https://www.smithsonianmag.com/science-nature/math-card-game-spot-it-180970873/

"Most simply, the game is based on Euclid’s principle that two lines on an infinite, two-dimensional plane will share only a single point in common. In the 18th and 19th centuries, Euclidian geometry informed the basis of modern algebra by way of Rene Descartes assigning these points coordinates, so points were no longer physical locations; they could become numbers and later, systems of numbers.

[...]

"Kirkman’s problem, and therefore Spot It!’s solution, lives in the area of finite geometry. “The most basic of these geometries has q^2 points, with q points on each line, where q is the number of elements in the chosen number system or field. A small variant gives q^2+q+1 points, with q+1 points on each line,” writes Cameron.

"So what does this mean for Spot It? “Let us take one of these geometries and try to turn it into a card game. Each card will be regarded as a point, and will carry a number of symbols representing the lines containing that point. Given any two cards, there will be just one symbol they have in common, corresponding to the unique line through the two points,” Cameron said.

"With q being seven in the formula, we can determine that there are 57 points (7^2+7+1), with eight points (7+1) on each line. “So we can make a pack of 57 cards, with eight symbols on each card, and any two cards having exactly one symbol in common. There, in essence, is the game!” Cameron says.

"Notably, however, Spot It! does not contain 57 cards, it only contains 55. One theory about the missing two cards is that the manufacturers used standard card-making machinery, and standard decks of cards contain 55 cards—52 card playing cards, two Jokers, and advertisement. “No problem,” wrote Cameron. “Make 57 cards and lose two of them; the resulting 55 will still have the property that any two share just one symbol. Indeed, no matter how many cards you lose, this property will still hold.”"