# Comments on On the game of Set

## Karl P (2004-08-03T10:46:32Z)

The six symbols seems simpler to me, because the idea of each symbol present occurring twice in a pset is simpler.

Playability can be improved by having each symbol in a fixed place on the card, which is blank if the symbol is absent from the card. One could have these places arranged in three rows of two with symbols in the same row having the same colour: E.g.

Diamond, Heart (red)
Circle, Star (blue)

Such an arrangement is similiar to Ruxor's, but has the advantage of the two symbol pset rule.
Diamonds

## Ruxor (2004-07-29T15:45:24Z)

Yes, it is quite obviously equivalent (just by grouping symbols by pairs). It's just that presenting it with just three slots for symbols would make the game much more "playable".

## Karl P (2004-07-29T13:00:08Z)

I think Ruxor's 63-card suggerstion is equivalent to the following.

Each of 63 cards has a different non-empty subset of 6 symbols (e.g. club, diamond, heart, spade, circle and star).
A 'pset' is three cards in which each symbol present occurs in exactly two of the cards.

Is it equivalent?
I think so. I count the same number of psets.

For any two symbols, there are 16 cards that have the first symbol and not the second symbol. So there are 32 cards that have either the first or second, not both the two symbols. This set of 32 cards has no psets, because if a set of 3 has none or two of one of the two symbols, it must have three or one of the other symbol.

## Ruxor (2004-07-28T14:59:42Z)

I think the most efficient way of searching for a set is as follows: first, find some parameter value which is taken by the smallest possible number of cards (ideally, only one: assume, for example, that there is only one green card). Now look for sets among the other values of that parameter (so, in our example, among red cards, then among blue cards; since the number of cards is then smaller, they are easier to spot). If none is found, one must look for sets containing the special card (the only green card): go through all red cards (say) and consider what the third element of the set would be and see whether it is there. This technique must reveal a set if there is one, but of course it takes quite a long time if there is no parameter value which is taken by a single card but by two or worse. (On the other hand, if some parameter value is taken by no card at all, then it one can quickly look for sets in the subsets defined by the two other parameter values - so, for example, among red cards and among blue cards.)

Now it seems that the most difficult parameter to handle is shading: while it is rather easy to mentally sort out all red cards, all green cards, all blue cards, all circles, all rectangles, all butterflies, all ones, all twos or all threes, it is quite difficult to mentally sort out cards according to shading.

## Karl P (2004-07-28T14:52:38Z)

I did for a short time think that each set correspondes to a line like in noughts and crosses in a 3x3x3x3 tessaract (4D hypercube), but then I realised that the ordering of each of the four dimensions matters for the tesseract but not for the cards. For 3, this is equivalent to wrapping the tesseract so that opposite cube hyperfaces are glued together.

It's more complicated for the extension of the game which has 256 cards and a 'set' of four cards.

## Karl P (2004-07-28T14:45:26Z)

I realise that finding a set in 12 cards is not so easy because every one of the 66 pairs of cards can belong to a set and checking them all can be exhausting. So I expect a skilled player will have a quicker way of finding a set.

## Ruxor (2004-07-28T08:33:22Z)

The authors of the paper I mention suggest a "projective" (rather than affine) version of "Set", namely five-dimensional projective space on the finite field with two elements, which I can describe briefly.

This deck would have 63 cards, each card being marked by one or none of three possible symbols in the top, middle and bottom parts (to avoid confusion, one would choose three different families of three symbols for the top, middle and bottom part, but this is inessential), and all combinations being possible except for the completely unmarked card (hence 4×4×4-1=63 cards); the rule for forming a pset (projective set) would be that in each of the three positions (top, middle and bottom) either all three cards bear different symbols (hence, all three different symbols for that position) or none is marked or two of them are marked with the same symbol and the third is unmarked. This gives a total of 651 psets among the 63 cards (each card belonging to 31 different psets). And the maximal number of cards without a pset is then 32 (this time, the result is quite easy to arrive at).

## Rrose (2004-07-28T07:52:31Z)

I had thought about Quarto too, but the underlying "space" is a bit different indeed. :-)

Also I was wondering: couldn't it be generalized to other spaces as well ? Well, perhaps the "small" version with only 9=3x3 cards would be interesting only for toddlers, and the 27=3x3x3 is a bit too simple too (however, the affine space structure perhaps is easier to see in that case)…

## ln (2004-07-27T21:45:54Z)

It reminds me another game called quarto.
Basically the subtle difference is… well maybe you wanna discover yourself.

## Ruxor (2004-07-27T17:28:10Z)

Not exactly "Morpion" (tic-tac-toe), because everything is modulo 3 whereas in tic-tac-toe there are boundaries (you can't randomly permute lines, or columns, for example).

But this suggests other possible games to be played with "Set" cards. For example: every player must put down a card from his hand, in turn, and the first one to form a set loses. Or something.

## Anonymous Coward #1200 (julio) (2004-07-27T16:57:07Z)

I was replying and I just saw Ruxor's reply… (Strange to think that you're typing in front of your computer exactly as I am ;))
I got an idea : You're playing "Morpion" in an hyper-cube, are'nt you ?

## Ruxor (2004-07-27T16:49:03Z)

AC#1199 → You're misparenthesizing this. I'm not saying “all of the parameters are different or all of the parameters are the same”, I'm saying “each one of the parameters is: either the different on all three cards or different on all three cards”. Of course, at least one of the four parameters has to be different on all three (else the three cards would be identical). But the three cards could have, for example, same count, same shape, different color and different shading; or perhaps: same count, different shape, same color and same shading. Or whatever.

## Anonymous Coward #1199 (2004-07-27T15:48:56Z)

"It is played with a deck of 81 cards, all unique … Now we say that three cards from the deck form a set when each of the four parameters (count, shape, color and shading)is either identical over the three cards …"
How is that possible ? I am probably missing something …

## Ruxor (2004-07-27T14:12:05Z)

Karl → Yes, it is 40 (this is easy to see because each of the 80 other points defines a line through the given point, and each such line contains two of the points (other than the given one). But I had already written that in the entry. ;-)

## Karl P (2004-07-27T13:55:11Z)

If you define a point as any single card and a line as any set, then we have a finite geometry.

Each pair of cards belongs to a unique set and hence each pair of distinct points has a unique line passing though it.

How many lines can pass through a single point?
I think it is 40.
This is equivalent to asking how many sets does each card belong?

## phi (2004-07-27T09:39:03Z)

Certainly what is patented is rather the representing the dimensions as number, form, colour, frame. Not even (?) the coding of coordinates into one subitized object. thus you could use for that products of powers of prime numbers. Bonus: you could avoid using GTK! btw it would be interesting to compare relative performance in various such versions of the game, like one with faces, or semantical role…

## Ska (2004-07-27T07:35:03Z)

Found the set. After at least 4 ou 5 minutes of hard searching :) That's pretty tough, when you're not used to it, but I guess the mind can quickly become accustomed to looking for patterns and associations.

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