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Combinatorics · Graph walks · Euler circuits · Parity · Degree counting · Extremal constructions

Problem 6, 2023

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Level 1 of the computer game Zakladnica takes place in an underground treasury built from \(13\) octagonal and \(12\) square rooms, arranged as in the figure. The only way into the treasury, and the only way out of it, is in the central room; each of the remaining \(24\) rooms contains one gold coin.

Any two rooms that share a wall are joined by a passage, and a passage closes for good the moment the player steps through it. Whenever the player leaves a room in which he has picked up a coin, a new gold coin appears in that room. A player who ends up trapped inside the treasury loses every coin he has collected and the game restarts.

What is the largest number of gold coins the player can bring out of the treasury?

The treasury from above. Every gap in a wall is a passage; the staircase in the middle is the entrance and the exit.

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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2023, 1. letnik, category A, problem B3. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source

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