Equivalence between games

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## Equivalence between games

In fact, both these games are equivalent (the technical word, taken from maths, is isomorphic, ``of the same form'') to each other, and to noughts-and-crosses. See below for how they correspond. This should begin to show you how apparently wildly different ways of representing and manipulating information can be, at a deeper level, the same.

Of course, we perceive the isomorphism because of the properties we're interested in. The 15 game has no analogue for the fact that one of the words in the word game is a different length. But for the properties that we consider essential to the game, they're isomorphic. This is not just a vague notion of similarity: we can define it exactly. Part, not all, of the meaning of isomorphism is that we can put the components of one game into one-one correspondance with those in the others:

##### ``` Noughts-and-crosses 15 game Word game Top left 1 HOT Top mid 2 TANK Top right 3 TIED Mid left 4 FORM Mid mid 5 HEAR Mid right 6 BRIM Bot left 7 WOES Bot mid 8 WASP Bot right 9 SHIP```

But there's more to isomorphism than that. After all, we can put the board positions in one-to-one correspondance with the planets, but that doesn't mean the Sun spends all its time playing Alpha Centauri at noughts-and-crosses. We can also map properties and actions to one another:

##### ``` Noughts-and-crosses 15 game Word game In same row Add up to 15 Share a letter Put a mark Add a number Initial a word Cross/ Your pile/ Your initials/ zero oppt's pile oppt's/initials```

That's the second part of isomorphism. It too is not enough. What's important is that objects, their properties, and the actions interact in the same way in different games. We can express this by trying to make rigorous the idea that if you start off by playing one game, and then change to another, nothing has really changed.

To give an example: you start a game of noughts-and-crosses by putting a cross in the top-left cell. Your opponent makes a move. Let's say he puts a zero in the bottom-left. Now you put a cross in the centre of the board. Then your opponent puts a zero in the top-right square. And finally you put a cross in the bottom right square. You have a row! Now, you translate this final state to the 15 game:

##### ``` Noughts-and-crosses 15 game You: top-left, centre, bottom-right You: 1, 5, 9 Oppt:bottom-left, top-right Oppt:7, 3 Row top-left to bottom-right Sum of 1+5+9 = 15```

What isomorphism means is that, if you'd played the 15 game all along, making moves that correspond to your noughts-and-crosses moves, you'd have got to the same state as if you'd played the noughts-and-crosses moves and then converted back to the 15 game.

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Jocelyn Ireson-Paine
Wed Feb 14 23:46:11 GMT 1996