The Integer Shuffle

A PuzzleCritic Original:

A sequence contains every positive integer exactly once, and no other terms. Must there exist, somewhere in the sequence:

(i) an odd number immediately followed by an even number;

(ii) a multiple of two immediately followed by a multiple of three?

Solution

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The Coin Collection

A PuzzleCritic Original:

The Museum of Mathematical Mysteries houses a peculiar collection of coins, each shaped like a polygon, with the two largest proper factors of n inscribed on either face, where n is the number of edges the given coin possesses. For example, there is a hexagonal coin on which the numbers 2 and 3 are inscribed. The museum’s curator examines one of the coins and sees the number 15 inscribed on one face. Determine all numbers that might appear on the other face.

Solution