# Fifteen

Players: 2
Ages: 6 and up
Cost: Free!
Math Ideas: Addition, Subtraction, deductive reasoning
Which number gets you yo fifteen?
Which number do I want to get? How should you block me?
Can the first player always win? Can the second player always block?

Sometimes, your kids have finished the maze and the crossword on the kids' menu and have started arguing about who is on whose side of the booth. Or maybe they are asking you when dinner will be ready in alternating 20-second intervals.

In either case, you just need a quick, simple game to occupy their minds. Something that doesn't involve a bunch of pieces or instructions.

If you find yourself in that situation, I'd like to recommend Fifteen.

## How to Play

Fifteen is perfect for these situations because all you need is a pencil, some paper, and the ability to add to fifteen.

Start by writing out the numbers from 1 through 9 on the paper. Then, each player takes turns capturing one of these nine numbers. The goal is to be the first player with a set of three numbers that adds to fifteen. For example, if you circle 3, 5, and 7, you win. Kids can either use two different colors, or one player can circle his numbers while the other player underlines hers.

The first player definitely has an advantage, so I would recommend that your kids alternate who goes first. Also, that means more games, which means more time off the clock before bedtime. Isn't that the most important measurement in parenting?

This game, by the way, isn't particularly fun for two adults; we're just too good at blocking each other's plans. But it can be really fun to play with someone in K-3rd grade, who is still becoming fluent with addition, but can also plan out a strategy for winning.

Fifteen clearly helps your kids count and add to fifteen, and for younger kids your questions should focus on strengthening those skills: How close are you to fifteen? What number do you want to circle to get fifteen? Do 5, 6, and 2 make fifteen?

Beyond addition, the game has some interesting mathematical ideas that older kids could use to think strategically about the game. For example, let's say you choose to circle 8 on your first move. That means that you need to get a total of seven from your other two numbers. So you could circle 1 and 6, 2, and 5, or 3 and 4.

It seems like 8 is a pretty good choice for a first move, then. Is it the best possible choice? Are there numbers that are particularly bad to choose as a first guess?

The second player, honestly, is mostly playing defense. That person has to think strategically about how to prevent the first player from achieving her goals. So if she has circled 6 and 7, what should we do to block her from winning on the next turn? Is there a best move for the second player's first turn, before he knows his opponent's strategy?

Finally, is all hope lost for the second player? Or is there a way to always prevent defeat? Read on to the next section for a hint...

## Bonus for Math Nerds

One way to think about the strategy of Fifteen is to list all the combinations of three numbers that add to fifteen. As it turns out, there are eight combinations:

1+5+9
1+6+8
2+4+9
2+5+8
2+6+7
3+4+8
3+5+7
4+5+6

So you can start to analyze the game by looking at these combinations. For example, 5 seems to be a powerful move, since it appears in four of the eight combinations. On the other hand, 9 only appears twice, making it less valuable.

But here's something even cooler: The nine numbers can be arranged into a square where every row, every column, and both diagonals add to fifteen.

Look familiar? It's a Magic Square, another free game I shared several weeks ago!

If you play Fifteen on a Magic Square, you realize that you win whenever you can capture an entire row, an entire column, or a diagonal. In other words, Fifteen is just a variation of Tic-Tac-Toe! In this arrangement, 5 is the crucial center square because it can be a part of four winning tic-tac-toe sequences, while 9 is stuck in one of those lame side squares.

This is why the game isn't particularly fun for adults; just as in Tic-Tac-Toe, the second player can always block the first player, which results in draw after draw. But the specific strategy for doing so is pretty tough to figure out. Unless, that is, you play on a Magic Square board, in which case the strategy is identical to Tic-Tac-Toe.

When you study math, you'll often find opportunities to make connections between topics that seem very different at first. Sometimes a particularly tough geometry problem can be solved by reframing it as an algebra problem, or vice versa. The connection between Fifteen and Tic-Tac-Toe is one of the plainest examples I've found of using mathematical structure to see the deep connections between two seemingly different ideas (or games).

Thanks to Math Frolic for showing me the connection between Fifteen and Magic Squares