I really enjoyed playing the game, and it got me thinking about what it would take to create a deck of my own. I also noticed that in all the cards provided, every card has the same number of images. The mathematician in me thought it would be nice to impose some additional symmetry: each image should occur on the same number of cards and any two images should occur on some card together. This led me to define the following variables

- $N=\text{#of cards;}$
- $P=\text{# of images;}$
- $r=\text{# of times each image occurs in the entire deck;}$
- $z=\text{# of images on each card.}$

Counting the total number of images (with the repeated occurrences counted) on all cards two ways (images per card, or cards per image) yields the equation $Pr=Nz.$

We use the symmetries assumed to count the number of images. Let $x$ be a particular image. For any card $S$, that $x$ appears on, there will be $z-1$ other images on $S$. Since there are $r$ cards that $x$ appears on, and any two images appear on exactly one card together, there must be $P=r(z-1)+1$ images.

Next we count the number of cards. Let $S$ be a particular card. For any image $x$ on card $S$, there will be $r-1$ other cards with $x$ on them. Since there are $z$ images on $S$, and any two cards only intersect at one point, there must be $N=z(r-1)+1$ cards.

Armed with our three equations

- $Pr=Nz$
- $P=r(z-1)+1$
- $N=z(r-1)+1$

If you've been exposed to projective geometry, this should begin to look very familiar. Viewed in a certain light, Spot It! is a carefully-crafted recreation of a finite projective plane where the images play the role of points and the cards play the role of lines. In fact, if you substitute "image" for "point" and "card" for "line" in the first two projective geometry axioms below, you'll get two of the assumptions above.

- Any two points lie on a unique line.
- Any two lines meet at a unique point.

If $r=z$ is one more than a prime power, then there is a method to create a plane – read "deck" – that we could play Spot It! with. I crunched the numbers, and when there are eight images per card, you can make a deck of 57 cards - two more than are in the Spot It! game. Fortunately, only the first axiom is important to play the game, so if you happen to lose a couple more cards, you can still have a fun time!

If you want to check Spot It! out for yourself you can find it on Blue Orange.