Using your Head is Permitted

December 2014 riddle

This month's riddle is a follow-up from last month and the month before it. It was suggested by Radu-Alexandru Todor and Lorenzo Gianferrari Pini. (Thanks!)

Spoiler alert: The following description contains hints regarding the solutions of the October and November riddles. If you still mean to tackle those other riddles, you may want to hold on before you continue to this one.

This month's riddle comes in two parts. Answer either or both parts. Separate solver lists will be kept for each part.

Part 1:

Let a "casino" be a point, p, on the interval [0,1], such that for any integer n≥2 and for any open neighbourhood N of p there is a point pn in N such that the n-sided dice can be simulated by a biased coin with probability pn in a finite number of throws.

Characterise exactly (with proof) the subset of [0,1] that is the set of casinos.

Part 2:

Let a "rational casino" be a point (not necessarily rational-valued), (p,q), in the set [0,1]2, such that for any integer n≥2 and any open neighbourhood N of (p,q), there is a rational-valued point (pn,qn) in N, such that a pair of biased coins with probabilities pn and qn, respectively, can simulate an n-sided dice in a finite number of throws.

Find (with proof) one such rational casino.

An asterisk will be given to solvers able to characterise exactly the subset of [0,1]2 that is the set of rational casinos.

List of solvers:

Part 1:

Li Li (16 December 10:28)

Part 2:

Harald Bögeholz (10 December 23:27)
Dan Dima (12 December 08:58)
Li Li (*) (16 December 10:28)

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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