Let’s play a game. First, you need an opponent. Next, you each take turns selecting a number from the spinner below. The first person to collect three numbers that add up to 12 wins.
Here is an example game:
Person A chooses 4. Person B chooses 7. Person A chooses 6. Person B chooses 3. Person A chooses 2 and wins, since 4 + 6 + 2 = 12
Give the game a try yourself to get the feel of it. You should realize that it is a fairly challenging game. Often, you get a couple number at the start, but are then forced to ‘block’ your opponent. Personally, a large number of my games ended in a draw because no one was able to get the right numbers to sum to 12.
As in any game, we want to develop a winning strategy. To start, we could write down all the sets of three numbers that add up to 12:
- 0 + 4 + 8
- 0 + 5 + 7
- 1 + 3 + 8
- 1 + 4 + 7
- 1 + 5 + 6
- 2 + 3 + 7
- 2 + 4 + 6
- 3 + 4 + 5
Next, we notice that the number 4 shows up in four of the sums, while the other numbers only show up two or three times. Hence, choosing 4 first would be a reasonable move. From there, we could try to keep our list of sums handy and use it to our wit our opponent. However, this strategy is cumbersome at best, and does not seem very intuitive. Perhaps there is a better way to understand the game…
The study of mathematics involves discovering structure and pattern. Frequently, a mathematician may find two objects that appear to have a similar structure or form. In our case, we wish to find a second object that has a similar structure to our spinner game. Since we need to add up three of the nine total numbers, we could consider a 3 X 3 grid:
Now we need to place our nine numbers somewhere in the grid. Since we want winning combinations, we should arrange all nine numbers in the grid so that every row, column, and diagonal adds up to 12 (our 8 different sums we listed above). Since the middle square is involved in four different sums (down, across, left diagonal, right diagonal) we place our most common number, the number 4, in the middle:
After some guessing and checking, we find the following placements of numbers appear to work for the diagonal sums:
We fill in the remaining spaces with our remaining numbers, ensuring the sum of each column and row is always 12. Voila:
For those of you who are interested, this type of construction has a special name, a magic square.
Back to our game. Now that we have our spinner in a different form, we can try using the grid to help us determine some strategy. Let’s represent Player A’s choice with an X and Player B’s choice with an O. Here is the first game from the beginning of the post:
Hmm, I think I have seen this game before… When a mathematician finds two things that have the same structure or form, they use a special word. That word is Isomorphic, which originates from the Greek iso, meaning “equal,” and morphic, meaning “shape” or “form.”
Due to our investigation above, we can confidently state that the spinner game and Tic-tac-toe are Isomorphic. This revelation was surprising to me. The spinner game seemed complicated and required a lot of thinking. Tic-tac-toe, on the other hand, is a child’s game that I mastered a long time ago. The fact that these two games are isomorphic shows just how useful looking at a problem from a different perspective can be.