The Largest Magic Deck

A deck contains N distinct cards, labelled 1 to N, with no suits or other features. An audience member picks any five cards. The assistant, who sees them, secretly chooses one to hide, then lays the other four face up in a row from left to right. The magician, who has not seen the five cards, looks at the row and names the hidden card. The assistant and magician agree on a deterministic scheme in advance, and the only information passed is which four cards are shown and in what order (no hidden signals of any other kind). What is the largest N for which such a scheme exists?

• combinations
• graph theory

Source: Based on Fitch Cheney's five-card trick (attributed to W. Fitch Cheney Jr.), popularised in recreational mathematics; the deck-size bound is a standard exercise. Statement written for AxiomIQ.