# Product Dice - The Sequel to Sum Dice

Players: 2
Ages: 7 and up
Cost: Free!
Math Ideas: Multiplication, probability, number theory
If I roll a 3, what numbers do you hope to roll?
What makes this game different from Sum Dice?
Is this game fair? Why or why not?

So last week, I wrote about Sum Dice, a simple and quick game that can be quite a fun introduction into the concept of fairness.

What makes a game fair? Essentially, a fair game should be one where both players have an equal chance of winning. Sometimes people like playing an unfair game, such as blackjack, because they can "beat the odds" and get the adrenaline rush of winning when they should have lost.

But mostly, people like to play fair games, and they definitely hate playing a game that seems fair but is actually rigged in one way of another.

So that's why Product Dice is such a fun and devilish game to play, once your child has played and thought about Sum Dice.

## How to Play

Remember how Sum Dice works? One player is Evens and the other player is Odds. They roll two dice and then add the numbers. If the sum is even, Evens gets a point. If the sum is odd, Odds gets a point.

Product Dice is exactly the same, except instead of adding the two numbers, you multiply them together. Then you award points in the same way: an even product is a point for Evens, while an odd product is a point for Odds.

So if you've played Sum Dice, you're wondering what makes them so different. I mean, sure, one involves addition and the other involves multiplication, but otherwise, how are they different?

Your child will probably be wondering the same thing! So tell them to trust you and play the game out. Just make sure you pick Evens...

## Where's the Math?

If you play the game ten or twenty quick times, you'll probably start to notice that Evens is probably winning! No doubt your kid is picking up on this as well and is wondering just what the heck is going on.

It seems strange, right? Sum Dice, which is almost the same game, was totally fair, as we proved last week. And yet there is something deep and important about multiplication that makes this game rigged. Check it out:

Let's say you roll a 1 with the green die. Now you roll the red die. If you roll one of the three odd numbers (1, 3, 5), the product will be odd. If you roll one of the three even numbers (2, 4, 6), the product will be even. Fair so far, right?

Now let's say you roll a 2 with the green die. No matter what you roll with the red die, the product will be even. After all, any whole number multiplied by 2 will result in an even product.

So it's hard to roll an odd product! If you roll an odd number with both dice, then the product will be odd. Any other outcome (green is even, red is even, both are even) will result in an even product. As a result, the Odds player will only get a point 25% of the time!

The properties of even and odd numbers are a delight to explore, especially in the context of games. These explorations also give your child an elementary version of a topic that mathematicians call number theory, which explores more complex versions of these same patterns and properties..

Once you and your child have explored Product Dice, next comes the best part - getting your kid to challenge an older sibling or a grandparent to a game! Your kid will love nothing more than explaining the rules to their grandfather and then sneakily saying "Let's play! I'll be Evens." [smirks devilishly]