Arithmetic Twister

Here we'll look at a physical activity in the vein of the four fours puzzle or 24 game.

Setup

For this activity, we'll need a number mat and a way of randomly producing numbers. I'll mostly look at a version where we modify a Twister mat and use foam dice.

• Twister Game ($18)

• Twister Ultimate Game ($22)

• Vinyl number stickers ($10, 20 copies of each digit)

• Foam dice set ($20)

I've toyed with a few possible layouts for the mat, but here's the one I've settled on:

Using just 1-4 so each circle is its own number wasn't as interesting, but would probably be fine for younger students. For that, I'd recommend an 8-sided die or a 1-8 spinner like the one linked above. The other layout below is one I liked because the numbers were more evenly distributed, but it seemed more helpful to students if the numbers had assigned colors like in the mat layout above.

Twister Ultimate has double the number of dots, so you can either duplicate your favorite mat pattern or make a variation.

Challenges

You can decide which Twister-like rules you want to adhere to. For example, in Twister, two players cannot both touch the same circle, so you can choose whether you want to enforce that or not. If there are only 2-3 students on a mat, that rule can make it more fun and silly.

Here are a few versions of the game, all assuming a mat that features the numbers 1-6. You'll want to choose one or two that fit your students' abilities.

1. Roll the 8-sided die and have each player place their feet on two circles that can be added or subtracted to get that number. From this point on, each round will consist of rerolling and letting players move at most one foot to a new circle so their two circles add or subtract to the rolled value. If a player cannot make the rolled value, they are out. The winner is the last player standing or any players left standing after 10 rounds if you want to keep it quick.

2. Roll the 8-sided die and have each player place their feet and a hand on three circles that can be added or subtracted to get that number. As above, each turn you will reroll the die and allow players to swap up to one hand or foot to a new circle to get three values that can be added and subtracted in some sequence to get the rolled value. Same win condition as above.

The rules for the following variants are as above, but changing the die, number of circles that must be touched simultaneously, and mathematical operations. We'll briefly describe them in terms of those:

3. 12-sided die; 2 circles; addition/subtraction

4. 12-sided die; 3 circles; addition/subtraction

5. 12-sided die; 3 circles; addition/subtraction/multiplication

6. 20-sided die; 3 circles; addition/subtraction/multiplication/division

Notes

• For 2 circles and addition/subtraction: With an 8-sided die the only danger is standing on \(1\) and \(1\) when \(8\) is rolled, so that should be avoided. With a 12-sided die, the dangers are

  • standing on \(4\) and \(4\) when a \(4\) is rolled

  • standing on \(5\) and \(5\) when a \(5\) is rolled

  • standing on \(6\) and \(6\) when a \(6\) is rolled

  • standing on numbers each \(\leq k\) when \(k+7\) is rolled

  Keeping one foot on a \(6\) at all times is a safe strategy. The only issue is if a \(12\) roll is followed by a \(6\), but all players not eliminated by the \(12\) will be eliminated by the \(6\).

• If we throw multiplication in, the strategy of keeping a foot on \(6\) now never leads to elimination, since \(6 \cdot 2 = 12\).

• For 3 circles and addition/subtraction: With an 8-sided die, two \(4\)'s cannot be used in making an \(8\), so the player should avoid touching three \(4\)'s at once. With a 12-sided die, there are a lot of potential pitfalls. One good strategy for both the 8- and 12-sided dice is to keep feet on \(3\) and \(4\), since any of \(1\) through \(12\) can then be made by moving a hand.

• For 3 circles and addition/subtraction/multiplication: The above strategy works for both 8- and 12-sided die without ever needing to multiply! For a 20-sided die, the tough roll is 19. Noting that \(3 \cdot 6+1 = 19\), we can look for a strategy of keeping feet on \(3\) and \(6\). You can check that moving your hand between \(1\), \(2\), and \(4\) will allow access to all rolls \(1\) through \(20\).

• The same strategy above works if we throw in division. Students likely won't realize division isn't necessary unless they've put some serious thought into it.

Extra Challenges

To push this more in the vein of four fours, here is a wilder challenge:

Random number generator for 1-100; 4 circles; any operations they know and can explain

You can always throw in new operations one at a time after division to get the wheels turning. Some good ones are:

• Concatenation (e.g., \(35\) from \(3\) and \(5\))

• Decimals (e.g., \(.4\) from \(4\))

• Exponents: \(x^y\)

• Square roots: \(\sqrt{x}\)

• More general radicals: \(\sqrt[x]{y}\)

• Logarithms: \(\log_x(y)\)

• Greatest common divisors: \(\gcd(x,y)\)

• Least common multiples: \(\textrm{lcm}(x,y)\)

• Ceilings (e.g., \(\lceil 5/3 \rceil = 2\) from \(3\) and \(5\))

• Floors (e.g., \(\lfloor 5/3 \rfloor = 1\) from \(3\) and \(5\))

• Bitwise operations like AND, OR, or XOR.

This post was sponsored by the Julia Robinson Mathematics Festival