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The upper part shows charts that you can use as filters, while

the lower part displays the images matching your query.

Click on any of the city map circles to see only selfies taken in that city.

Clicking on the map background will remove the filter.

You can navigate through the results grid using the arrow buttons on the left and right.

Note how the other charts react when you filter the data! If you select female selfies only, you can see that women are younger (age chart) and are more numerous in every city (especially in Moscow).

Click and drag the mouse to create a filter. You can also move the filter by clicking and dragging. Remove the filter by clicking anywhere on the background of the chart.

The gender filter works the same as the map: Just click on a circle to filter. (The circle marked with a question mark contains all selfies, where our Mechanical Turkers did not come up with a clear, unanimous result.)

Demographics is location (captured by Instagram app) and gender and age (tagged and guesses by Mechanical Turk workers.)

Pose is the direction a person is facing (up/down, left/right, head tilt). This was automatically extracted by the Rekognition API.

Features is an estimate of facial attributes by Rekognition, such as the presence of glasses, or if mouth or eyes are open. Question marks indicate unclear results.

Mood is the estimated mood of a person - again automatically extracted by Rekognition.

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The upper part shows charts that you can use as filters, while the lower part displays the images matching your query. Click on any of the city map circles to see only selfies taken in that city. Clicking on the map background will remove the filter.

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Welcome! Explore these patterns with your students and watch their natural tendencies to see patterns morph into powerful algebraic thinking and reasoning. It's an ideal routine to foster mathematical practice #7 - look for and make use of structure.

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Explore these patterns with your students and watch their natural tendencies to see patterns morph into powerful algebraic thinking and reasoning. It's an ideal routine to foster mathematical practice #7 - look for and make use of structure.

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https://mathgameswithbaddrawings.com/games

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https://people.math.harvard.edu/~knill/mathmovies/index.html

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Lambada [IMDb link]

Kevin Laird, a Beverly Hills school teacher talks about complementary angles, the most useful angle, the identity cos(x) ² + sin(x) ²=1 and the Cartesian coordinate system. (Thanks to Aleksandra Ravas for the suggestion).

1990

Old School [IMDb link]

This comedy not only shows shots of Harvard college but also some rather tough math test problems: like Harriot's method of solving cubics (named after Thomas Harriot, you find Harriot's method for finding the three solutions x=e-b ²/e,e,b ²/e of the equation x ³+3 b ² x = 2 c ³ here). Also Diophantine equations or integration problems appear in this test.

2003

Alice in Wonderland [IMDb link]

Riddle: Why is a raven like a writing desk? Carols own answer given in 1896: Because it can produce a few notes, tho they are very flat; and it is nevar put with the wrong end in front! (Thanks to Tania Moloney for the suggestion).

1951

Good Will Hunting, Cayley formula [IMDb link]

Will Hunting writes down Cayley's formula stating that the number of labeled trees on n nodes is n n-2. Then lists 8 different unlabeled trees with 10 nodes. (Thanks to George Lowe for the suggestion)

1997

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Scenes in Movies in which some Mathematics appears

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https://eyes.nasa.gov/apps/asteroids/#/home?time=2017-11-03T02:31:31.312+00:00&rate=-79200

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Explore the 3D world of Asteroids, Comets and NEOs. Learn about past and future missions, tracking and predicting orbits, and close approaches to Earth.

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This is Which One Doesn't Belong?, a website dedicated to providing thought-provoking puzzles for math teachers and students alike. There are no answers provided as there are many different, correct ways of choosing which one doesn't belong. Enjoy!

Which One Doesn't Belong?

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This is Which One Doesn't Belong?, a website dedicated to providing thought-provoking puzzles for math teachers and students alike. There are no answers provided as there are many different, correct ways of choosing which one doesn't belong. Enjoy!

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Students love to play games in math class. So we've curated (and created!) the best math games we know and put them in one place, to make sure that students will be learning math and engaged through play.

These games are notable for their:

1. Student Choice
2. Mathematics is the engine of the game
3. Easy to learn and quick to play - perfect for classroom use

Want more? Check out the Mathematical Games Minicourse. A 115-minute recording with accompanying 45 page booklet of favorite games for pre-K - 2nd grade.

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Students love to play games in math class. So we've curated (and created!) the best math games we know and put them in one place, to make sure that students will be learning math and engaged through play. These games are notable for their: 1. Student Choice 2.

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https://drrajshah.com/math-treats/

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Math Treats are videos for kids to explore at home. Watch the first video where I introduce some cool math, then try to figure it out. I'll post hints and sometimes I'll post solutions. The goal is to PLAY and EXPLORE. Ask WHY? and WHAT IF? Don't worry about just getting the answer.

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Collection of Free Printable Puzzles for the Math Classroom (or any Classroom)

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​ You may need to click on an image to see it fully.

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Welcome to Oops, I forgot... ! This page represents my ongoing attempt to curate a collection of ideas of a task structure that I invented while playing with teaching moves that instigate, amplify, and sustain student activity in open teaching spaces.

This landing page is designed to:

Provide some key advice for using Oops, I forgot... with mathematics learners

Provide a space to collect the examples submitted by the math teacher community

If this is your first time here, please start by reading my post where I introduce the routine. This will describe, in a touch more detail, the origins, influences, intentions, and justifications for Oops, I forgot... (OIF) tasks.

Each OIF consists of two pieces: a launch and a list. The launch is the starting point for the routine, and the list is a series of shifts in the original prompt that are "conveniently remembered" as the routine unfolds. (Again, see this post for an elaboration).

Some key advice and reminders for teachers:

1. The launch of an OIF typically asks learners to create an object. This is a demanding starting point all in itself, and I like to facilitate this type of thinking in collaborative settings-usually groups of three.

2. Consider the lists of new constraints in each example as a productive suggestion. There is no perfect list or sequence; this structure should embolden teacher creativity. Design lists that-in your anticipations-will sponsor activity with your learning goals.

3. Focus on the difference created through the convenient amnesia. Students should justify why their new creation satisfies the scenario, but also what needed to be adjusted and what stayed the same.

4. Make the move in the moment. If/when you anticipate that convenient remembering will further mathematical activity, make the play. There is no prescription.

5. Students may need different things at different times. There is no law that each group needs to get the new constraints at the same time. Use your judgement; keep thinking going.

6. As they catch on, let students predict what the next shift might be. This is actually a beautiful window into the mathematics they know.

Examples are collected in the table below. Each is formatted into a downloadable word document. This format helps with preparation time and also encourages edits of the constraints to fit your specific classroom context. These tasks are published under this Creative Commons licence.

Take this as your open invitation to implement, adapt, share student thinking, and contribute new ideas. You can download this template to build your own and then communicate with me via Twitter (using the hashtag #OIFMath) or the Contact Page!

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Welcome to Oops, I forgot... ! This page represents my ongoing attempt to curate a collection of ideas of a task structure that I invented while playing with teaching moves that instigate, ampl...

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Quizizz | Free Online Quizzes, Lessons, Activities and Homework

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Motivate every student to mastery with easy-to-customize content plus tools for inclusive assessment, instruction, and practice. Engage students with Quizizz today!

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https://docs.google.com/document/d/1EMq1w2gdI--oSN5cRUrItqSzvjQq7VMXwkOu2O1Wz3Q/mobilebasic?usp=gmail_thread

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Duane's Collection of Non-Curricular Tasks

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https://drive.google.com/file/d/1rOMY-RuhVbEThlxaPMDkvjhepS6zV7LQ/view

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Basketball Statistics & History of Every Team & NBA and WNBA Players | Basketball-Reference.com

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Checkout the statistics, scores & history of every team & NBA and WNBA players and more on Basketball-Reference.com

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Kahoot! | Learning games | Make learning awesome!

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Kahoot! is a game-based learning platform that brings engagement and fun to 1+ billion players every year at school, at work, and at home.

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Interactive logic puzzles for your web browser.

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What is Scorigami?

Scorigami is a concept thought up by Jon Bois. It is the art of building final scores that have never happened before in NFL history. Due to the unique nature of how points are scored in (American) Football, where it is impossible to score 1 point on its own, as well as the rarity of the 2 point safety and 8 point touchdown and 2 point conversion, there are a lot of scores that are possible, but have never happened. For more info, check out the video made by Jon Bois about Scorigami below.

How do I read this chart?

The winning team's score is along the top (horizontal axis) and the losing team's score is along the right (vertical axis). Scores marked in green have happened, scores in white have not yet happened, scores in black are impossible. The bottom half is all black because the losing team can not score more points than the winning team. You can find out more information by clicking on a cell. You can also use the switches and dropdown on the top of the page to see more visualizations of the data.

Why isn't the 1 row all black?

I highly recommend you watch the video above to answer this question as having it charted out will help out a lot (also the video is amazing). The critical part to answer this question is at the 18:23 mark. If you can't watch the video, the gist is that it is possible for the defense to score 1 point on a PAT attempt by getting a safety.

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Scorigami is a concept thought up by Jon Bois. It is the art of building final scores that have never happened before in NFL history. Due to the unique nature of how points are scored in (American) Football, where it is impossible to score 1 point on its own, as well as the rarity of the 2 point safety and 8 point touchdown and 2 point conversion, there are a lot of scores that are possible, but have never happened.

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Steve Wyborney\\\'s Blog: I\\\'m on a Learning Mission.

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For my educator friends who have access to Google Slides, but do not have access to PowerPoint, this post is for you. Since Google Slides does not have the animation that makes the Splats become transparent, the Splat! reveal does not work in Slides. Now - with these materials - it will. I've rewritten the first 20 Splat! lessons and the first 20 Fraction Splat! lessons with a different method so they will now include the reveal when using Google Slides.

The content of the lessons is the same as the original lessons.

Visit these links to see the original videos about how to use the and the Fraction Splat! slides.

Quick Note: The file sharing power of Google Drive is impressive and highly useful. However, for the purposes of sharing these resources, I need to use a method that I can personally maintain, update and support - and I've learned that the best way to provide resources from my blog is through the download links and the sign-up forms.

I hope you enjoy these Splats! formatted to work in Google Slides!

Splat Through 10
How many dots are under the Splat?
SPLAT 1.1 - numbers through 10 (Google Slides compatible version)
SPLAT 2.1 - with numbers 12-20 (Google Slides compatible version) SPLAT 2.2 - with numbers 12-20 (Google Slides compatible version) SPLAT 2.3 - with numbers 12-20 (Google Slides compatible version) SPLAT 2.4 - with numbers 12-20 (Google Slides compatible version) SPLAT 2.5 - with numbers 12-20 (Google Slides compatible version) SPLAT 1.2 - numbers through 10 (Google Slides compatible version)SPLAT 3.1 - multiple splats (Google Slides compatible version) SPLAT 3.2 - multiple splats (Google Slides compatible version) SPLAT 3.3 - multiple splats (Google Slides compatible version) SPLAT 3.4 - multiple splats (Google Slides compatible version) SPLAT 3.5 - multiple splats (Google Slides compatible version)
SPLAT 4.1 - instant multiple splats (Google Slides compatible version) SPLAT 4.2 - instant multiple splats (Google Slides compatible version) SPLAT 4.3 - instant multiple splats (Google Slides compatible version) SPLAT 4.4 - instant multiple splats (Google Slides compatible version) SPLAT 4.5 - instant multiple splats (Google Slides compatible version) Note: These lessons use a single splat along with a mixture of unit fractions and whole numbers. In Level 11, you control the Splat! SPLAT 11.1 - fraction splats (Google Slides compatible version) SPLAT 11.2 - fraction splats (Google Slides compatible version) SPLAT 11.3 - fraction splats (Google Slides compatible version) SPLAT 11.4 - fraction splats (Google Slides compatible version) SPLAT 11.5 - fraction splats (Google Slides compatible version) SPLAT 1.3 - numbers through 10 (Google Slides compatible version)Note: These lessons use a single Splat! which appears at the same moment as the whole numbers and fractions. Each question instantly presents missing information. SPLAT 12.1 - Instant Fraction Splats (Google Slides compatible version) SPLAT 12.2 - Instant Fraction Splats (Google Slides compatible version) SPLAT 12.3 - Instant Fraction Splats (Google Slides compatible version) SPLAT 12.4 - Instant Fraction Splats (Google Slides compatible version) SPLAT 12.5 - Instant Fraction Splats (Google Slides compatible version) Multiple Splats! with Fractions Note: These lessons include multiple Splats! and you control the timing of the Splats! The entire screen can be seen, the total can be determined, and then Splat! Remember: Each Splat! that is the same color must be covering the same number. SPLAT 13.1 - Multiple Fraction Splats (Google Slides compatible version) SPLAT 13.2 - Multiple Fraction Splats (Google Slides compatible version) SPLAT 13.3 - Multiple Fraction Splats (Google Slides compatible version) SPLAT 13.4 - Multiple Fraction Splats (Google Slides compatible version) SPLAT 13.5 - Multiple Fraction Splats (Google Slides compatible version)
Note: These lessons use multiple Splats! which appear at the same moment as the whole numbers and fractions. Each question instantly presents missing information. Remember: Each Splat! that is the same color must be covering the same number. SPLAT 14.1 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 14.2 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 14.3 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 14.4 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 14.5 - Instant Multiple Fraction Splats (Google Slides compatible version) SPLAT 1.4 - numbers through 10 (Google Slides compatible version)
SPLAT 1.5 - numbers through 10 (Google Slides compatible version)

Splat Through 20
How many dots are under the Splat?

Multiple Splats
Note: Splats that are the same color must cover the same number.

Instant Multiple Splat
Note: An additional feature of this level is the opportunity to ask, "What could the total be?"

Instant Fraction Splat!

Multiple Instant Splats! with Fractions

After reading this blog post, take a look at THE 12 MOST POPULAR MATH STRATEGIES AND DOWNLOADS ON THIS BLOG.

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Splat! for Google Slides - 40 Lessons - Steve Wyborney's Blog: I'm on a Learning Mission.

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For my educator friends who have access to Google Slides, but do not have access to PowerPoint, this post is for you. Since Google Slides does not have the

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This year, I'm writing a book about mathematical games. For my research, I've been reading old collections, scrounging Board Game Geek, hosting play-test parties, attending game design conventions, and conscripting students as guinea pigs (which works much better than the reverse: conscripting guinea pigs as students).

Nowhere in all my research have I come across a mind quite like that of Walter Joris.

Walter generates games, puzzles, and pencil-and-paper experiments with such intensity and regularity that he must be a kind of pulsar: some heretofore unknown astronomical object, emitting what I admiringly call Joris Radiation.

You and I see right angles. Walter sees a doodle game.

You and I see paper. Walter sees the Incredible Paperman.

You and I see a cube. Walter sees... well, to be honest, I don't know what Walter sees, but I can't help wanting to see it too.

His book 100 Strategic Games for Pen and Paper is the most bizarre and marvelous thing I've read this year. "Nearly all the games have been invented by me," he writes in the introduction, and it's true: his fingerprints are on every page.

From those hundred, I picked out half a dozen to share here. Each is for two players; each requires only pens and paper; and each has surprising strategic depths to plumb.

May the wondrous light of Joris Radiation shine upon you in these strange times!

( See the bottom of the post for an interview with Walter.)

1.
Magical Squares

In an actual magic square, every row, column, and diagonal has the same sum. In Walter's game, you won't achieve that, but the goal is to get as close as possible.

Each player begins with a blank square, then secretly places numbers in the four corners. You may use whatever numbers you like (including repeats).

Then, you reveal your squares to each other. Now, whatever numbers your opponent put in her corners, you must put in your edges ( in whatever order you like).

Finally, you can choose whatever number you like for the center.

The goal: have as many rows, columns, and diagonals as possible sharing the same sum.

A pair with the same sum scores 1 point; a trio scores 2 points; a quartet scores 3 points; and so on. (This sample game ended in a 3-3 tie.)

What happens if you start with identical numbers in your corners (e.g., 7, 7, 7, 7)? What if you pick radically different numbers (1, 10, 100, 1 million)? Is there a best strategy? If so, is it deterministic or probabilistic? The field is open for exploration!

This is, in one sense, a standard game of territory control. It's like dozens of others I've encountered. Yet I can't get it out of my head. It's something about those juicy grapes, the silly theme, and the lovely drawings that result.

First, draw a bunch of grapes. Make it clear which grapes share a border.

Then, by turns, each player picks a grape on which their "fly" begins, and marks it with a colored dot.

Then, take turns moving. (Whoever placed their fly second should begin.)

On each move, your fly consumes the grape it's on (shown by fully coloring in the grape), then moves to an adjacent grape.

Whoever winds up unable to move, because there are no adjacent grapes available, is the loser. Here, three more moves have been made:

The strategy seems straightforward, but the grapes can trick the eye, lending an element of suspense. (You may have less territory left than you think!) Also, whereas most pencil-and-paper games leave the paper coated with crisscrossing gibberish, this one ends up like a page from a coloring book.

Here, purple wins! (Green fly made some bad life choices.)

Ideally played while snacking on grapes.

This has more the flavor of a puzzle than a game. It's a puzzle I've yet to solve.

You begin by drawing a pyramid of 21 circles. Draw six on the bottom row, five on top of that, four on top of that, and so on.

Then, take turns writing a 1 in a circle of your choice.

After that, take turns writing 2, 3, and so on, in order. (You must write your numbers in order; no skipping ahead.)

When you have each written your 10, there will be one circle left blank: the black hole.

The black hole destroys all its neighboring circles. Whoever has a greater sum of numbers left over - that is, whoever loses a smaller sum to the black hole - is the winner.

If this speedy game is still too slow for your overheated 21st-century attention span, then you can eliminate the bottom row, and play using just 15 circles, where each player writes the numbers from 1 to 7.

(Or if you want to kill an extra ten minutes, add three more rows, for a total of 45 squares, so that each player writes the numbers from 1 to 22.)

Anyway, after a few rounds, I still have very few strategic intuitions about this game, but I love the simplicity of the design.

You begin with a 6 by 6 grid. Players take turns filling in pairs of adjacent squares, as if covering them with a domino.

These covered squares belong to nobody. Rather, you are fighting for control of the other squares, which you claim when your domino completes a fence closing off a region with odd number of squares.

Close off 2, 4, 6, or 8 squares? That's useless. Close off 1, 3, 5, 7, or 9? You can claim them. (More than that, though, and it doesn't count.)

The winner is whoever claims more squares by the end.

Some games, like "Bunch of Grapes" and "Black Hole," feel so simple that they must have existed all along, as if their designer merely "discovered" them. Other games, like this one, are so quirky that they can only have been invented.

Or perhaps I should say "bred." To my eye, this game has genetic traces of Nim, Cram, Dots and Boxes, and more.

Many pencil-and-paper games feel airy and abstract. Why dots? Why boxes? Why tics, tacs, and toes? That's why I love the cute and highly literal theme of this one; the game resides right there in the name.

Snake fight!

Begin with a 5-by-5 array of dots, and draw in the outline. The players begin their snakes in opposite corners.

The goal is to cross the enemy snake as many times as possible.

You take turns extending your snake via vertical, horizontal, or diagonal lines. Your snake can never cross or touch itself, and cannot trace over a segment that has already been drawn (by you, your opponent, or the border).

The game continues until neither player can move. Make sure to keep score as you go; otherwise, you won't be able to tell who crossed whom!

(Fine print: Passing through the enemy's "head" counts as a crossing. So does the last move above for orange, where the head reaches the enemy, but doesn't pass through.)

I've saved for last the game that is perhaps the simplest - not to mention the deepest.

Begin with a 6 by 6 grid. On each turn, you mark any box you like, but you must also eliminate an empty neighboring box.

Eliminating a diagonal neighbor is allowed.

The winner is whoever creates the largest group of connected marks. (Diagonal connections count.)

Play until no more moves are possible.

I have no idea why this one is called "Collector." Why not "Connector"? Or "Barrier"? No matter: a pleasant mystification is a natural byproduct of Joris Radiation.

The gameplay here reminds me of Amazons, a classic 1988 territorial game in which each move involves the annihilation of a square. This creates a "the world is falling away!" flavor of drama.

A Brief Interview with Walter
How did you develop an interest in designing games?
I always liked board games. And in my youth, there was a kind of a culture in pen paper games, long before computers were there. And then there was Martin Gardner and his math puzzles, which fascinated me.
How do you go about designing a game? What's the process like?
The first ones are the hardest. But once you give your mind the task: "invent games", it obeys. It starts to be creative. And then, whenever there is a kind of inspiration, a special pattern you see, a combination, your creativity will turn it into a game.
Where do you farm for ideas? As in: what board games do you play? What books do you read?
Well, mathematical puzzles of course, but also puzzles and pastimes for children. Also: existing board games in the world, and there is a lot of them. I myself play Go. And reading, I've read an enormous amount of books. Must be thousands. All kind of genres, philosophy, science fiction, novels, also the classical ones, from Dostoevsky to Rimbaud. Strange books likes the ones of Madame Blavatsky. Dali. Marinetti, Dada...Popular science also, but mostly in magazines. Art, history, cultural history.... I speak Dutch, French, English, German, and a bit of Spanish, In the first 4 languages I can read books rather easily. So, living in Belgium, in the middle of those cultures, you can imagine what an enormous amount of books and rare books you can find. With the fast train, you are from Brussels in about 1 hour in Paris, 1,5 in London, Cologne in Germany. Where I live, I can go by tramway from the Netherlands to France.
Do you think of your games as fundamentally mathematical?
Yes, I think of all games as fundamentally mathematical. I call myself a "matheist."
Lots of games in your book are adapted from board games. How do you decide if a game is suitable for adaptation?
It has to be able to become a pen and paper game. And that is fundamentally different. In material board games, it's all about the empty spaces; in a pen and paper game, about the occupied ones.
What are your favorite games (not of your own making)?
I played them all once. Now, I've limited myself to only Go.
What are your favorite games (of your own making)?
Since I invented them, I like them all. But Sequentium is definitely my favorite.

Thanks to Walter for answering my questions. You can find his work on Facebook, at Deviant Art, and of course in his book.

And at his request, here's Walter's explanation of his crown jewel, Sequencium:

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Six Strategic Pen-and-Paper Games (from a Strange and Bottomless Mind)

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This year, I'm writing a book about mathematical games. For my research, I've been reading old collections, scrounging Board Game Geek, hosting play-test parties, attending game design conventions, and conscripting students as guinea pigs (which works much better than the reverse: conscripting guinea pigs as students).

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smarter every day science education "how to"

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I explore the world using science. That's pretty much all there is to it. Watch 2 videos. If you learn something AWESOME, please subscribe if you feel like I earned it. Contact: http://www.smartereveryday.com/contact. I currently get lots of correspondence so please forgive me if I'm unable to reply.

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When students become active doers of mathematics, the greatest gains of their mathematical thinking can be realized. The process of sense making begins when we create classrooms full of curious students' thoughts and ideas.

By asking "What do you notice? What do you wonder?" we help our students see problems in big-picture ways.

Self-confidence, reflective skills, and engagement soar, and students discover that the goal is not to be "over and done," but to discover multiple strategies for tackling a problem.

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Notice and Wonder - National Council of Teachers of Mathematics

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What is Notice and Wonder? When students become active doers of mathematics, the greatest gains of their mathematical thinking can be realized. The process of sense making begins when we create classrooms full of curious students' thoughts and ideas. By asking " What do you notice? What do you wonder?"

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Article by Jennifer Piggott and Jo Scholten

This article invites you to get familiar with a strategic game called "sprouts" which appears here on the NRICH site. The game is simple enough for younger children to understand, and has also provided experienced mathematicians with significant food for thought. After a description of the basic game, this article will suggest a few ways to explore it.

The game

The game of Sprouts was invented in 1967 by two mathematicians John H. Conway and Michael S. Paterson, when they were both at the University of Cambridge in the UK. The game was popularised by one of Martin Gardner's "Mathematical Games" columns in Scientific American. Here is a quote from Conway:
"The day after sprouts sprouted, it seemed that everyone was playing it, at coffee or tea times, there were little groups of people peering over ridiculous to fantastic sprout positions."
Sprouts is a game for two players. All you really need is paper and a pencil.. The game starts by drawing any number of spots. In this example we are going to look at 3 spots.

The first player has a turn by joining two of the spots and marking a new spot in the middle of the line. Or the line may start and end on the same spot.

Rules:

You are not allowed to draw a line which crosses another line. This is important to remember!

A spot cannot have more than three lines leading to or from it. For example, in the game below, spots A and B cannot be used any more because they already have three lines.

The idea is to make it impossible for the other player to draw a line.

So the last person to draw a line is the winner.

Over to you!

Find a friend to play the game with. We suggest that you limit the number of spots to 3. Try playing several games to get the hang of it. Here are a few questions to focus your mind, but don?t try to answer them until you have played the game a few times, and feel confident with the rules.

Does the person who goes first or second tend to win?

Can you explain any winning tactics?

The game must end after a limited number of moves. Explain why. (Hint: you might find it useful to explain this in terms of "liberties". The liberties of a spot equals 3 minus the number of lines drawn from that spot.)

Conway notation:

How did you explain the winning tactics? In discussing this game, several enthusists have found it helpful to adopt a form of notation to communicate their ideas and strategies. We often use notation to explain our ideas in mathematics, and you might find it interesting to learn a little about Conway notation, the official notation of the WGOSA (The World Game of Sprouts Association). It began to evolve in 1999 in a discussion on the mathforum website.

Extending the game to different numbers of spots:

The strategy becomes more complex as you increase the number of spots, although WGOSA suggests that live spots at the end of the game are the key to making the game fun to play. (A "live" spot is a spot that is not "dead". A "dead" spot has 3 lines attached to it and is therefore out of the game.) These live spots at game's end are called "survivors".

Suppose a normal sprouts game begins with an even number of spots. Which player is likely to win and what can you say about the number of survivors (1)? If the game started with an odd number of spots what would you expect to happen and what could you say about the number of survivors (2)?

Proofs to be explored:

There is quite a lot of nice mathematics that can be gleaned from the game. What follows is a look at a few proofs. Stephan Duerr has summarised some of these, and added his own at the web site given below.

Enthusiasts have come up with a terminology that is useful to the proofs. These are all given below:

The "liberties" of a spot equal three minus the number of lines emerging from the spot.

A spot is "dead" if it has no liberties.

A "survivor" is a live spot at game's end.

A "guard" is a dead spot that is one of the two nearest neighbours of a survivor (see more details below).

A "pharisee" is a dead spot that is not a guard.

A "region" is an area of white space on the paper bounded by lines. The region extending off to the horizon is called "outer region", all other regions are "inner regions". At game's start, there is one region.

A "cluster" (or "clump") consists of all spots that are directly or indirectly connected by lines. An "indirect" connection means that there are other spots along the connection. The smallest possible cluster consists of just one unattached spot. At the beginning of a game with n spots there are n clusters.

The lines emerging from a spot divide the area around the spot into "sites". Spots have as many sites as lines, except for no lines, where there is one site.

Let:

$n=$ number of spots at the beginning of the game

$d=$ number of dead spots

$m=$ number of moves

$s=$ number of survivors

$p=$ number of pharisees

$c=$ number of clusters

$r=$ number of regions

$L=$ number of liberties

A few to get your teeth into?

All of these proofs have been taken from work by contributors to the maths forum.

Prove that the number of liberties at any time can be expressed as $L=3n-m$

Each spot has $3$ liberties at the beginning of the game. There are $n$ spots. Hence at the beginning of the game there are $3n$ liberties.

Each move reduces the number of liberties by $1$. This is because the two spots that are joined result in $2$ liberties being lost, whilst the spot that is added results in $1$ liberty being added.

Hence $L=3n-m$.

Prove that the maximum number of moves can be expressed as $m\leq 3n-1$.

At the game's end, each survivor has exactly $1$ liberty (otherwise you could connect the survivor to itself), hence $s=3n-m$.

$s\geq 1$ hence $3n-m\geq 1$.

Rearranging this gives $3n-1\geq m$ or $m\leq 3n-1$.

Prove that the minimum number of moves can be expressed as $m\geq2n$.

At the end of a game, each survivor is surrounded by $2$ guards.

These guards are spots that are dead in one of two ways:

If any guard were live, the game could be continued by connecting the guard to the survivor. Each guard has at least two of its sites accessing a region that is also accessible to the survivor. Since no two survivors can access the same region (otherwise we could connect the survivors), no spot can be a guard for two different survivors.

A pharisee is a dead spot that is not a guard.

The number of pharisees is the total number of spots minus the number of survivors and minus the number of guards. Written algebraically:

$$p=n+m-s-2s$$ $$p=n+m-3s$$ $$p=n+m-3(3n-m)$$ $$p=n+m-9n+3m$$

Hence $p=4m-8n$

Rearranging this equation gives us $m=p/4 + 2n$.

If the game is played so that there are no Pharisees, that is a game is played with the minimum number of moves, then $m=2n$.

So we can say that $m\geq 2n$.

Prove that the maximum number of regions can be expressed as $r\leq 2n+1$

There are exactly two types of moves in sprouts. One type merges two clusters, the other type creates a new region.

Because the number of regions either increases by $1$ or the number of clusters decreases by $1$, we can deduce that each move increases $r-c$ by $1$, or that $r-c-m$ must be constant throughout the game.

At the beginning of a game, $r=1$, $c=n$ and $m=0$.

From this we can deduce that $r-c-m=1-n-0$ which can be rearranged to give: $$r=1-n+c+m$$

At the end of the game, as we have already seen, $s=3n-m$, or $m=3n-s$.

Substituting this equation into the above we have: $$r=1-n+c+3n-s$$ which simplifies to: $$r=2n+1+c-s$$ Every cluster must contain at least one live spot, namely the spot of the cluster that was last to be placed on the paper. From this we can say $c\leq s$.

As $c\leq s$, then $c-s\leq 0$. We can substitute this into $r=2n+1+c-s$ to give: $$r\leq 2n+1$$.

In conclusion

Sprouts is an intriguing game which contains some interesting mathematics. Have fun with the game, and may sprouts continue sprouting!

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Article by Jennifer Piggott and Jo Scholten This article invites you to get familiar with a strategic game called "sprouts" which appears here on the NRICH site. The game is simple enough for younger children to understand, and has also provided experienced mathematicians with significant food for thought.

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Same But Different is a powerful routine for use in math classrooms. The images can be used for teaching a way of thinking: grayscale thinking, categorical thinking; helping students to build a network of ideas and an approach for learning and thinking about mathematics.

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Original Single Digits Edition card

Object of the game: Make the number 24 from the four numbers shown.You can add, subtract, multiply and divide. Use all four numbers on the card, but use each number only once. You do not have to use all four operations. All number nines have a red center, so you can tell a nine from a six. We show one solution to the card below, can you find another?

5 x 5 = 25
4 - 3 = 1
25 - 1 = 24

This is an easy card, called a "1 Dot," because it has one white dot in the corner. Cards of medium difficulty have two red dots, and tough cards have three yellow dots. To try another Single Digits card, click here. Other 24 game editions play differently. To try them, go to Catalog, and click on 96 card decks to choose a game. Each game page includes sample cards and a brief explanation.

Video Tutorial

Download printable SD/DD Instructions: English | Spanish

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The 24 Game is a unique mathematics teaching tool proven to successfully engage students in grades 1-9 from diverse economic and social backgrounds. Knowing the answer is always 24 alleviates a classic type of math anxiety-getting the right answer-and instead puts the emphasis on the method behind the math.

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Example rules

Standard

Double the number and add two; eg 11 -> 24

Write the number in reverse; eg 13 -> 31

Write the number twice; eg 42 -> 4242

Count the number of characters in the word for the number; eg 19 -> 8

Advanced

The number of the number's prime factors; eg 16 -> 4

The sum of the squares of the number's digits; eg 13 -> 10

The number written in base three; eg 22 -> 211

Try and guess Eliza's rule

Let Eliza guess your rule

One player thinks of a mathematical rule and the other player tries to guess it. Play against our mathematical wizard Eliza!

About the game

The Chooser thinks of a rule that converts one whole number (integer) into another. A mathematician would call this a "function".

The Guesser then tries to guess the rule by testing it with different numbers.

For example, if the Chooser thought of the rule "double it and add one", the game might proceed as follows:

Standard game

Rules can use arithmetic and everyday concepts.

Advanced game

Rules can use mathematical concepts, including: squares, powers, bases, factors, divisors, modulo, prime numbers, and piecewise functions.

Additional points

For some rules it may not be possible to give an answer for all guesses. For example, if the rule is "The number of times that the number divides into 100 (ignoring any remainder)", no answer is possible for the guess "0". In such cases the Chooser says "I can't answer it", or "Choose a larger number".

Some rules may have alternative ways of expressing them; for example "Add one and double it" is the same as "Double it and add two", which you can see by writing the rule as an equation:

(x+1) x 2 is the same as 2x+2.

Either of these is therefore a valid answer to this game. If there is any doubt the Chooser can give the Guesser a number to make sure they can arrive at the correct answer themselves.

March 2024 Fiendish Function

Coming soon!

Last Month's Answer

Here is the answer to last month's Fiendish Function.
February 2024

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Double the number and add two; eg 11 -> 24 Write the number in reverse; eg 13 -> 31 Write the number twice; eg 42 -> 4242 Count the number of characters in the word for the number; eg 19 -> 8 The number of the number's prime factors; eg 16 -> 4 The sum of the squares of the number's digits; eg 13 -> 10 The number written in base three; eg 22 -> 211 The Chooser thinks of a rule that converts one whole number (integer) into another.