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An arithmagon is a polygon with numbers at its vertices which determine the numbers written on its edges.

Check you understand how the values at the vertices determine the values on the edges in the arithmagons generated by the interactivity:

If you are given the values on the edges, can you find a way to work out what values belong at the vertices?

Use the interactivity below to test out your strategies.

There are four different challenge levels to try.

Once you are confident that you can work out the values at the vertices efficiently, here are some questions you might like to consider:

Can you describe a strategy to work out the values at the vertices irrespective of the values given for the edges?

By looking at the numbers on the edges, how can you predict whether the numbers at the vertices will all be positive?

By looking at the numbers on the edges, how can you predict whether the numbers at the vertices will all be whole numbers?

What happens to the numbers at the vertices if you increase one or more of the numbers on the edges by 1? By 2? ... By n?

What relationships can you find in square arithmagons? Or pentagonal arithmagons?

A poster of this resource is available here.

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An arithmagon is a polygon with numbers at its vertices which determine the numbers written on its edges. Check you understand how the values at the vertices determine the values on the edges in the arithmagons generated by the interactivity: If you are given the values on the edges, can you find a way to work out what values belong at the vertices?

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The idea of this task is to complete the four by four grid so that the numbers 1, 2, 3 and 4 appear only once in each row and only once in each column.

You will see that some squares, or groups of squares, are outlined with a thick black line. These groups of squares are called 'cages'.

In the corner of one of the squares in a cage, you will see a small target number and usually a mathematical operation too. For example, if a cage of three squares has '8x' in the corner of one of the squares, it means that the three numbers in that cage multiply together to make 8.

If the cage is made up of only one square, which has a number in its corner but not an operation, that number is telling you exactly what goes in that square.

Have a go at the grid in the interactivity below. Clicking on a square allows you to type in a number, or choose a number from the options given. You can change your mind just by typing/choosing another number instead.
How would you convince someone else that the number you have put in a square must be correct? Please do share your reasoning with us.
Click on the Settings menu (the purple cog in the top right-hand corner) to select another grid to try. You can increase the level of difficulty in the Settings menu too.

If you would prefer to work away from the computer, you could print out these grids (the numbers refer to the level of difficulty): Mathdoku1.pdf, Mathdoku2.pdf, Mathdoku3.pdf.

You may also like to take a look at this feature, which includes also 3 by 3, 5 by 5 and 6 by 6 Mathdokus.

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The idea of this task is to complete the four by four grid so that the numbers 1, 2, 3 and 4 appear only once in each row and only once in each column. You will see that some squares, or groups of squares, are outlined with a thick black line.

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Showing All 20 Triangles Created from 6 Intersecting Straight Lines - Step by Step Solution

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session 14 - method in multiplying madness (5612)

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If you are a teacher, click here for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on...
If you had to work out $23 \times 21$ how would you do it?
What if you needed to work out $246 \times 34$?

Here are eight videos showing four different methods for working out the two multiplications. Can you make sense of them?
Grid Multiplication Column Multiplication Multiplying with Lines Gelosia Multiplication
Once you have watched the videos, make up some multiplication calculations of your own and have a go at answering them using all four methods. Check that you get the same answer each time!
Here are some questions to consider:
Why does each method work?

What do the methods have in common?

What are the advantages and disadvantages to each method?
Extension challenge
Here is a video of another multiplication method, one where no writing down is needed along the way. Can you make sense of the video, and explain how this method works?

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md number operations and calculation methods

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If you are a teacher, click here for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on... If you had to work out $23 \times 21$ how would you do it? What if you needed to work out $246 \times 34$?

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session 14 - escape from the castle (7501)

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session 13 - where are the primes (14955)

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Please note - this has been adapted to make it KS2 friendly!

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Here are some prime numbers:
$5, 17, 23, 59, 89, 101$

They are all odd. What else do they have in common?

Add $1$ to each. What do the answers all have in common now?

Here are some more prime numbers:
$13, 19, 31, 37, 61, 67$

They are also all odd. What else do they have in common?

Subtract $1$ from each. What do the answers all have in common now?

Can you find any primes greater than $3$ which are not one more, or one less, than a multiple of $6$?

Charlie thought that it wasn't possible to find any primes which were not one more, or one less than a mutiple of $6$. He thought that he might be able to use a number grid to help him prove this.

Why do you think some numbers are shaded in red, blue or yellow?
What do you notice about the numbers in circles?

Claire also though that it wasn't possible to find any primes which were not one more, or one less than a mutiple of $6$. She thought that she might be able to use some general expressions to help her prove this.

Any number can be written in the form $6n$, $6n+1$, $6n+2$, $6n+3$, $6n+4$ or $6n+5$.

We know that $6n$ is a multiple of $6$, so cannot be prime.
We can write $6n+2=2(3n+1)$, and so we know that $6n+2$ is a multiple of $2$. The only multiple of $2$ which is prime is $2$ itself, which is not greater than $3$.

Can you complete Claire's method to show that primes numbers greater than $3$ must be of the form $6n+1$ or $6n+5$?

Can you show that you can write $6n+5$ in the form $6m-1$, and so show that numbers of the form $6n+5$ are one less than a multiple of $6$?

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Here are some prime numbers: $5, 17, 23, 59, 89, 101$ They are all odd. What else do they have in common? Add $1$ to each. What do the answers all have in common now? Here are some more prime numbers: $13, 19, 31, 37, 61, 67$ They are also all odd.

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Can you select the missing digit(s) to find the largest multiple?

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Skippy and Anna are imprisoned in a castle and need to find their way out of five locked rooms to escape.

To unlock each room, they need to find the solution to a problem. In each case, the solution provides the key to that room.

Can you help them to get out?

The first room:

There is a pile of dice.

Three of them are put in a row. The numbers on the top of these three add to 8.
What do the hidden numbers on the bottom add to?

This number is the first key.

The second room:

Skippy and Anna have the first key number. They have gone through the first locked door.

There are ten cards numbered from 0 to 9. Five of these are face down in a row on the table.

The numbers on the first two cards add to the first key number.
The numbers on the second and third cards add to 9.
The numbers on the third and fourth cards add to 11.
The numbers on the fourth and fifth cards add to 16.

What number is on the last card? This number is the second key.

The third room:

Skippy and Anna have the second key number. They have gone through the second locked door. Here there is a diagram.

There are two overlapping circles inside a rectangle. The rectangle is the second key number of centimetres long and 5 centimetres high.

How far apart are the centres of the two circles?

Square this answer and subtract one. This will give you the third key number.

The fourth room:

Skippy and Anna have the third key number. They have gone through the third locked door.

On the floor there is a strange diagram and the numbers from 1 to 8 on eight cards. The diagram is a square with eight boxes arranged round it.

Skippy and Anna have to arrange the numbers in the boxes so that each side of the square adds to the third key number.

To find the fourth key number, add the numbers on all the corner boxes and then subtract 10.

The fifth room:

Skippy and Anna have the fourth key number. They have gone through the fourth locked door.

On the table there are some jam tarts.

The tarts were baked in two identical trays. The trays hold the fourth key number of tarts each.

There are enough tarts to fill one tray but not enough to fill the other one as well.

If the tarts are counted in fours there are three left over.

If they are counted in threes there is one left over.

How many tarts are there altogether? This number is the fifth and last key.

The last question:

Before they can leave the castle you must answer a last question.
What is Skippy's real name?
Using the code 1 = A, 2 = B, 3 = C, ... 26 = Z translate all the key numbers into letters.

These letters will give you an anagram of Skippy's real name. When you have worked this out, Skippy and Anna will be free!

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Skippy and Anna are locked in a room in a large castle. The key to that room, and all the other rooms, is a number. The numbers are locked away in a problem. Can you help them to get out?

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Coded Hundred Square printable sheet

This hundred square is written in code.

It starts with one and ends with a hundred.

Can you build it up? How did you do it?
Can you build it up in a different way?
Talk to a friend who has also tried building up the hundred square. How did they do it? What do you like about their method?

You can use the interactivity below or print and cut out the pieces from these sheets.

You may be interested in the other problems in our Number Sense and Place Value Feature.
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An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

Use the 'double-3 down' dominoes to make a square so that each side has eight dots.

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This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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Off-the-shelf Primary Mathematics Masterclass all about Mobius Bands. Play with simple strips of paper for this fun mathematical activity that will blow the minds of both the young students and attending adults alike. Wow them with this lovely introduction to the mathematical field of topology.

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This challenge is based on the game Frogs which you may have seen before.

There are two blue frogs and two red frogs.

A frog can jump over one other frog onto an empty lilypad or it can slide onto an empty lilypad which is immediately next to it.
Only one frog, at a time, is allowed on each lilypad.

Now the idea is for the blue frogs and red frogs to change places. So, the red frogs will end up on the side where the blue frogs started and the blue frogs will end up where the red frogs began.

The challenge is to do this in as few slides and jumps as possible.

You could use the interactivity below to help you try out your ideas.

How do you know you have found the smallest possible number of slides and jumps?

Why not try three red frogs and three blue?
What is the smallest number of slides and jumps now?

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Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

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Maze 100 printable sheet
In this maze there are numbers in each of the cells. You go through adding all the numbers that you pass. You may not go through any cell more than once.

Can you find a way through in which the numbers add to exactly 100?

What is the lowest number you can make going through the maze?

What is the highest number you can make going through the maze? (Remember you may not go through any cell more than once.)

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Can you go through this maze so that the numbers you pass add to exactly 100?

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You may like to have a look at Domino Sets before trying this problem.
Amy has a box containing ordinary domino pieces but she does not think it is a complete set.

She has 24 dominoes in her box and there are 125 spots on them altogether.

Which of her domino pieces are missing? How do you know?

If you do not have any dominoes, you might find our interactive Dominoes Environment useful.

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You may like to have a look at Domino Sets before trying this problem. Amy has a box containing ordinary domino pieces but she does not think it is a complete set. She has 24 dominoes in her box and there are 125 spots on them altogether.

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If you don't complete this problem in the run-up to Christmas, just change the context (birthdays etc) and use - it's still a great problem!

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An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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If you are a teacher click here for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on ...
Nine squares with side lengths $1, 4, 7, 8, 9, 10, 14, 15$ and $18$ cm can be fitted together with no gaps and no overlaps, to form a rectangle.

What are the dimensions of the rectangle?

Once you've had a chance to think about it, click below to see how three different pupils began working on the task.

This is how Anna started:

Here is what Brendan tried:

Here is Chandra's initial approach to the problem:

Can you take each of these starting ideas and develop them into a solution?

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If you are a teacher click here for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on ... Nine squares with side lengths $1, 4, 7, 8, 9, 10, 14, 15$ and $18$ cm can be fitted together with no gaps and no overlaps, to form a rectangle.

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Three dice are sitting in the corner with the simple rule that where two faces touch they must be the same numbers.
So, in the first picture above there are $3$'s at the bottom of the red dice and on the top of the middle green and there are $4$'s on the bottom of the green dice and the top of the white dice. The numbers on the seven faces that can be seen are then added and make $21$.

In the second picture above there are $4$'s at the left of the red dice and on the right of the green dice and there are $3$'s on the left of the green dice and the right of the white dice. The numbers on the seven faces that can be seen are then added and make $23$.

Use your own dice (you could use two or three or more...)
What total have you made?
Can you make a different one?
How many different ones can you make?

Now for a challenge - arrange dice (using at least $2$ and up to as many as you like) in a line in the corner, so that the faces you can see add up to $18$ in as many ways as possible.

Each line of dice must be along or up a wall (or two walls). A line going up is counted the same as a line going along. Remember the dice must touch face to face and have the same numbers touching. The dice must be all in one line, so this arrangement below is not allowed;

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Three dice are sitting in the corner with the simple rule that where two faces touch they must be the same numbers. So, in the first picture above there are $3$'s at the bottom of the red dice and on the top of the middle green and there are $4$'s on the bottom of the green dice and the top of the white dice.

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What Nrich look for in submitted solutions

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What Nrich look for in submitted solutions

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In this article, members of the NRICH team explain the process they go through when they choose solutions to publish on the site. To have a go at the current live problems, visit the Primary or Secondary live problem pages, or to see the most recent solutions, visit the Primary or Secondary solution pages.
Live Problems

Each time you visit the site you will see that there are some "live problems" which are open for solutions. This means we are inviting students to send in thoughts, ideas, workings out and solutions, and then we'll publish them after a suitable time so that our archives are complete with both problems and solutions. On average problems remain live for about six weeks. They have a "Submit a solution" link to the left of the Problem title.

To check which problems have been recently solved you can visit the "Recently solved" pages.

As well as wanting to showcase the hard work and effort that has been put into solutions, we know that lots of students, teachers and parents find it useful to look at the solutions to the problems in our archive.

Students might want to check that their answers and methods agree with what someone else has done.

Teachers might want to show their classes different approaches to solving a problem. Parents might want to see how problems can be solved in order to support their children.

We publish solutions that are clearly explained so that students, teachers and parents can understand them.

We receive solutions in a variety of formats. Here are a few examples:

Answer or Solution?

The first decision to make is whether a solver has sent us an "answer" or a "solution". An answer is brief and just contains answers to the questions we've posed. A solution may not be complete, but will include the solver's ideas, why they chose a certain method, and explanations of what they noticed. We want someone looking at the published solution to a problem to get some understanding about the sort of thought processes that led to the answer, so they can learn from someone else's work and become a better problem-solver.

Mathematically Correct

Secondly, we need to make sure that the solution is mathematically correct. If there are one or two minor errors, we will correct them and publish the solution, but if a solution showed a major misconception or the solver didn't really explain their method, we wouldn't publish their solution. Of course if a solution began very clearly but had lots of errors towards the end, we might publish just the first part of that solver's work.

How we organise solutions

Once we've eliminated all the submitted solutions that are just answers without explanation, we then think about how to organise the solutions.

At Stages 1 and 2, to begin with we look out for solutions that perhaps tackle just the first part of the problem and we often publish these at the top of the page. We would then choose solutions that address the question more fully and we often order them to reflect increasingly sophisticated methods. We don't want to put anyone off having a go at the problem, which is why solutions using higher-level mathematics come nearer the bottom of the page.

For example, in the published solutions to the Stage 1 task, The Add and Take-away Path, Jack and Ellie answer the first and second parts of the problem, giving us the scores for each grid. Then, the other solutions go on to explain why this always happens.

In a similar way, the solutions for the Stage 2 problem Three Dice begin with some observations and the suggestion of a "rule". Several learners then offer very clear explanations for why this always happens. Finally, we also include solutions which look at more than three dice and go on to generalise the problem.

For the Stage 3 and 4 solutions, we start by looking for solutions which make a good start on the problem but perhaps don't answer the whole question. For example, quite a few problems involve numerical relationships which can be analysed using algebra, so we regularly receive solutions from younger students who made lots of progress with the numerical part of the problem but don't always proceed to the algebraic parts. We then pick out the solutions which develop the next part of the problem, and feature those next. If someone has used some particularly advanced mathematics (perhaps using ideas not normally met until Stage 5), we usually publish their solution nearer the bottom of the page. This means that anyone reading the solution can read as far as they understand and then take a look at what might come next as they learn new mathematical techniques.

Take a look at the solutions to the March 2011 problem Multiplication Arithmagons. Lots of students came up with the method described by Alexander, so we chose to feature his very clear description and mention everyone else who had worked in the same way. Fionn's algebraic approach came next, and Francesco's solution added some extra detail (each solution could be multiplied by -1). At the bottom of the page, we provided links to those solutions we had received as uploaded attachments.

At stage 5 many of the problems are reasonably easy to start but somewhat more difficult to complete. We very often receive partial solutions to the problems and use these to piece together a full solution. We will correct minor mistakes and very often paraphrase solutions. As with other stages, we want to know what was going through your mind as you constructed your solution, as this is useful to share with other solvers, as well as making the solutions more entertaining! One particular feature of many stage 5 problems is that they involve a great deal of mathematical symbolism. Taking care to format this appropriately will increase your chances of the solution appearing on the site. Sometimes, if a solution is good enough, we simply publish the entire contribution as an additional download from the solutions page: see Escape from Planet Earth as an example of this.

Toughnut!

If after a suitable time we haven't received a solution to a problem, we mark it as a "Toughnut".

You can submit solutions to toughnuts at any time, and then after a few months the team will revisit the problem to see if any solutions have been received. Once a solution is received, we publish it and remove the problem from the toughnuts list. If you submit a solution to the toughnut, be patient - it may not appear on the site for quite a while, as we usually revisit the toughnuts during the summer months.

Five tips for students on how to get your solutions published

Finally, here are five tips for students on how to get your solutions published:

1. Check your solution carefully, and perhaps ask someone else to look over it to spot any mistakes you might have missed.

2. Try to explain simply and clearly what you did to solve the problem. It doesn't have to be a long explanation, but we like solutions that share the mathematical journey of the solver.

3. If you made any diagrams or used a spreadsheet to help you, upload it with your solution. Make sure you explain clearly what you are referring to if you talk about your diagram or spreadsheet in your solution.

4. If you upload your solution as a document, keep it as simple as possible. If you send a solution with lots of formatting, your solution might not appear on our screens as you intended. If you use a less common file type, we might not be able to read it easily, but .doc and .pdf files are fine.

5. Read through other people's solutions to previous problems on the site. This will give you a flavour for the sorts of solutions we publish, and help you to develop your own mathematical style for writing up your work. Learning how to communicate mathematically is an important part of learning to be a mathematician!

When we publish solutions, we don't publish full names or ages, so if you submit your solution as a pdf, just write your first name.

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In this article, the NRICH team describe the process of selecting solutions for publication on the site.

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submit a solution and be published on Nrich!

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submit a solution and be published on Nrich!

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How could you arrange at least two dice in a stack so that the total of the visible spots is 18?

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Place the digits 1 to 9 into the circles so that each side of the triangle adds to the same total.