open when you solve the equation for the window that opens to the black unicorn stallion.

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Welcome to Whimsical Windows. It is here, behind one of the 15 windows, where you can find the long lost black unicorn stallion. Each window, however, is locked and only a special equation can open it. To find the equation, you'll need to look at the relationship between the ordered pairs etched into each window.

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Tangram | Free Virtual Manipulatives | Toy Theater

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Free online tangram puzzles | Mobile + tablet friendly | Designed using physics, these tangrams give the feel of the real thing with no pieces to clean up!

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Welcome to ZooDesigner. You have been hired to design five enclosures for the animals at a local zoo. You have to use your knowledge of how to calculate area and perimeter to design the correct enclosures and to earn your ZooDesigner Points. Use the blueprint area to sketch out the dimensions (area, perimeter, or area and perimeter) of the enclosure. If you design the enclosures incorrectly, the animals will escape and the zoo visitors will run for their lives. You, of course, will be fired!

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Welcome to ZooDesigner. You have been hired to design five enclosures for the animals at a local zoo. You have to use your knowledge of how to calculate ar

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Math Games: Fruit Splat Fractions Addition

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Practice adding fractions in this fun free arcade flash math game.

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Math Games: Fruit Splat Fractions Subtraction

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Practice subtracting fractions in this fun free arcade flash game.

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In Ratio Rumble, players choose from among 18 different characters as their avatar in the tournament.

Ratio Rumble is all about the player re-creating each ratio recipe given to them by the game. In this case, the game asks for two blue potions and one red potion. The player used double the original ratio. This is possible because the ratio is still the same even if the quantities are doubled.

Players can use health potions found on the gameboard to heal their characters.

Potions with lightning symbols, when played as part of a ratio, will weaken the opponent. Potions with numbers are time bombs that tick down each turn. It's important to use these potions in your ratios. If they are not used before they finish their countdown, they explode and weaken the character's health.

In Ratio Rumble, players learn to create ratios following numerical recipes. Ratios get more complicated as the game progresses and as the player masters new levels of difficulty.

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Students engaged in Math Snacks in class learned more math than those with access to just classroom instruction.

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Reading time: ~30 min Reveal all steps

Let's start simple: a triangle is a closed shape that has three sides (which are It also has three line segments) and three vertices (the points where the sides meet). internal angles, and we already know that the sum of them is always °.

We can classify triangles by the size of their angles:

For convenience, we always label triangles in the same way. The vertices are labelled with capital letters A, a, α, β and γ ("alpha", "beta" and "gamma"). b and c, and the angles are labelled with Greek letters B and C, the sides are labelled with lowercase letters

The side that lies angle that lies right next to The same pattern works for opposite vertex B/ b/ β and for C/ c/ γ. A is labeled A is labelled α. a, and the

Medians

Here you can see a triangle as well as the midpoints of its three sides.

A median of a triangle is a line segment that joins a vertex and the midpoint of the opposite side. Draw the three medians of this triangle. What happens as you move the vertices of the triangle?

It seems like the medians always This point is called the intersect in one point centroid. have the same length divide each other in the middle.

Medians always divide each other in the ratio 2:1. For each of the three medians, the distance from the vertex to the centroid is always twice three times exactly as long as the distance from the centroid to the midpoint.

The centroid is also the "balancing point" of a triangle. Draw a triangle on some cardboard, cut it out, and find the three medians. If you were accurate, you can now balance the triangle on the tip of a pencil, or hang it perfectly level from a piece of string that's attached to its centroid:

This works because the weight of the triangle is evenly distributed around the centroid. In physics, this point is often called the center of mass.

Perpendicular Bisectors and Circumcircle

Recall that the perpendicular bisector of a line is the perpendicular line that goes through its midpoint endpoints.

Draw the perpendicular bisector of all three sides of this triangle. To draw the perpendicular bisector of a side of the triangle, simply click and drag from one of its endpoints to the other.

Like before, the three perpendicular bisectors meet in a single point. And again, this point has a special property.

Any point on a perpendicular bisector has the same distance from the two endpoints of the lines it bisects. For example, any point on the blue bisector has the same distance from points A and C and any point on the red bisector has the same distance from points A and B A and C B and C.

The intersection point lies on all three perpendicular bisectors, so it must have the same distance from all three vertices sides of the triangle.

This means we can draw a circle around it that perfectly touches all the vertices. This circle is called the circumcircle of the triangle, and the center is called the circumcenter.

In fact, this means that if you are given any three points, you can use the circumcenter to find a circle that goes through all three of them. (Unless the points are collinear parallel, in which case they all lie on a straight line.)

Angle Bisectors and Incircle

You're probably getting the hang of this now: we pick a certain construction, do it three times for all sides/angles of the triangles, and then we work out what's special about their intersection.

Recall that the angle bisector divides an angle exactly in the middle. Draw the angle bisector of the three angles in this triangle. To draw an angle bisector, you have to click on three points that form the angle you want to bisect.

Once again, all three lines intersect at one point. You probably expected something like this, but it is important to notice that there is no obvious reason why this should happen - triangles are just very special shapes! Continue

Points that lie on an angle bisector have the same distance from the two lines that form the angle. For example any point on the blue bisector has the same distance from side a and side c, and any point on the red bisector has the same distance from sides a and b a and c b and c.

The intersection point lies on all three bisectors. Therefore it must have the same distance from all three sides vertices of the triangle.

This means we can draw a circle around it, that lies inside the triangle and just touches its three sides. This circle is called the incircle of the triangle, and the center is called the incenter.

Area and Altitudes

Finding the area of a rectangle is easy: you simply multiply its width by its height. Finding the area of a triangle is a bit less obvious. Let's start by "trapping" a triangle inside a rectangle. Continue

The width of the rectangle is the length of the bottom side of the triangle (which is called the base). The height of the rectangle is the perpendicular distance from the base to the opposite vertex. Continue

The height divides the triangle into two parts. Notice how the two gaps in the rectangle are exactly as big as the two parts of the triangle. This means that the rectangle is twice as three times as exactly as large as the triangle.

We can easily work out the area of the rectangle, so the area of the triangle must be half that:

A=1 2× base × height

To calculate the area of a triangle, you can pick any of the three sides as base, and then find the corresponding height, which is the line that is perpendicular parallel to the base and goes through the opposite vertex.

In triangles, these heights are often called altitudes. Every triangle has altitudes.

Like the medians, This is called the perpendicular bisectors and orthocenter of the triangle. angle bisectors, the three altitudes of a triangle intersect in a single point.

In acute triangles, the orthocenter lies inside lies outside is a vertex of the triangle.

In obtuse triangles, the orthocenter lies outside lies inside is a vertex of the triangle.

In right-angled triangles, the orthocenter is a vertex of lies inside lies outside the triangle. Two of the altitudes are actually just sides of the triangle.

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Midsegments and Similarity

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Properties of Triangles - Triangles and Trigonometry - Mathigon

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Reading time: ~30 min Reveal all steps Let's start simple: a triangle is a closed shape that has three sides (which are It also has three line segments) and three vertices (the points where the sides meet). internal angles, and we already know that the sum of them is always °.

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Number square starter to promote reasoning skills in the classroom.

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Reading time: ~35 min Reveal all steps

different parts of an object look But using transformations, we can give a much more precise, mathematical definition of what symmetry Symmetry is everywhere around us, and an intuitive concept: the same in some way. really means:

An object is symmetric if it looks the same, even after applying a certain transformation.

We can reflect this butterfly, and it looks the same afterwards. We say that it has reflectional symmetry.

We can rotate this flower, and it looks the same afterwards. We say that it has rotational symmetry.

Reflectional Symmetry

A shape has reflectional symmetry if it looks the same after being reflected. The line of reflection is called the and it splits the shape into two halves. Some figures can also have more than one axis of symmetry.

Draw all axes of symmetry in these six images and shapes:

Many letters in the alphabet have reflectional symmetry. Select all the ones that do:

Here are some more shapes. Complete them so that they have reflectional symmetry:

Shapes, letters and images can have reflectional symmetry, but so can entire numbers, words and sentences!

For example "25352" and "ANNA" both read the same from back to front. Numbers or words like this are called Can you think of any other palindromes?

If we ignore spaces and punctuation, the short sentences below also have reflectional symmetry. Can you come up with your own?

But Palindromes are not just fun, they actually have practical importance. A few years ago, scientists discovered that parts of our DNA are palindromic. This makes them more resilient to mutations or damage - because there is a second backup copy of every piece.

Rotational Symmetry

A shape has The rotational symmetry if it looks the same after being rotated (by less than 360°). center of rotation is usually just the middle of the shape.

The You can also think about it as the order of symmetry is the number of distinct orientations in which the shape looks the same. number of times we can rotate the shape, before we get back to the start. For example, this snowflake has order .

Find the order and the angle of rotation, for each of these shapes:

Now complete these shapes, so that they have rotational symmetry:

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Symmetry Groups and Wallpapers

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Symmetry - Transformations and Symmetry - Mathigon

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Reading time: ~35 min Reveal all steps different parts of an object look But using transformations, we can give a much more precise, mathematical definition of what symmetry Symmetry is everywhere around us, and an intuitive concept: the same in some way.

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Reading time: ~30 min Reveal all steps

At the beginning of this course we defined regular polygons as particularly "symmetric" polygons, where all sides and angles are the same. We can do something similar for polyhedra.

In a Polyhedra with these two properties are called regular polyhedron all Platonic solids, named after the Greek philosopher Plato. faces are all the same kind of regular polygon, and the same number of faces meet at every vertex.

So what do the Platonic solids look like - and how many of them are there? To make a three-dimensional shape, we need at least faces to meet at every vertex. Let's start systematically with the smallest regular polygon: equilateral triangles:

If we create a polyhedron where three It is called a equilateral triangles meet at every vertex, we get the shape on the left. Tetrahedron and has faces. ("Tetra" means "four" in Greek).

If four equilateral triangles meet at every vertex, we get a different Platonic solid. It is called the Octahedron and has faces. ("Octa" means "eight" in Greek. Just like "Octagon" means 8-sided shape, "Octahedron" means 8-faced solid.)

If triangles meet at every vertex, we get the Icosahedron. It has faces. ("Icosa" means "twenty" in Greek.)

If triangles meet at every vertex, something different happens: we simply get a tessellation a quadrilateral another Icosahedron, instead of a three-dimensional polyhedron.

And seven or more triangles at every vertex also don't produce new polyhedra: there is not enough space around a vertex, to fit that many triangles.

This means we've found Platonic solids consisting of triangles. Let's move on to the next regular polygon: squares.

If squares meet at every vertex, we get the The cube is sometimes also called cube. Just like dice, it has faces. Hexahedron, after the Greek word "hexa" for "six".

If squares meet at every vertex, we get another tessellation a tetrahedron another cube. And like before, five or more squares also won't work.

Next, let's try regular pentagons:

If pentagons meet at every vertex, we get the Dodecahedron. It has faces. ("Dodeca" means "twelve" in Greek.)

Like before, four or more pentagons don't work are possible because there is not enough space.

The next regular polygon to try are hexagons:

If three hexagons meet at every vertex, we immediately get a tessellation polyhedron hexahedron. Since there is no space for more than three, it seems like there are no Platonic solids consisting of hexagons.

The same also happens for all regular polygons with more than six sides. They don't tessellate, and we certainly don't get any three-dimensional polygons.

This means that there are just Platonic solids! Let's have a look at all of them together:

Notice how the number of faces and vertices are These pairs of Platonic solids are called swapped around the same for cube and octahedron, as well as dodecahedron and icosahedron, while the number of edges dual solids. stays the same are different.

We can turn a polyhedron into its dual, by "replacing" every face with a vertex, and every vertex with a face. These animations show how:

The tetrahedron is dual with itself. Since it has the same number of faces and vertices, swapping them wouldn't change anything.

Plato believed that all matter in the Universe consists of four elements: Air, Earth, Water and Fire. He thought that every element correspond to one of the Platonic solids, while the fifth one would represent the universe as a whole. Today we know that there are more than 100 different elements which consist of spherical atoms, not polyhedra.

Archimedean Solids

Platonic solids are particularly important polyhedra, but there are countless others.

They are named after another Greek mathematician, Archimedean solids, for example, still have to be made up of Archimedes of Syracuse, and there are 13 of them: regular polygons, but you can use multiple different types.

Applications

Plato was wrong in believing that all elements consists of Platonic solids. But regular polyhedra have many special properties that make them appear elsewhere in nature - and we can copy these properties in science and engineering.

Many viruses, bacteria and other small organisms are shaped like icosahedra. Viruses, for example, must enclose their genetic material inside a shell of many identical protein units. The icosahedron is the most efficient way to do this, because it consists of a few regular elements but is almost shaped like a sphere.

Many The most famous example is molecules are shaped like regular polyhedra. C 60 which consists of 60 carbon atoms arranged in the shape of a Truncated Icosahedron.

It was discovered in 1985 when scientists researched interstellar dust. They named it "Buckyball" (or Buckminsterfullerene) after the architect Buckminster Fuller, famous for constructing similar-looking buildings.

Most crystals have their atoms arranged in a regular grid consisting of tetrahedra, cubes or octahedra. When they crack or shatter, you can see these shapes on a larger scale.

Tetrahedra and octahedra are incredibly rigid and stable, which makes them very useful in construction. Space frames are polygonal structures that can support large roofs and heavy bridges.

Platonic solids are also used to create because of their symmetry, every side has the dice. probability of landing facing up - so the dice are fair.

The Truncated Icosahedron is probably the most famous polyhedron in the world: it is the shape of the football.

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Platonic Solids - Polygons and Polyhedra - Mathigon

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Geometric shapes are everywhere around us. In this course you will learn about angels, polygons, tessellations, polyhedra and nets.

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Reading time: ~25 min Reveal all steps

We have now reached an important point in geometry - being able to state and understand one of the most famous theorems in all of mathematics: Pythagoras' Theorem. It is named after the ancient Greek mathematician Pythagoras of Samos.

In any right-angled triangle, the square of the length of the Pythagoras' Theorem
hypotenuse (the side that lies opposite the right angle) is equal to the sum of the squares of the other two sides. In other words,

a b c 2 2 = 2 +

The converse is also true: if the three sides in a triangle satisfy a 2 right-angled acute obtuse. +b 2=c 2, then it must be

Right angles are everywhere, and that's why Pythagoras' Theorem is so useful.

Here you can see a The bottom of the ladder is 6m long ladder leaning on a wall. 1m away from the wall. How far does it reach up the wall?

Notice that there is a right-angled triangle formed by the ladder, the wall and the ground. Using Pythagoras' theorem, we get

Whenever you've got a right-angled triangle and know two of its sides, Pythagoras can help you find the third one.

Proving Pythagoras' Theorem

Pythagoras' theorem was known to ancient Babylonians, Mesopotamians, Indians and Chinese - but Pythagoras may have been the first to find a formal, mathematical proof.

There are actually many different ways to prove Pythagoras' theorem. Here you can see three different examples that each use a different strategy:

Rearrangement

Have a look at the figure on the right. The square has side length a+b, and contains four right-angled triangles, as well as a smaller square of area a−b a+b .

Now let's rearrange the triangles in the square. The result still contains the four right-angles triangles, as well as two squares of size and .

Comparing the area of the red area before and after the rearrangement, we see that

a 2+b 2=c 2.

This is the original proof that Pythagoras came up with.

Algebra

Here we have the same figure as before, but this time we'll use algebra rather than rearrangement to prove Pythagoras' theorem.

The large square has side length a a 2+b 2 . +b and area

It consists of four triangles, each with an area of 1 2 a b 1 2 a+b , and one square of area a×b .

If we combine all of that information, we have

a+b 2

=4×1 2 a b+c 2

a 2+2 a b+b 2

=2 a b+c 2

a 2+b 2

=c 2

And, once again, we get Pythagoras' theorem.

Similar Triangles

Here you can see another right-angled triangle. If we draw one of the altitudes, it splits the triangle into two smaller triangle. It also divides the hypotenuse c into two smaller parts which we'll call x and y. Continue

Let's separate out the two smaller triangles, so that it's clearer to see how they are related... Continue

Both smaller triangles share one angle with the original triangle. They also all have one right angle. By the AA condition, all three triangles must be similar congruent right-angled.

Now we can use the equations we already know about similar polygons:

Continue

But remember that c=x + y . Therefore

c=a 2 c+b 2 c

c 2=a 2+b 2

Once more, we've proven Pythagoras' theorem!

Much about Pythagoras' life is unknown, and no original copies of his work have survived. He founded a religious cult, the Pythagoreans, that practiced a kind of "number worship". They believed that all numbers have their own character, and followed a variety of other bizarre customs.

The Pythagoreans are credited with many mathematical discoveries, including finding the first irrational number, 2 . Irrational numbers cannot be expressed as a simple fraction - a concept the Pythagoreans found deeply troubling and (unsuccessfully) tried to cover up!

Calculating Distances

One of the most important application of Pythagoras' Theorem is for calculating distances.

On the right you can see two points in a coordinate system. We could measure their distance using a ruler, but that is not particularly accurate. Instead, let's try using Pythagoras. Continue

We can easily count the horizontal distance along the x-axis, and the vertical distance along the y-axis. If we draw those two lines, we get a right-angled triangle.

Using Pythagoras,

d 2

=${b.x-a.x} 2+${b.y-a.y} 2

d 2

=${(b.x-a.x)**2 + (b.y-a.y)**2}

d

=${(b.x-a.x)**2 + (b.y-a.y)**2} =${round(distance(a,b),4)}

This method works for any two points:

If you are given two points with coordinates ( The Distance Formula
x 1, y 1) and (x 2, y 2), the distance between them is

d 2=x 2−x 1 2+y 2−y 1 2

d=x 2−x 1 2+y 2−y 1 2

Pythagorean Triples

As you moved the However there are a few examples of right-angled triangles where the lengths of vertices of the triangle in the previous step, you might have noticed that in most cases, the length of the hypothenuse all three sides happens to be whole numbers. d ended up being a decimal number fraction integer.

One famous example is the 3-4-5 triangle. Since 3 2+4 2=5 2, any triangle with sides of length 3, 4 and 5 must be right-angled.

The ancient Egyptians didn't know about Pythagoras' theorem, but they did know about the 3-4-5 triangle. When building the pyramids, they used knotted ropes of lengths 3, 4 and 5 to measure perfect right angles.

Three integers like this are called Pythagorean Triples. (3, 4, 5) is one example of a Pythagorean triple. If we multiply every number by 2, we get another Pythagorean triple: (6, 8, ).

We can think of these triples as grid points in a coordinate systems. For a valid Pythagorean triples, the distance from the origin to the grid point has to be a whole number. Using the coordinate system below, can you find any other Pythagorean triples?

Do you notice any pattern in the distribution of these points?

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Isosceles and Equilateral Triangles

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Pythagoras' Theorem - Triangles and Trigonometry - Mathigon

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Reading time: ~25 min Reveal all steps We have now reached an important point in geometry - being able to state and understand one of the most famous theorems in all of mathematics: Pythagoras' Theorem. It is named after the ancient Greek mathematician Pythagoras of Samos.

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Tilted Squares printable sheet

It's easy to work out the areas of squares drawn on a grid if they are oriented in the usual way:

Can you find a quick and easy method to work out the areas of tilted squares?

Here are some squares with a tilt of 1:

See the hint for suggested ways to calculate their areas.

Notice anything special about their areas?
Can you predict the areas of other squares with a tilt of 1?

What about squares with a tilt of 2? Or 3? Or 4? Or...?
Notice anything interesting?

Can you make any conjectures about the areas of tilted squares?
Can you prove your conjectures?

You might like to use the interactivity below to help you to draw tilted squares.

Full Screen tablet/mobile friendly version

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md angles polygons and geom proof - stage 3

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pm - posing questions. making conjectures

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Tilted Squares printable sheet It's easy to work out the areas of squares drawn on a grid if they are oriented in the usual way: Can you find a quick and easy method to work out the areas of tilted squares?

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Pascal’s Triangle – Sequences and Patterns – Mathigon

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Learn about some of the most fascinating patterns in mathematics, from triangle numbers to the Fibonacci sequence and Pascal’s triangle.

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Prime Numbers: The Sieve of Eratosthenes

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Prime Numbers: The Sieve of Eratosthenes

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When calculating these factor pairs, it can happen that a number doesn't have any factors except for the first pair. One example is 13 - its only factors are 1 and 13 itself. These special numbers are called They can't be broken up into products of smaller numbers, which, in a way, makes them the "atoms of numbers". Prime numbers.

Note that 1 itself is not a prime number, so the first few prime numbers are 2, 3, 5, 7, 11, 13, ...

Any number which is not prime can be written as the product of prime numbers: we simply keep dividing it into more parts until all factors are prime. For example,

Now 2, 3 and 7 are prime numbers and can't be divided further. The product 2 × 2 × 3 × 7 is called the prime factorisation of 84, and 2, 3 and 7 are its prime factors. Note that some primes, like 2 in this case, can appear multiple times in a prime factorisation.

Every integer has a prime factorisation and no two integers have the same prime factorisation. Furthermore, there is only a single way to write any number as a product of primes - unless we count different orderings of the primes. This is called the Fundamental Theorem of Arithmetic (FTA).

Using the FTA can make many problems in mathematics much easier: we divide numbers into their prime factors, we solve the problem for the individual primes, which can often be much easier, and then we combine these results to solve the initial problem.

The Sieve of Eratosthenes

It turned out to be quite difficult to determine if a number is prime: you always had to find Instead, the Greek mathematician the all its prime factors, which gets more and more challenging as the numbers get bigger. Eratosthenes of Cyrene came up with a simple algorithm to find all prime numbers up to 100: Sieve of Eratosthenes.

First we need to write down all numbers up to 100.

We know that 1 is not prime, so we delete it.

The smallest prime number is 2. Any multiple of 2 can't be prime, since it has 2 as a factor. Therefore we can cross out all multiples of 2.

The next number in our list is 3 - again a prime number. All multiples of 3 can't be prime, since they have 3 as a factor. Therefore we can cross these out as well.

The next number, 4, is already crossed out so we move on to 5: it is a prime number and again we cross out all multiples of 5.

The next prime number must be , since 6 is crossed out. Once more, we cross out all of its multiples.

The next prime number is . Notice, however, that all of its multiples are already crossed out multiples of 3. The same is actually true for all other remaining numbers. Therefore all these remaining numbers must be prime.

Now we can count that, in total, there are prime numbers less than 100.

How many Prime Numbers are there?

Of course we can also use the Sieve of Eratosthenes to find larger prime numbers. There are 21 primes between 100 and 200, 16 primes between 200 and 300, 17 primes between 400 and 500 and only 11 between 10,000 and 10,100.

The primes seem to keep getting more and more spread out, but do they ever stop? Is there a biggest or a last prime number?

The ancient Greek mathematician Euclid of Alexandria first proved that there are infinitely many prime numbers, using the following argument:

1. Suppose there were only finitely many prime numbers.

2. Let us multiply all of them together, to get a very large number which we call N.

3. Now let's think about Any prime number that divides And since all prime numbers we have found so far divide N + 1. N can't also divide N, none of these can also divide N + 1. N + 1.

4. We know from the Either Fundamental Theorem of Arithmetic that N + 1 is itself prime, or there is some other new prime P' that divides N + 1. N + 1, must have a prime factor.

5. In both cases we've found a new prime not in our original list - but we assumed that all primes were in this list.

6. Clearly something went wrong! But since steps 2-4 were definitely valid, the only possibility is that our initial assumption in 1 was wrong. This means there must actually be infinitely many primes.

Euclid's explanation is one of the first examples in history of a formal mathematical This example is often called proof - a logical argument that shows a statement must definitely be true. proof by contradiction: we start with an assumption, deduce something impossible, and thus know that our assumption must be incorrect.

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Prime Numbers - Divisibility and Primes - Mathigon

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Prime numbers represent both the most basic properties and the most complex unsolved problems. Here you can learn about the building blocks of mathematics.

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Sequences and Patterns | Fibonacci Numbers

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Learn about some of the most fascinating patterns in mathematics, from triangle numbers to the Fibonacci sequence and Pascal's triangle.

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Take an equilateral triangle and cut it into smaller pieces. What can you do with them?

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2023 MrNussbaum.com is a copyright of the Nussbaum Education Network,LLC.

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This innovative game requires students to “purchase” as many of the world’s most important landmarks as possible with ten billion dollars. Students must pu

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3D Solids: Spheres, Cones, and Cylinders

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3D Solids: Spheres, Cones, and Cylinders

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Reading time: ~50 min Reveal all steps

In the previous sections, we studied the properties of circles on a flat surface. But our world is actually three-dimensional, so lets have a look at some 3D solids that are based on circles:

A cylinder consists of two congruent, parallel circles joined by a curved surface.

A cone has a circular base that is joined to a single point (called the vertex).

Every point on the surface of a sphere has the same distance from its center.

Notice how the definition of a sphere is almost the same as the definition of a - except in three dimensions!

Cylinders

Here you can see the cylindrical Gasometer in Oberhausen, Germany. It used to store natural gas which was used as fuel in nearby factories and power plants. The Gasometer is 120m tall, and its base and ceiling are two large circles with radius 35m. There are two important questions that engineers might want to answer:

How much natural gas can be stored? This is the of the cylinder.

How much steel is needed to build the Gasometer? This is (approximately) the of the cylinder.

Let's try to find formulas for both these results!

Volume of a Cylinder

The top and bottom of a cylinder are two congruent circles, called bases. The of a cylinder is the perpendicular distance between these bases, and the of a cylinder is simply the radius of the circular bases.

We can approximate a cylinder using a ${n}-sided As the number of sides increases, the prism starts to look more and more like a cylinder:

Even though a cylinder is technically not a prism, they share many properties. In both cases, we can find the volume by multiplying the area of their base with their height. This means that a cylinder with radius r and height h has volume

In the examples above, the two bases of the cylinder were always directly above each other: this is called a right cylinder. If the bases are not directly above each other, we have an oblique cylinder. The bases are still parallel, but the sides seem to "lean over" at an angle that is not 90°.

The volume of an oblique cylinder turns out to be exactly the same as that of a right cylinder with the same radius and height. This is due to Cavalieri's Principle, named after the Italian mathematician Bonaventura Cavalieri: if two solids have the same cross-sectional area at every height, then they will have the same volume.

Imagine slicing a cylinder into lots of thin disks. We can then slide these disks horizontal to get an oblique cylinder. The volume of the individual discs does not change as you make it oblique, therefore the total volume also remains constant:

Surface Area of a Cylinder

To find the surface area of a cylinder, we have to "unroll" it into its flat You can try this yourself, for example by peeling off the label on a can of food.

There are two , one at the top and one at the bottom of the cylinder. The curved side is actually a large .

The two circles each have area .

The height of the rectangle is and the width of the rectangle is the same as the of the circles: .

This means that the total surface area of a cylinder with radius r and height h is given by

Cylinders can be found everywhere in our world - from soda cans to toilet paper or water pipes. Can you think of any other examples?

The Gasometer above had a radius of 35m and a height of 120m. We can now calculate that its volume is approximately and its surface area is approximately

Cones

A cone is a 3-dimensional solid that has a circular base. Its side "tapers upwards" as shown in the diagram, and ends in a single point called the vertex.

The radius of the cone is the radius of the circular base, and the height of the cone is the perpendicular distance from the base to the vertex.

Just like other shapes we met before, cones are everywhere around us: ice cream cones, traffic cones, certain roofs, and even christmas trees. What else can you think of?

Volume of a Cone

We previously found the volume of a cylinder by approximating it using a prism. Similarly, we can find the volume of a cone by approximating it using a

Here you can see a ${n}-sided pyramid. As the number of sides increases, the pyramid starts to look more and more like a cone. In fact, we could think of a cone as a pyramid with infinitely many sides!

Notice the similarity with the equation for the volume of a cylinder. Imagine drawing a cylinder around the cone, with the same base and height - this is called the circumscribed cylinder. Now, the cone will take up exactly of the volume of the cylinder:

Note: You might think that infinitely many tiny sides as an approximation is a bit "imprecise". Mathematics spent a long time trying to find a more straightforward way to calculate the volume of a cone. In 1900, the great mathematician David Hilbert even named it as one of the 23 most important unsolved problems in mathematics! Today we know that it is actually impossible.

Just like a cylinder, a cone doesn't have to be "straight". If the vertex is directly over the center of the base, we have a right cone. Otherwise, we call it an oblique cone.

Once again, we can use Cavalieri's principle to show that all oblique cones have the same volume, as long as they have the same base and height.

Surface Area of a Cone

Finding the surface area of a cone is a bit more tricky. Like before, we can unravel a cone into its net. Move the slider to see what happens: in this case, we get one circle and one .

Now we just have to add up the area of both these components. The base is a circle with radius r, so its area is

The radius of the sector is the same as the distance from the rim of a cone to its vertex. This is called the slant height s of the cone, and not the same as the normal . We can find the slant height using Pythagoras:

Finally, we just have to add up the area of the base and the area of the sector, to get the total surface are of the cone:

Spheres

A sphere is a three-dimensional solid consisting of all points that have the same distance from a given . This distance is called the of the sphere.

You can think of a sphere as a "three-dimensional circle ". Just like a circle, a sphere also has a , which is the length of the radius, as well as chords and secants.

In a previous section, you learned how the Greek mathematician Eratosthenes calculated the radius of Earth using the shadow of a pole - it was 6,371 km. Now, let's try to find the Earth's total volume and surface area. Continue

Volume of a Sphere

To find the volume of a sphere, we once again have to use Cavalieri's Principle. Let's start with a hemisphere - a sphere cut in half along the equator. We also need a cylinder with the same radius and height as the hemisphere, but with an inverted cone "cut out" in the middle.

As you move the slider above, you can see the cross-section of both these shapes at a specific height above the base:

Let us try to find the cross-sectional area of both these solids, at a distance above the base.

If you compare the equations for the volume of a cylinder, cone and sphere, you might notice one of the most satisfying relationships in geometry. Imagine we have a cylinder with the same height as the diameter of its base. We can now fit both a cone and a sphere perfectly in its inside:

Notice how, if we the volume of the cone and the sphere, we get exactly the volume of the cylinder!

Surface Area of a Sphere

Finding a formula for the surface area of a sphere is very difficult. One reason is that we can't open and "flatten" the surface of a sphere, like we did for cones and cylinders before.

This is a particular issue when trying to create maps. Earth has a curved, 3-dimensional surface, but every printed map has to be flat and 2-dimensional. This means that Geographers have to cheat: by stretching or squishing certain areas.

Here you can see few different types of maps, called projections. Try moving the red square, and watch what this area actually looks like on a globe:

To find the surface area of a sphere, we can once again approximate it using a different shape - for example a polyhedron with lots of faces. As the number of faces increases, the polyhedron starts to look more and more like a sphere.

COMING SOON: Sphere Surface Area Proof

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Spheres, Cones and Cylinders | Circles and Pi

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Reading time: ~50 min Reveal all steps In the previous sections, we studied the properties of circles on a flat surface. But our world is actually three-dimensional, so lets have a look at some 3D solids that are based on circles: A cylinder consists of two congruent, parallel circles joined by a curved surface.

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Reading time: ~20 min Reveal all steps

The circle is one of four different shapes which can be created using "slices" through a cone. This can be demonstrated using the light cone of a torch:

If you point the torch vertically downwards, you see a If you tilt the cone, you get an If you tilt it even further, you get a circle ellipse oval of light. ellipse. parabola or a hyperbola.

Collectively, these four shapes are called conic sections. Even though they all look very different, they are closely related: in fact, they can all be generated using the same equation!

Conic sections were first studied by the ancient Greek mathematician Apollonius of Perga, who also gave them their unusual names.

In later courses, you'll learn much more about parabolas and hyperbolas. For now, let's have a closer look at the ellipse.

Ellipses

An ellipse just looks almost like an "elongated circle". In fact, you could think about it as a circle with Just like every point on a circle has the same distance from its center, every point on an ellipse has the same two centers - these are called focal points. sum of distances to its two focal points.

If you have a long string connected to two fixed points, you can draw a perfect ellipse by tracing the maximum reach of the strings:

Coming soon: Ellipses drawing interactive

There are many other physical representations of how you could draw an ellipse:

Planetary Orbits

You might remember from the very beginning of this course, that ancient Greek astronomers believed that the Earth is at the centre of the universe and that the sun, moon and planets move around Earth on circular orbits.

Unfortunately, astronomical observation of the sky didn't quite support this. For example, the sun appeared larger during some parts of the year and smaller during others. On a circle, every point should have the same an increasing a decreasing distance from its center.

To fix this, astronomers added Epicycles to their model of the solar system: planets move on a large circle around Earth, while simultaneously rotating on a smaller circle. While very complicated, this was the most widely accepted model of our universe for more than 1000 years:

Over time, people realised that Earth was just one of many planets orbiting the sun (the Heliocentric model), but it wasn't until 1609, that the astronomer Johannes Kepler discovered that planets actually move on elliptical orbits.

The sun is in one of the two focal points of these ellipses. The planets speed up as they get closer to the sun, and slow down as they move further away.

A few decades later, Isaac Newton was able to prove Kepler's observations, using his newly developed laws of gravity. Newton realised that there is a force between any two masses in the universe - similar to the attraction between two magnets.

Gravity is what makes everything fall to the ground and gravity is also what makes the planets move around the sun. It is only the great speed at which planets move, that prevents them from falling directly into the sun.

Using Newton's laws, you can derive the path that objects take when moving under the force of gravity. It turns out that planets move on ellipses, but other objects like comets can travel on parabolic or hyperbolic paths: they fly close to the sun before turning around and shooting off into the universe, never to come back.

According to legend, a falling apple inspired Newton to think about gravity. He was one of the most influential scientists of all time, and his ideas shaped our understanding of the world for nearly 300 years - until Albert Einstein discovered relativity in 1905.

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Conic Sections - Circles and Pi - Mathigon

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Reading time: ~20 min Reveal all steps The circle is one of four different shapes which can be created using "slices" through a cone. This can be demonstrated using the light cone of a torch: If you point the torch vertically downwards, you see a If you tilt the cone, you get an If you tilt it even further, you get a circle ellipse oval of light.

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Reading time: ~15 min Reveal all steps

We have already used graph theory with certain maps. As we zoom out, individual roads and bridges disappear and instead we see the outline of entire countries.

When colouring a map - or any other drawing consisting of distinct regions - adjacent countries cannot have the same colour. We might also want to use as few different colours as possible.

Some simple "maps", like a chessboard, only need two colours (black and white), but most complex maps need more.

When colouring the map of US states, 50 colours are obviously enough, but far fewer are necessary. Try colouring the maps below with as few colours as possible:

All of these maps can be coloured with only four different colours, but it is not hard to imagine that other, very complicated maps might need many more colours. In fact, some maps need at least four colours, whenever they contain four countries all connected to each other.

Like before, we can convert a map with countries and borders into a planar graph: every country becomes a vertex an edge a face, and countries which share a border have the same colour get connected by an edge:

Now we want to colour the vertices of a graph, and two vertices must have a different colour if they are connected by an edge.

In 1852, the botany student He observed that four colours seemed to suffice for any map he tried, but he was not able to find a proof that worked for This turned out to be an extremely difficult problem, and became known as the Francis Guthrie had to colour a map of counties in England. all maps. four colour theorem.

During the following 100 years, many mathematicians published "proofs" to the four colour theorem, only for mistakes to be found later. Some of these invalid proofs were so convincing that it took more than 10 years to discover errors.

For a long time, mathematicians were unable to either prove that four colours are enough, or to find a map that needed more than four colours.

Little progress was made on the four colour problem until 1976, when Wolfgang Haken and Kenneth Appel used a computer to finally solve it. They reduced infinitely many possible maps to 1936 special cases, which were each checked by a computer taking over 1000 hours in total.

The four colour theorem is the first well-known mathematical theorem to be proven using a computer, something that has become much more common and less controversial since. Faster computers and a more efficient algorithm mean that today you can prove the four colour theorem on a laptop in just a few hours.

The four colour theorem only works for maps on a flat plane or a sphere, and where all countries consist of a single area.

However, mathematicians have also looked at maps of empires, where countries can consist of multiple disconnected components, and at maps on differently-shaped planets, such as a torus (doughnut shape). In these cases you may need more than four colours, and the proofs become even more difficult.

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The Travelling Salesman Problem

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Map Colouring - Graphs and Networks - Mathigon

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Reading time: ~15 min Reveal all steps We have already used graph theory with certain maps. As we zoom out, individual roads and bridges disappear and instead we see the outline of entire countries. When colouring a map - or any other drawing consisting of distinct regions - adjacent countries cannot have the same colour.

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Reading time: ~10 min Reveal all steps

Welcome to the most spectacular game show on the planet! You now have a once-in-a-lifetime chance of winning a fantastic sports car which is hidden behind one of these three doors. Unfortunately, there are only goats behind the other two doors. Select one to make your choice!

Are you sure about that? You can still change your mind and select a different door...

I'm sure!

A great choice, but let me make life a little easier for you. I'll open one of the other doors with a goat, so that there are only two doors left for you to pick from. Do you want to stick with your choice, or do you want to swap?

I want to stay! I want to swap!

Ok - let's see how you did...

Open the doors...

Looks like you made the right choice. Congratulations, you just won a beautiful new sports car!

Replay game

If you play this game many times, you'll notice that you're more likely to win if you swap don't swap after the first door is opened, rather than sticking with your initial choice.

But how can this be - surely the car is equally likely to be behind each of the two remaining doors?

The explanation is very subtle. When you pick the initial door, the probability of being correct is 1 3 and the probability of being wrong is 2 3.

Artboard 20

After the game master opens one of the other doors, the probability of being wrong is This means that swapping doors still doubles triples halves your chance of winning. 2 3, except now all this probability is on just one door.

Artboard 21

Even if this doesn't seem very intuitive, we can prove that it is correct - simply by listing all different possibilities:

Out of the 9 possibilities need you to switch doors, to win. This gives a chance of 6 9=2 3 like before.

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The Monty Hall Problem - Probability - Mathigon

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Reading time: ~10 min Reveal all steps Welcome to the most spectacular game show on the planet! You now have a once-in-a-lifetime chance of winning a fantastic sports car which is hidden behind one of these three doors. Unfortunately, there are only goats behind the other two doors.

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Horrendous Soup is a fun game in which students have to make a recipe for the most disgusting soup they can imagine. Students progress through the game by adding ingredients to the soup. Some of the ingredients are measured in capacity, some are measured in mass, and some are measured in distance (for example, the recipe might call for a certain amount of millimeters of toothpaste or tree branch). Given this is a bit far-fetched, but still a perfect way to reinforce metric conversion skills. To successfully print out the recipe, students must demonstrate the ability to convert the amounts of ingredients to other metric units. For example, liters to milliliters and vice versa, grams to milligrams and vice versa, and millimeters to centimeters to meters. See demonstration video for more information.

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2023 MrNussbaum.com is a copyright of the Nussbaum Education Network,LLC.

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Mr. Nussbaum - Horrendous Soup - Online Game

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Horrendous Soup is a fun game in which students have to make a recipe for the most disgusting soup they can imagine. Students progress through the game by

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A 6-inch tall flower casts a 4-inch shadow at 5:00pm. One month later, when the sun was at the same position in the sky as it was at 5:00pm one month earlier, the shadow cast was 5 ¹/ ₃ inches long. How many inches taller was the flower when it cast the 5 ¹/ ₃-inch shadow than it was when it cast the 4-inch shadow?

Two years ago Melanie planted two trees. Tree A was 24 inches tall when she planted it and Tree B was 40 inches tall when she planted it. If Tree A grows at a rate of 8 inches per year and Tree B grows at a rate of 6 inches per year, after how many more years will Tree A be as tall as Tree B?

Last spring, Fredrico's favorite topiary bush formed a perfect sphere with a diameter of 16 inches. Over the past year the bush's diameter has increased by one inch. By what percent has the volume of the bush's sphere increased? Express your answer to the nearest tenth.

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A 6-inch tall flower casts a 4-inch shadow at 5:00pm. One month later, when the sun was at the same position in the sky as it was at 5:00pm one month earlier, the shadow cast was 5 1/ 3 inches long.

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Help your students fall in love with math through challenging and puzzle-like problems. Order the grade-specific Problem of the Week bundle for the entire school year! Freshly updated in Summer 2023!

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We need help to keep Puzzle of the Week running! Please consider donating through our Patreon page or volunteering some time and joining the team. See the Donate page for more information.

Puzzle of the Week is a free international puzzle competition for schools. Students submit answers to a puzzle which is published weekly on Mondays. School and student performances are recorded and leaderboards are published on the results page. You can sign your school up for free using the button above or by clicking here.

All previous puzzles are available with detailed answers including extensions and background information on the puzzle library page.

All Time School Leaderboard
This shows the top 20 schools in the competition since its inception in 2016.

The leaderboard updates live as entries are submitted, just refresh the page to see the most recent data.

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We need help to keep Puzzle of the Week running! Please consider donating through our Patreon page or volunteering some time and joining the team. See the Donate page for more information. Puzzle of the Week is a free international puzzle competition for schools. Students submit answers to a puzzle which is published weekly on Mondays.

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This puzzle features a series of computations involving, among other factors, the student's own age and birth month. After the computations are completed, students are asked to find a pattern in their answers. This is best done in groups and is not difficult if no computational errors have been made. After students have discovered the pattern they can confound unsuspecting friends by telling them their ages and birth months if the the friends share the same results of the same computational process.

The real challenge in this puzzle, however, is to find out why it works. We hope that you will give this week's puzzle a try.

Here is an interesting puzzle for you. If you solve it, you will be able to guess people's ages and birth months. Do the following steps carefully.

1. Write the number that corresponds to the month of your birth:

January =1, February =2, March =3, etc. _____________

2. Multiply the above number by 4 and record the product. _____________

3. Add 12 to the above product and record the sum. _____________

4. Multiply this sum by 25 and record the product. _____________

5. Add your age to the above number and record the sum. _____________

6. Add 13 to the above number and record the sum. _____________

7. Subtract 365 from the above number. Record the difference. _____________

8. Add 52 to the above number and record the sum. _____________

9. Look at this final number. What do you notice?

10. If you have been able to discover a pattern in the above answer, you will be able to tell others their ages and birth months by having them tell you their answer when they do the above steps.
Solution
Click the arrow below to view the solution.

The Age Game featured a series of computations involving , among other things, inputting student's ages and birth-months. After the computations were completed, students were asked to find a pattern in their answers. They should have discovered that their answers were three- or four-digit numbers with their ages given by the tens and ones digits (for students younger than 10 the tens digit is zero.) and their birth-months given by the hundreds digits (or the thousands and hundreds digits if their birth months are October, November, or December). For example, an eleven-year-old student born in February would have an answer of 211, while an eight-year-old student born in October would have an answer of 1008.

The real challenge in this puzzle is to find out why it works. If we let M represent the month of birth and A the age, then steps 1-4 produce ( 4M + 12 )25 which simplifies to 100M + 300. Steps 5-8 produce A + 13 - 365 + 52 which simplifies to A - 300. When the results of steps 1-4 are combined with those of steps 5-8, you get 100M + 300 + A - 300. When this is simplified, the +300 and the -300 cancel out and you are left with 100M + A. Thus, when a student's month of birth and age are input, a three- or four-digit number is produced with the age in the tens and ones place and the month of birth in the hundreds place (or the thousands and hundreds place if the month is October, November, or December).

One extension of this activity would be to challenge students to change the calculations and still have it work out. Another extension would be to get the answer to show the month of birth in the ones and tens digits and the age in the hundreds and thousands digits.

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This puzzle features a series of computations involving, among other factors, the student's own age and birth month. After the computations are completed, students are asked to find a pattern in their answers. This is best done in groups and is not difficult if no computational errors have been made.

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Home > Olympiad Program > Contests > Samples >Problem of the Month >Awards > Enrollment > MOEMS Board of Directors

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Practice your knowledge of geometry and angles in this game. Calculate the numbers in your head or use a calculator. Then use your keyboard to type in your answer into the box and hit "Submit." Speed counts, so try to solve the problems as quickly as possible before time runs out!

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Practice your knowledge of geometry and angles in this game. Calculate the numbers in your head or use a calculator. Then use your keyboard to type in your answer into the box and hit "Submit." Speed counts, so try to solve the problems as quickly as possible before time runs out!

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Before students finalize their answers, ask them to consider why their mathematical answers may not match the answer in the world. If they're going to adjust their answer, should they adjust it down or up? Once they have their final answer, show them the answer.

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Dan Meyer asked: How long will it take the dog to pop all of the balloons?? What is the first question that comes to your mind?

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1. If you wanted to eat enough Mini-Bears to match the Super-Bear, how many would it take? What if you ate Regular-Bears?

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1. Guess how long the other bubble wrap pieces will take to pop.

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Tinkercad | From mind to design in minutes

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Tinkercad is a free, easy-to-use app for 3D design, electronics, and coding.