ABSTRACT
A bird in the hand is worth two in the bush. Do you stick with a sure thing or risk it all and switch, hoping for something of greater value? It’s a common dilemma inspiring the US television game show Let’s Make a Deal. As mathematical recreations, envelope problems have been in circulation since the mid 1950s when Belgian mathematician Maurice Kraitchik published “The Paradox of the Neckties” in Mathematical Recreations, the English-language edition of La Mathématique des Jeux [Kraitchik 53, pp. 133-134]:
Kraitchik gives a short discussion but offers no satisfactory resolution. Martin Gardner, perhaps the world’s best-known author of mathematical recreations, further popularizes the problem as “The Wallet Game” in Aha, Gotcha: Paradoxes to Puzzle and Delight, using wallets rather than neckties in the formulation [Gardner 82, p. 106]. Whether it’s neckties, wallets, or cold hard cash, the idea is the same. Stick with what you’ve got or exchange for another item of unknown value. We begin with the classic envelope problem, offer a solution, and then move on to variations with highly unexpected consequences.
