Here’s a game with the sort of simplicity that may lead to its becoming a craze at some point. The rules are easy and the apparatus simple: nine playing cards will do fine.
In “Triptych” you will need nine distinctive pieces: cards ace to nine in one card suit will be fine. The object of this two-handed game is to pick up a single card in each turn until you have three cards that total to fifteen. (You may of course end up with four or even five cards, but you get to identify any three in your hand that total exactly fifteen). The game only lasts a short time: the longest game finishes when the first person has taken the last card; that will be his fifth turn. If by then neither person identifies a winning triplet, the game is drawn.
If you become expert at Triptych, you will easily win most of the time. If however two experts play the best they can, the result will always be a draw. The fun lies in that in-between time, when you are an expert and you can still find opponents who are not. You stand to have a winning advantage for a fair time (unless you teach somebody the winning strategy) because it seems quite complex to find it out experimentally.
Let’s simulate a few games of Triptych: I shall draw the SIX; your first draw is, shall we say, the EIGHT. My second choice will be the TWO; you, canny player that you are, see the danger: if I pick the SEVEN for my third card, I will have won – so you make sure I don’t by securing the SEVEN for yourself. (You now have fifteen in your hand, but to win it must be the count on three cards, not two). I choose FOUR as my next card, leaving you with SEVEN and EIGHT, and me with TWO, FOUR and SIX. You want to stop me getting a triptych, but either FIVE or NINE will do for me (4-5-6 or 2-4-9). You can block the choice of either, but not both at the same time – so I can win on the next move, whatever you do.
You start the second game: say you choose SEVEN: my first choice may be TWO. You then choose THREE, which forces me (do you see why?) to select FIVE. In turn you must choose EIGHT to prevent my winning, and I choose FOUR (to prevent you getting 3-4-8 and winning). This puts you in a quandary, because I can win in two different ways with the next turn: SIX (for 4-5-6) or NINE (for 2-3-9), and you can only stop one or the other. Again I can win on the next move.
If both players are expert and play their best, the game will always end in a draw: but unless your opponent knows the strategy well, you are likely to win by waiting for him to make a poor choice.
But what could that strategy possibly be? It does NOT depend on the first move: if you go first, you can progress toward a win (or at least a draw) with ANY first choice. Certain choices are stronger than others because there are fewer counters to your first move. For example, if your opening choice is FOUR, your opponent had better choose FIVE or he will lose to your expertise! Of the nine possible choices, this applies to four of them (that there is only one proper reply). In the case of the other five possible “first choices” exactly half the remaining numbers are good replies.
Analysing the Game
Let’s start our analysis of the game by compiling a list of all possible winning triptychs: start with ONE: the only remaining pair that sums to 14 is [FIVE and NINE]. If a triptych has TWO as its lowest component, the other pair must be [FIVE and EIGHT], or [SIX and SEVEN]. When the lowest card is THREE, the remaining pair must add to twelve, so we have [FOUR and EIGHT] or [FIVE and SEVEN]. Starting with FOUR, the remaining pair can only be [FIVE and SIX]. So there are eight possible wins, and FIVE is the card appearing most often among the winning triptychs. Setting all nine cards into a matrix we have a simple pattern (figure i) where all rows, columns and diagonals represent a triptych involving FIVE. Many of these triptychs actually total fifteen already, and with only slight adjustment we can represent all the winning triptychs: notice that missing winners involve the numbers in peculiar but regular order: ONE-SIX-EIGHT involves one corner of the matrix and the middle of the two enclosing sides. THREE-FOUR-EIGHT involves the same pattern in a different corner. So also for the other two corners: TWO-SIX-SEVEN and TWO-FOUR-NINE. We can do a little re-shaping and get this matrix very neat indeed.
First we must tip it on one corner, moving it 45 degrees to the left. Now convert this back to a square and we have half-completed the job (figure ii).
Now interchange the outside cells in the middle of each side with their opposite (figure iii). These four numbers (ONE-THREE-SEVEN-NINE) used to be corner cells in the original form of the matrix. Now we have a square matrix with TWO-SEVEN-SIX as its top row.
In this matrix all the rows and columns, plus the two diagonals, each sum to fifteen, and these represent all the winning triptychs in our game. Say.. this matrix looks familiar! And this property of all rows and columns and diagonals adding to the same value – where have we heard that before? No doubt, somewhere in your memory you will have record of this: is it not a so-called “magic square”? That sure was unexpected!
Well, let’s press on: if we are to program this game we could now substitute our cards with this matrix, a three-by-three magic square, and the object now becomes to select (by turns) until you complete a triptych from the matrix – to make a row, column or diagonal.
A Eureka! Moment
I don’t know about you, but when I run into fascinating connections like this I have a “eureka!” moment. Archimedes, who lived in the third century BC, solved a pressing problem with such a connection: he was trying to establish some way of measuring the mass of an ornate metal wreath so that it could be established whether or not it was composed of pure gold. He dare not melt down the wreath. As he settled in his bath, he got to thinking about the water his own body displaced. He is reported as having dashed naked through the streets around the public baths, shouting “Eureka!” (“I have found it!”). As a registered nitpicker, I must observe that this is wrong: he must have shouted “Heureka!” (as related to the word “heuristic”). Or maybe Sicilians of the third century BC dropped their Hs.
Maybe you, too, had a “eureka!” moment when we found the connection between the three-by-three magic square and our game “Triptych”. If so, I will wait for a moment while you calm down so that you can best savour the surprise to follow. You see, playing the “Triptych” game with the magic square is equivalent to playing naughts and crosses – you win when you get three in a row. Eureka again! So we have a new game which combines a magic square with the old childhood game of naughts and crosses.
Not everybody is familiar with the best strategies for naughts and crosses. If you are to go first, you can choose any cell in the three-by-three grid, and be assured that you will either win or draw.. Similarly if you are to go second, you can guarantee that you will at least draw.
You have an extra advantage when you play Triptych: your opponent is not very likely to realise he is playing naughts-and-crosses with you. For example, if you start by taking the NINE, your opponent is doomed unless he chooses ONE, TWO, FOUR or FIVE. Even if he stumbles upon one of these choices, he is hardly likely to be able to anticipate your overall strategy.
It will take a little practice to “see” the magic square in your mind so that you can mentally play naughts-and-crosses while your opponent struggles with Triptych, but it will be worth it.