At the Games Night for the recent Math Prize for Girls event, Gili Rusak and I ran a treasure hunt that aimed to explore the lighter side of math. One of our activities: Mental Madness. It’s a game that can push one’s mental computational abilities to the limit.

Here are the rules we used for the Math Prize for Girls version.

1. Players take turns going around a circle modifying a number N.
2. The value of N starts at 20.
3. The current player rolls the icosahedral die and the pair of regular dice.
4. Let S be the sum of the numbers rolled on the regular dice.
5. The current player must modify N as follows:

If N = 0, then add S to N.
Otherwise, use the following table to determine the operation:

 Icosahedral Die Result Operation Name Operation Description 10, 20 Add $N + S$ 6, 12, 18 Subtract $N - S$ 11, 13, 14, 15, 17, 19 Multiply $NS$ 3, 5, 7 Floor Divide $\lfloor N/S \rfloor$ 2 Root $\lfloor \sqrt{\vert N \vert}\rfloor$ 1, 4, 9, 16 Square $N^2$ 8 Cube $N^3$
1. If the current player errs, the value of N is reset to 20 and the next player becomes the current player.
2. The current player may pass, in which case treat the situation as an error.
3. The only assistance that other participants may give when they are not the current player is to remind the current player what the current value of N is.
4. All computations must be done entirely in the head! No paper, pencils, pens, abacuses, slide rules, calculators, computers, or any other computational aides are allowed!

Goal: The players must make it once around the full circle of participants without an error or a pass.

* * * * *

To get a statistical profile of the likely sizes of numbers that players would have to work with, we ran computer simulations. We knew that Math Prize for Girls contestants could handle taking the square root of five digit numbers and we wanted moments where they’d be nearly full circle only to be hit with the perfect cube operation. At the same time, we didn’t want the game to go on forever and simulations suggested that there is about a 10% chance that a round will not involve any numbers bigger than 20,000.

### Rule Modification

The rules are easy to change and the game can be adapted to many situations and used for many purposes. We encourage modification!

For example, suppose you are working with students that are uncomfortable with fractions. We’ll change the rules so that the game helps students to realize that fractions can actually be one of your best friends. In fact, the only rules we will change are Rules 2 and 5 (along with the corresponding adjustment to rule 6):

1. The value of N starts at 1.
1. The current player must modify N as follows:

If N = 0, then add S to N.
Otherwise, use the following table to determine the operation:

 Icosahedral Die Result Operation Name Operation Description 2, 3, 4, 5 Add $N + S$ 6, 7, 8, 9 Subtract $N - S$ 12, 14, 16, 18, 20 Multiply $NS$ 11, 13, 17, 19 Divide $N/S$ 10, 15 Floor Divide $\lfloor N/S \rfloor$ 1 Square $N^2$
1. If the current player errs, the value of N is reset to 1 and the next player becomes the current player.

* * * * *

With these changes, there is about a 50% chance that the round will never see numbers bigger than 1000.

The reason this version can help students learn to be more accepting of fractions is because there is a decent chance that students will have to divide by a number that does not go evenly into the current number and where the result does not have a simple decimal expression. If, for instance, the first player had to divide by 7, probably the player would prefer to say “one seventh” and not “zero point one four two eight five seven repeating.” Not only would the fraction representation be simpler to state, it will also likely prove easier to compute with.

Here’s how the first ten operations came out in one round of this version:

#### Subtract 8 Add 10 Add 7 Divide by 6 Divide by 7 Multiply by 10 Subtract 8 Multiply by 2 Add 11 Add 11 Divide by 8

Starting at 1, go through this round in your head. If you are wedded to decimals, it’s going to be rough seas. Although there are still challenges, use of fractions gives real hope of making it through. (If you made it to 113/84, you did it!)

We're a math club for girls.
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