Chapter 37: Harary's Generalized Ticktacktoe

This chapter gives a different approach to play one of the most well known games in existence: Ticktacktoe. This generalized way that the chapter details about was first created by Frank Harary. The concept of ticktacktoe was simple: two people play on a 3x3 grid taking turns placing their marks on the grid. The objective was to line up three of your symbols (X or O) either vertically, horizontally, or diagonally. Older variations date back to the ancient times where two players are given three counters and would move them around the grid until they matched their three counters in a row. But Harary devised a new approach to the game.

He wanted to find out two things: What is the smallest in which a player can force a win? and "In how few moves can the first player win?". It introduces the concept of cell polyominoes. An example of a polyomino is a domino, where it has two squares together. That is classified as two cells. This variation of the game can be played on any polyomino, whether it is one cell, three cells, four cells etc. The shape varies depending on how many cells there are. To make it simple to understand, think of it like Tetris where you can change the shape of the blocks. That applies to this game in that you can use a different board for each game to play by changing the shape.

In order to know which board you are using, Harary coined the term "animal" cells where each board is given a name. Now the first task is to determine the "animal's" board number which is the length of the side of the smallest square in which the first player can, by playing the best strategy, force a win. If there is a number, the animal is called a "winner" and if not, then it is called a "loser". If two people play a board that is a "loser" then the second player can always force a draw but can never win. Once you know the board number, then you find the fewest moves it takes the first player to win. On a one cell board its one move and on a two cell board its always two moves. For three cell boards and up, it gets tricky to analyze it. The chapter itself illustrates the boards with the board number and moves for each as well as the "losers" of the boards. As in regular Ticktacktoe, you play marking the cells with square matrixes with crosses and noughts (X's and O's). It gets more complicated in determining which ones are winners and which ones are losers and the chapter goes into further details on that but what I wanted to show you is the basic setup in how to play this generalized way of Ticktacktoe.

To me, this chapter at first glance before reading it, I thought it would be something simple to understand because it was ticktacktoe but after reading it, I did come across some parts that were a bit confusing. It was interesting in how he developed a different way to play the game, but trying to understand it took me a while and I still have questions about it. I wanted to know when playing on a board how can you tell that the animal is a "loser"? What surprised me is how this simple game can be transformed into a strategy to be able to win on different boards and to know when you are unable to. I'm sure everyone has played Ticktacktoe before but this new approach can be difficult to understand at first, but after rereading it a few times you can get a better understanding of it.