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?
(i) Yes. Find the number 1 in the sequence. Only finitely many even numbers appear before it, so some even number will appear after it. Just keep moving through the sequence from the number 1 until you first hit an even number. This term is even, and the one immediately before it will be odd.
(ii) No. Consider the usual ordering of the positive integers; now imagine the sequence split into blocks of six terms: 1-6, 7-12, 13-18 and so on. Then, in each block, swap the second and third terms. The new sequence looks something like this:
1, 3, 2, 4, 5, 6, 7, 9, 8, 10, 11, 12, …
Now, every multiple of three immediately follows an odd number, and we have our counterexample.