Friday, November 8, 2013

The Math Behind Our Game

Dear Reader,

By popular demand, here's an explanation of the math behind our Squirrel Match card game!

On the right is an example of what our typical playing cards look like. The tricky thing is that for every two cards there is exactly one squirrel in common. How do we achieve that? ... Hold onto your hats, to understand we're going to venture to infinity and back :)

To start, here's some basics: As everyone knows, two lines on a flat surface always intersect at exactly one point ...

Except when they don't!

But anyone who's ever looked down a set of train tracks knows that parallel lines really do intersect, if only you could walk all the way out to infinity.

Mathematicians like all things equal if and when they kind of should be, so here's tricky step number 1:

To each line we add a point "out at infinity" that corresponds to the slope of that line. We've now made what mathematicians call "projective space" and parallel lines all intersect at their corresponding slope point out at infinity!

Soooo ... how does this relate to our squirrels? Each card correspond to the points on one a line. ... But there's only 8 squirrels on a card! So here's tricky step number 2:

Imagine that the whole plane is represented by a square, sort of like one of those old computer games - if you go off the right, that means you just come back on the left. Off the bottom, you come back on the top. So the line in the square below is one line.

This is all well and good, but there's still infinitely many points on that line I hear you say. Well yes, sure, but suppose we only worry about the whole number points represented by the dots. I've chosen a square with 7 dots across.

Now putting this all together is the last tricky part:

Because 7 is prime, a line that goes through two points in this square will go through seven points, add in the point at infinity corresponding to the slope of this line and you have the eight squirrels on a card! And the kicker!! Every pair of lines meet at exactly one point, ie every pair of cards have exactly one squirrel in common.

Now who said math isn't cool?!

Chin chin
Ruthless Whims

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