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 …

Baby Poker is a two-player dice game that fits on a napkin and still manages to be a real bluffing game. Each player rolls a single die concealed in a cup. There is one round of betting. That is it. And yet the optimal strategy involves mixed frequencies, a value range, a bluffing range, and a call/fold cutoff. It is the smallest game I know of that has all the moving parts of full poker.

Our working notes: Baby Poker.

The Problem

This one comes from the FiveThirtyEight Riddler column (now defunct, but happily still on the Wayback Machine …

Project Euler problem 227 - "The Chase" - is a nice example of a problem where the brute force approach is completely reasonable to reach for first, completely infeasible to actually finish, and directly points you at the right mathematical reformulation.

This is what we ended up with. Our full working notes are on the wiki at Project Euler/227.

The Problem

An even number of players sit around a table. Two players sitting directly opposite each other each start with a die. Each round, both players roll:

• Roll a 1 → pass the die to your left neighbor
• Roll a 6 → pass …

Here is a nice little problem from Paul Nahin's Will You Be Alive 10 Years From Now?. It is a problem from the 1654 correspondence between Antoine Gombaud (the Chevalier de Méré) and Blaise Pascal - one of the founding episodes of probability theory, dressed up as a gambling puzzle.

(This one lives on our wiki as FMM21, part of our Friday Morning Math Problem series.)

The Two Games

Game 1. You toss a single die \(N_1\) times. What should \(N_1\) be to make the probability of seeing at least one 6 greater than \(\tfrac{1}{2}\)?

Game 2. You toss …