How do siblings split a cake so no one envies anyone else? Solved for two. Open for many.
Play It
Characterization
Fair division is the problem of dividing a resource among multiple parties so that each party considers their share to be fair. For two players, the ancient solution — "you cut, I choose" — guarantees envy-freeness: neither party prefers the other's piece. For three players, the Selfridge–Conway protocol (discovered independently by John Selfridge and John Horton Conway around 1960) achieves an envy-free division in a bounded number of steps. For four or more players, the problem explodes. Haris Aziz and Simon Mackenzie proved in 2016 that an envy-free division exists for any number of players and found a protocol to achieve it — but the protocol requires a number of steps that grows as a tower of powers of n, rendering it practically useless. Whether a bounded envy-free protocol exists for four or more players — one whose step count is a polynomial or even a fixed tower of exponentials — remains one of the central open problems in mathematical economics. The field traces to Hugo Steinhaus, Stefan Banach, and Bronisław Knaster, who studied fair-division procedures in 1940s Poland; it was formalised by Duncan Foley's concept of envy-freeness (1967) and has since expanded to encompass rent division, the allocation of indivisible goods, and the fair distribution of public resources. The Academy hosts Fair Division in the World School because it is the oldest game siblings play — and the mathematics of making it fair are still being written.
Lineage
Hugo Steinhaus, "The Problem of Fair Division," Econometrica 16(1), 1948. The Selfridge–Conway protocol, c. 1960 (unpublished; described in Jack Robertson and William Webb, Cake-Cutting Algorithms, 1998). Steven Brams and Alan Taylor, Fair Division: From Cake-Cutting to Dispute Resolution (Cambridge University Press, 1996). Haris Aziz and Simon Mackenzie, "A Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents," FOCS, 2016. Ariel Procaccia, "Cake Cutting: Not Just Child's Play," Communications of the ACM 56(7), 2013.
Quests
The Magus
Implement or simulate an envy-free division protocol for four or more agents — Aziz-Mackenzie, or a protocol of your own design. You may use code, physical objects, or pencil-and-paper simulation. Record the number of steps required and identify the bottleneck. Compare the practical step count with the theoretical bound and reflect on what the gap reveals about the distance between existence proofs and usable algorithms.
The Adventurer
Divide a real, physical resource — a cake, a pizza, a bag of items, or a shared rent — among at least three people using a formal fair-division protocol. You may use cut-and-choose for two, the Selfridge-Conway protocol for three, or the last-diminisher method for more. After the division, ask each participant whether they envy any other share. Record the protocol, the participants, and the outcome.
The Sage
Trace the history of envy-free division from Steinhaus, Banach, and Knaster in 1940s Poland through Selfridge and Conway to the Aziz-Mackenzie breakthrough of 2016. Explain what "envy-free" means, why it is harder for n players than for two, and what remains open. Cite at least three primary sources.