Using your Head is Permitted

August 2014 riddle

This month's question was sent in by Jan Fricke. (Thanks, Jan! More credits will be given on the solution page.)

Consider the arithmetic sequence x, 2x, 3x,..., up to n×x, for an irrational x.

If you round the numbers to the nearest integer, some of them will have been rounded up and some rounded down. Let d be the number of those rounded down and u be the number of those rounded up.

For which x values is the difference |d-u| bounded as n approaches infinity, and for which is it unbounded?

As usual: prove your answer.

List of solvers:

Yury Volvovskiy (13 August 23:24)
Yuping Luo (17 August 22:41)
Radu-Alexandru Todor (20 August 11:08)

Elegant and original solutions can be submitted to the puzzlemaster at Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.

The solution will be published at the end of the month.


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