The Game of 15 (Part 2)

Published on Thursday, August 16, 2012 in , , , , ,

nneonneo's Optimal decision tree for player X in Tic-Tac-ToeIn the previous post, you learned about a disguised form of tic-tac-toe known simply as 15, and how to avoid losing if you go first.

In this post, you'll learn about the best strategy to use when you go second.

The game of 15 was written about repeatedly by Martin Gardner. His original Scientific American column on it is available in his book Mathematical Carnival. Gardner also discusses it briefly in Aha! Insight. In both sections, he also discusses other interesting ways to disguise tic-tac-toe.

You can also find out more variations of tuc-tac-toe, including 15, in the Games column of the August 1979 issue of OMNI magazine.

To start, you should understand that going second effectively puts you on defense. In this version of tic-tac-toe, as with regular tic-tac-toe, the first player has roughly twice as many opportunities to win than the second player does.

When going second in 15, the first move is simple. If the other player takes the 5 for their first move, your response is to take any even number. In this post, I'll assume you always take the 4, but the strategies can be adapted to any even number. If the other player takes anything EXCEPT the 5 for their first move, then you must take the 5.

Where do we go from here? Obviously, that depends on the other player's second move. We'll start by assuming they took something other than the 5.

Other person goes first, first move is anything EXCEPT a 5: The first thing you need to watch out for is whether that second card, combined with their first, can make a total of 15. If so, you need to block that potential win by taking that card. For example, if they took the 4 first, you took a 5, then they took a 3, you have to realize that their 4 and 3 can be a win with an 8, so you need to take the 8.

After the other player makes their third move, you need to check for a threatened win and block that, as well. If this move doesn't threaten a win, take any odd numbered card (except 5, of course). At this point, you'll have the 5 and an odd numbered card (such as the 9). If they're smart, they'll see this and block your win (Seeing your 5 and 9, they take the Ace, for example). If they do this, all you can do is block and draw. If they miss it, you've got a win!

Most cases are going to wind up as the game above. If they take two even cards that require a 5 to complete a 15, which would be strange as you've taken the 5 on your first move, take any of the remaining odd cards. They'll either block, or give you the win unknowingly. From here, it's the same as above.

The best possible situation is when their second move gives them two odd cards, neither of which is a 5 (effectively, 2 edge squares, as you have the 5). If their 2 cards are such that your 5 would be required to make 3 in a row, take any remaining available odd card. They'll have to block you, and there will only be 4 even cards remaining.

Take a look at the 2nd card you drew, and think about what two even cards would make 15 with it. This is where it helps to be able to recall the whole board as taught in the first post. If your 2nd card is a 9, the even cards would be the 4 and 2. If it's the Ace, the even cards will be 8 and 6. 7 is in line with 6 and 2, and 3 is in line with 8 and 4.

Whatever two cards you come up with, take either one of those. The other person will have to block 1 of your possible wins, but there will still be 1 way available for you to win, so you take that:

Other person goes first, first move is a 5: As mentioned above, if they take a 5, you simply take any even number. From here, the most likely scenario is that you'll be blocking repeatedly and winding up in a draw, similar to what has already been described. Once again, you can take advantage of any mistakes to win, but otherwise, you'll draw.

To brush up on your tic-tac-toe strategy, check out:

wikiHow: How to Win at Tic Tac Toe
chessandpoker.com: Tic Tac Toe Strategy Guide
Buzzle: Tic-Tac-Toe Strategy Guide
learnplaywin.com: Tic Tac Toe: Strategy

If you're concerned about not being able to win every time, you can set up the challenge by saying, “If you win, I'll...(explain your losing wager here)..., but if I don't lose, I'll...(explain your winning wager here)....” That way, if you draw, you can remind people that you bet you wouldn't lose, and since there was a draw, you didn't lose!

I hope you enjoyed this mini-series of posts of 15. Try it out, and let me know what you think of it.

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