The Game of 15 (Part 1)

Published on Sunday, August 12, 2012 in , , , , ,

nneonneo's Optimal decision tree for player X in Tic-Tac-ToeIn this post, you'll be introduced to a simple new game to play. Even better, it's a game you'll never lose.

It's not another version of Nim. This time, it's even sneakier!

Here's the rules of the game of 15:

1) Nine playing cards, with face values from Ace to nine, are face up on the table. The Ace always has a value of 1 in this game.

2) Players alternate taking turns, and on a given player's turn, they must take 1 card from the available group on the table. Neither player make take a card that has already been removed from the main pile.

3) The winner is the first person to obtain exactly 3 cards that add up to 15.

What kind of strategy would you use to win this game, or at least prevent losing?

You might be surprised to learn that you probably already know this game. It's a disguised version of (or in math terminology, it's isomorphic to) tic-tac-toe! How exactly does this relate to tic-tac-toe? Imagine the numbers 1 throught 9 arranged as a classic 3 by 3 magic square, so as to total 15 horizontally, vertically and diagonally:

This might seem like a hard arrangement to keep in memory, but it's easier if you picture the arrangements of even and odd numbers separately:

Now you can clearly see how the game relates to tic-tac-toe, and why it's played to 15. Since the other person doesn't realize what they're playing, this gives you an advantage.

In this version, however, the Ace through nine are laid out in a straight line in order, not the traditional crisscross pattern, so you can't see things as clearly as you would in a regular game. So, exactly what strategy should be used?

The proper strategy depends on whether you're going first or second. Let's start by assume you're going first. Following the classic strategies for X, as taught at chassandpoker.com and Wikihow, you'll want to take a corner square, which in this game equates to any even card (2, 4, 6, or 8).

When first learning this version of the game, always take a 4 when you go first. As you become more proficient in the game, you can start with any even card, but always starting with a particular card at first will help you get familiar with the essential.

There's only 2 different replies the other player can make:

1) They choose a 5: This is akin to taking the center square. You must reply by taking the 6 (the diagonally opposite corner). This might seem strange, as you'll have a 6 and 4 with no possibility of a 5, but you're setting a trap for them. If their 2nd move involves taking either the 8 or the 2 (a corner square, in other words), you've just won!

How? You take the sole remaining even number, which simultaneous blocks their possible win, and opens up 2 ways to win for you! When they block 1 way, you simply play the other to win.

Below is an animation of how the game looks in the standard form of tic-tac-toe. If you arrange the cards in the form of a magic square above, you'll be able to better follow along as I teach the strategies.

Remember, in actual play the cards are laid out in straight line Ace through 9, but laying cards out in the magic square form during practice will help you learn the strategies more quickly.

There's another possibility here. If you've take the 4, they've responded by taking the 5, then you've taken the 6, they could possibly take an odd card (equivalent to an edge square). In that case, you'll have to block by taking the 1 card that would total 15. For example, if they now have the 9 and 5, you'll want to take the Ace (9 + 5 + 1 = 15).

Here's how that kind of game looks in tic-tac-toe:

However the game proceeds from this point, just make sure you either wind up with the 3rd even card and block as needed, unless you hav take advantage of any mistakes they make by completing a row of 3. As you can see, the second player's best move is to take the 5 followed by any odd-numbered card, as it's possible to play you to a draw.

2) They choose anything EXCEPT a 5: In response, you need to take either the 8 or the 2, whichever one they haven't blocked. If they took the 3 or the 8, then the 4-3-8 (leftmost) column is blocked, meaning you have to take the 2. If they took the 9 or the 2, then the 4-9-2 (bottommost) row is blocked, so you'll need to take the 8. It's also possible that neither the 8 nor the 2 is blocked, and you have a free choice.

They should recognize that you need a particular card at this point, and take that card next. If you've take the 4 and the 8, it's not hard for them to figure out that they need to take the 3. If you've take the 4 and the 2, they'll go for the 9. If they don't make either of these proper responses, they've just handed you the win by mistake!

Assuming they don't hand you the win by mistake, first ask yourself if they can win by taking the 5. If they can win with a 5, take it! This will block their win, and set up two possible wins for you. All they can do at this point is to block you in one corner, and you win by taking the other:

If you don't need to block them with a 5, you'll need to take an even card (corner). If there's only 1 even card remaining, take it. Otherwise, you'll have two possibilities and you'll need to make sure that the one you take isn't blocked. To do this, simply ask yourself whether it's possible to make 3 in a row with your cards and the remaining cards. If so, then it's not blocked, and you can safely take that even card.

At this point, you'll have 3 even cards (corners), and two ways to win, so you simply wait for them to block one way, then you play the other to win:

That covers all the possibilities for when you go first. In the second part of this series, I'll delve into what happens when you're the second player. For now, simply practice as the first player. Remember, try the strategies out with cards in the magic square arrangement above, then get used to playing with the numbers in line, as you would in a real game.

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