Friday, February 14, 2014

Educating through the Game Fifteen

Nine cards are marked with digits 1 through 9 and placed face-up on a table between two players. The players take turn picking one card at a time. The winner is the first player to get the sum of exactly 15 from among any three of his or her cards.
  This game was proposed by our professor Marteen during our first meeting, as an early morning mind-exercise.  It was fun and quite a work out for brain, but he made it harder for us by demanding what he called as “the winning strategy.” Basically, he asked us whether or not there is a strategy that can ensure us to always win this game, and if such strategy exists, to find it. We worked in group of 5 people, and none of us could come up with that kind of strategy. I did note the following cases though:

1.    We made a list of all possible combinations of numbers that sum up as 15, in which 5 has the most frequent appearance. So we used that as strategy and always started with 5, but it did not ensure us to always win the game.

2.   There was this check-mate like situation in which whatever cards your opponent pick, you will always win. For example, I start first and I pick 8. You pick 4. Then I choose 6. You definitely have to pick 1 to prevent me from winning with 8+6+1=15. Then I choose 2. Now I will always win since I have 2 opportunities: 8+2+5=15 and 6+2+7=15, while you only have 4 and 1. The problem is to find out how to recreate such situation. 

3.    When our opponent picked a certain card, it was almost automatic for us to do quick calculation and pick a card that would prevent them for getting the sum of 15, therefore blocking them. It was only when Marteen brought it to our attention that we realized that it was not us being smart, but our opponent controlling us. 

It turned out that there is a winning strategy, only slightly went out of what it is defined to be.


First thing though, I am not digressing. This actually has something to do with what I was talking about. 

Magic square is an arrangement of numbers (usually integers) in a square grid, where the numbers in each row and in each column, and the numbers in the forward and backward main diagonals, all add up to the same number.  The size of a magic square is denoted by its number of columns and rows; magic square with 3 rows and 3 columns is said to be of size 3x3. There are 9 grids and the numbers in each row and in each column as well as the numbers in the diagonals, all add up to 15.

It is easy to see that playing this game here is like playing tic-tac-toe on a magic square.  Now we just have to ensure that we can always win in a tic-tac-toe.


Winning strategy in general means just like what Marteen said. But in tic-tac-toe, there are no sure-win strategies; a winning strategy is simply one that will maximize the chance of a player winning (Yeo, 2012). So you will either end up a winner or in a draw. Some winning strategies for tic-tac-toe follow. 

Imagine a tic-tac-toe game where player 1 uses X and player 2 uses O.
Case 1: player 1 starts in the center. The game will always end in a draw, assuming each player makes optimal moves.

Case 2: player 1 put the first x in a corner. If player 2 does not put the first O in the center, they will lose. For example, if player 2 put the first O in the corner opposite of the first X, player 1 can place second X in any of the remaining corners. There are two choices right? Player 2 will be forced to put an O in between two Xs to prevent player 1 from winning. Player 1 can win by placing an X in the last corner.

It is easy to figure out how to win if player 2 put the first O in any for the edge squares.
Case 3: player 1 put the first x in a corner. If player 2 put the first O in the center, there is still a way to win, but only if player 1 puts the second X at the corner opposite the first X and player 2 puts the second O in any of the remaining corners.  Then player 1 can win by placing the third X in the last corner.

Case 4: player 1 put the first X in an edge-square. If this is the case, then the game is wide open. Either player can win, although the game will most often end in a draw. Below are example of this case.

Therefore, the winning strategy for player 1 is to place the first X at the corner. For player 2, pray that player 1 will not start at the corner and if they do, put the first O in the center. 

Back to game we were talking about before, it is clear that any player can win the game by following this tic-tac-toe strategy on a magic square. That is why we never won when we started with 5, because 5 is located at the center. Although it will take extra effort since the players must visualize the magic square; otherwise, opponents will be aware of the strategy.


Yeo (2012) introduced this game as Fifteen and stated that it can help students learn mathematics and develop heuristic skills such as examining all possible scenarios and systematic listing, spatial visualization, and thinking skills, including predicting, conjecturing, generalizing and checking. The question is, to children of what age should we introduce this game? 

This game is exciting because it requires the players to work his best and develop a strategy to win the game. But not every student will employ that particular mental action against the game. How children respond to games is apparently linked to their developmental levels. Kamii and Nagahiro (2008) summarize these developmental levels as follows. 

- Level 0: the child does not try to win. They are focused on doing what they are supposed to do, here by getting a sum of 15.
- Level 1: the child tries to win. They know they are supposed to get a sum of 15 and do it before the other player does.
- Level 2:  the child tries to block the opponent.
- Level 3: the child is at the most advanced level; they compare the consequences of winning the game now with blocking the opponent or foresee that the opponent has two possible ways of winning.

Most 4-year-old are at level 0, while 6 and 7-year-olds are generally at the highest level. Therefore, it will best to introduce this game to students in 6th grade or higher. 

Aside from all those technical matters, we teachers know that most students love to play games and from the beginning I already though that this game will be a fun addition in the classroom. Fifteen is an interesting and thought-provoking game that helps students learn mathematics at the same time.

Yeo, J.B.W. 2012. "Fifteen: Combining Magic Squares and Tic Tac Toe". Mathematics Teacher, vol 106, no. 1. USA: NCTM.

Kamii, et. al. 2008. "The Educational Value of Tic-Tac-Toe for Four to Six-Year-Olds". Teaching Children Mathematics. USA: NCTM.

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