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# The Last Set

## Question

The description of game Set in Wikipedia contains the following statement:

If 26 Sets are drawn from a collection of 81 cards, the remaining 3 cards form a Set too.

Can you provide a nice and smooth proof for it?

Edited by witzar

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We can simplify the problem to a single dimension, so let's just look at the colors.

Any combination of complete sets will either contain the same number of cards of each color or the number of cards will differ by a multiple of 3. This is easy to see if you start from nothing and begin adding sets, you will either always be adding the equal number of cards of each color or you will be adding 3 cards of the same color.

Therefore, since we have 27 cards of each color to use, 26 complete sets will either have 26 cards of each color leaving 3 cards with one of each, or we'll have 27 cards of one color, 27 of another and 24 of the third leaving 3 cards of the third color in the pile.

The same exact reasoning can be applied to the shapes, numbers and shading.

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Well done, k-man.

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