# A Probability Puzzle and Game

One of the nice things about probability games is that they can offer a high level of replay while simultaneously advancing math education objectives. Here’s a simple one that I think you’ll find as a good alternative or supplement to a traditional lesson. This game can span many levels — from elementary school all the way to a college math class.

Here’s how the game works. It is two player, but can be made to be solitaire since only one player is making decisions while the other one follows along. For materials, some paper that you can cut up is enough.

### Rules

1. Start with the digits $$1,2,3,4,5,6,7,8,9$$. (Nine pieces of paper, equally sized, is enough.)
2. Player 1 picks five digits of their choosing so that they are all distinct AND that they add to 25.
3. Player 2 gets the remaining four digits and a (an additional) 5 (one more piece of paper).
4. Both players will now have a total sum of 25.
5. Player 1 picks the five digits in a random order and constructs the five digit number that results. For example, if Player 1 had the digits $$1,3,4,8,9$$ and drew them as $$4,8,3,9,1$$, then Player 1’s number is $$48391$$.
6. Player 2 does the same. Continuing with the example above, Player 2 would have (by force) $$2, 5, 5, 6, 7$$ (yes, two 5s). So Player 2, might randomly construct the number $$56527$$.
7. Whoever, has the larger number wins! In this case, Player 2 wins since $$56527 > 48391$$

### So What’s The Puzzle?

Player 1’s part is strategic. Thus, their goal is to find the five-digit combination that will lead to the greatest chance of winning. This isn’t as simple as it sounds nor is the optimal choice obvious. And proving it is a different matter. But the puzzle part is finding the best five digits to choose.

### So What’s the Education?

There are a few things here that we can get out of this.

1. For elementary students (and maybe even older!) can we identify all the possible choices? Remember the five digit combination has to add up to 25! Hint, there are only 12 possibilities! Do you know how much non-forced addition practice they will get?
2. For elementary students this is a great simple way to get them practice integer inequalities, especially with “large” numbers.
3. Because there is a decision to be made, this is exactly the perfect opportunity for students to discuss their reasoning! It may very well be that they will be able to argue for the best choice.
4. For more advanced students (say those taking a probability course), can they prove / deduce the best choice of five digits (ie the choice with the highest win probability)? Can they identify what that win probability is?
5. For all students, this is a great way to talk about probability and statistical significance. All twelve choices are within 5 percentage points of each other. So if you are playing this out live, it is going to feel like a 50-50 split because getting statistical significance will require a large sampling. Ambiguity is the playground for math debate!
6. Got any students craving to program? This is a great programming exercise and an introduction to simulation methods!

To be clear, this is not to replace an entire lesson. Rather it’s an activity that is out-of-the-box.

### One Example

If Player 1 chooses $$(2, 4, 5, 6, 8)$$, then this is roughly a 50-50 “hand” to play.

### Want Me To Make A Small Web App Of This?

If this sounds like something you would want in your classroom, just subscribe at the top of this page. Leave a note that you want something for this probability game. I’m happy to do it if there are enough people who request! 