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Context and examples

When teaching mathematics to a new class in high school, I often open up with some mathematical games that can be played cooperatively and with minimal requirements (good old chalk and blackboard). On the one hand, this helps break the ice with the class (so to speak) and I get to quickly learn their names, and on the other hand, by trying different games I get to partially gauge their understanding of certain fundamental topics (such as parity, symmetry, prime numbers, ...) and their technical skills (multiplication table, visualisation skills, posing and solving simple equations, ...).

The games I use are simple enough in premise that even when there is a language barrier (such as welcoming classes for recently arrived refugees), kids will be able to quickly grasp the idea and participate.

Game of multiples and divisors

One good example is the Juniper-Green game, or simply the game of multiples and divisors: you set an upper bound (say 30), and then

  • 1st student chooses a number between 1 to 30.
  • 2nd student chooses either a multiple or divisor (still within 1 to 30) of the 1st student's number.
  • 3rd student chooses similarly based on the 2nd student's number, so on and so forth.
  • As they choose these numbers I write them on the blackboard with their name next to it. This is important because no number can be used twice.
  • The last player who can still choose a valid number wins. The game is fun because they not only brush up on their multiples/divisors but after a few runs they also discover the tactical role of prime numbers in this game.

Color of a square on the chessboard

Another example is drawing a chessboard on the blackboard but only coloring the first row, rest being a blank grid. Then asking the students to guess the color of a random square I choose. At first they show their visual skills by instantly finding paths connecting to the that square from a known square and then using the alternating color rule.

  • So if e5 is the target, one student tells me: I start from b1 which is a light square, then move to c1-dark, c2-light, d2-dark, d3-light, e3-dark, e4-light and e5 must therefore be dark.

Others find more efficient ways of rerouting to the target square and guess the color (e.g., moving diagonally). Then I would tell them, what if the board was much larger, not 8 by 8, but 100 by 100? Surely, it would take a long time to find the color of a random square with these path-finding methods and assuming we don't make mistakes along the way. What if there was a mathematical rule that would associate the coordinates of a square to its color?

As soon as I would say it and also substitute the file letters a-to-h with numbers 1-to-8 (like rows), they would discuss in small groups and within minutes hands would be raised! They've got it: if the sum of the coordinates of a square is an even number the square is dark, if the sum is odd the square is light. Suddenly, they discover on their own the power of mathematically modelling a problem: their predictions are no longer bound by how far they can visualize or how deep a sequence can be memorized.

Questions

  • Do you know of other similarly spirited mathematical games (or more logic oriented) that can readily be played in classrooms? Ideally, accessible to most high school students.

Admittedly, this is a rather open question meaning that all proposals are welcome, whether they be games/puzzles you've used or seen used before, your own personal ideas for designing a new game, or simply examples you've read about before which may be fitting here. The latter brings me to the next question:

  • Equally of interest would be puzzle themed book (or other resources) recommendations with similar mathematical games that would be particularly suited for teaching.
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    $\begingroup$ I've had students discover Euler's $V-E+F=2$ empirically, but this requires a large collection of polydrons. $\endgroup$ Aug 15 at 12:02
  • $\begingroup$ @JosephO'Rourke That's beautiful!! Any chance you've come up with other playable variants of the game, one not requiring polydrons? $\endgroup$
    – user929304
    Aug 15 at 15:25
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    $\begingroup$ @SueVanHattum It works wonders, students easily get hooked and truly enjoy the game. In my experience 40 (instead of 30) is an optimal upper bound in terms of playability, and you can also have them figure out what number the first player should choose to win (under perfect play) in a two-player version of the game. $\endgroup$
    – user929304
    Aug 15 at 20:33
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    $\begingroup$ I wonder if I'm missing something. I'm first. I pick 29. 2nd player picks 1, and I pick 23, and I win. $\endgroup$
    – Sue VanHattum
    Aug 15 at 21:22
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    $\begingroup$ Shoot. So the 2-player version would not be very interesting for college students. $\endgroup$
    – Sue VanHattum
    Aug 15 at 23:59

3 Answers 3

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Board Games

  • Prime Climb
  • Katamino (pentominoes, for 2 or as a puzzle for 1)
  • Blokus (shapes using 1 to 5 squares, strategy)

Card Games

  • Set (logic)

A game on the board

It's a bit like yours, but the teacher is the Divisor Miser or the Tax Collector. It is possible for the students to win, if they strategize well.

  • Numbers 1 to (perhaps) 40 are on the board.
  • Students pick one. You cross it off and add it to their score.
  • You add all its factors to your score.
  • They cannot pick a number that will give you nothing.
  • You get all that's left at the end.

Logic Puzzles Rush Hour and Chocolate Fix

Geometry Puzzles

  • Geometry Snacks, by Ed Southall and Vincent Pantaloni
  • Also Catriona Shearer has lots of good ones on twitter

(There might be others you like at my blog.)

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    $\begingroup$ Thank you Sue, these are excellent recommendations! I also quite like your blog which I have discovered through your post. Honestly, a treasure box of wisdom goodies :) $\endgroup$
    – user929304
    Aug 15 at 15:17
  • $\begingroup$ Indeed interesting blog! Unfortunately many of the links don’t work anymore. $\endgroup$ Aug 16 at 13:19
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    $\begingroup$ If you comment at my blog, I can fix the ones you care about. I'll try to fix more when I have time. $\endgroup$
    – Sue VanHattum
    Aug 16 at 16:40
  • $\begingroup$ Thanks! I don't usually listen to podcasts, though. (Mainly not enough time. Also, I guess I prefer to read.) $\endgroup$
    – Sue VanHattum
    Aug 30 at 23:41
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In a comment I mentioned discovering Euler's Formula using polydrons, which requires quite a collection of polydrons.

However, Euler's Formula holds for planar graphs (the $1$-skeleton of a polyhedron is a planar graph). So you could discover it with paper and pencil, or better, on a blackboard, drawing a random planar graph and counting. Needs introduction to explain what is a connected, simple, planar graph; what is a vertex, an edge, a face. One tricky part is to make sure to count the outer face. Once $V-E+F=2$ is established, you can show a few polyhedra, and connect the $1$-skeleton of the polyhedron with the corresponding planar graph.


   

   Icosahedron graph: $V=12$, $E=30$, $F=20$. Wikipedia image.


Then once they know what is a planar graph, you'd have an excuse to mention the $4$-color theorem :-).

   4-coloring

   Image from mathpages.com.

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Logic

Tatham's Puzzles, such as Loopy (Cairo) and Tracks, both of which are endless sources of puzzles where pigeonhole principle and parity are very useful tools for solving them. Tatham's version of Minesweeper is also guaranteed to be solvable with pure logical deduction (i.e. no guessing), but Loopy and Tracks encourage a much wider variety of combinatorial deductions.

Loopy LogoTracks Logo

Cosmic Express, which some people solve by just trying randomly, but which can be solved quite systematically if you are sufficiently good at logic and abstraction.

Cosmic Express Logo

Programming

Manufactoria, which was originally a free Flash game with an incredibly steep rise in difficulty (most people cannot complete the hardest levels), but apparently has a gentler version released in 2022.

Manufactoria 2022 Logo

Robozzle, which is a more restricted type of programming puzzle than Manufactoria, but has simpler puzzles. The "Campaign" should be a good place to start.

Robozzle Logo

Geometry

ScienceVSMagic Geometry, which is a collection of compass-straightedge challenges.

ScienceVSMagic Geometry Logo.

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