The fictional Acme Insurance Corp employs a team of recreational mathematicians, people who study the fun parts of math (game theory, combinatorics, and cryptography), and puts them to work as actuaries. They take the most entertaining problems in mathematics and turn them into dry, boring actuarial tables.

This week, two fast food chains want Acme to insure the prizes in their collect-the-set contests. Burger Slop and Chicken Star will each print ten million game pieces, and almost none of them will win anything big. When a customer completes a set, Acme pays the prize. Each chain is choosing among three …

Enumeration Solitaire is a one-player card game with a two-sentence rule set and a genuinely nontrivial answer. You shuffle a deck, deal cards face up, and count "ace, two, three, ..., king" over and over as you go. If the rank you speak ever matches the rank of the card you just dealt, you lose. Get through all 52 cards without a match and you win.

It is a four-copy generalization of the classical probleme des rencontres (the derangement problem), and the solution is one inclusion-exclusion argument compressed into one generating function.

Our working notes: Enumeration Solitaire.

The Problem

This one …