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An open-ended question is a question that cannot be answered with a "yes" or "no" response, or with a static response. Open-ended questions are phrased as a statement which requires a longer answer. They can be compared to closed-ended questions which demand a “yes”/“no” or short answer. [1]
It turns out that these proportions hold in general unless the first envelope contains 1. Denote by a the amount we imagine finding in Envelope A, if we were to open that envelope, and suppose that a = 2 n for some n ≥ 1. In that case the other envelope contains a/2 with probability 3/5 and 2a with probability 2/5.
In 2, it is stated that either the first envelope contains A and the second ½A, or the first envelope contains A and the second contains 2A. Therefore, if we open both envelopes, either we find A + ½A = 1½A, or we find A + 2A = 3A. In either case, the total is X, so 1½A = X and 3A = X. Why "and" and not "or"? --NeoUrfahraner 13:17, 13 July ...
Prisoner 2 opens drawers 2, 4, and 8 in this order. In the last drawer they find their own number, 2. Prisoner 3 opens drawers 3 and 6, where they find their own number. Prisoner 4 opens drawers 4, 8, and 2, where they find their own number. This is the same cycle that was encountered by prisoner 2 and will be encountered by prisoner 8.
[2] [3] An important open mathematics problem solved in the early 21st century is the Poincaré conjecture. Open problems exist in all scientific fields. For example, one of the most important open problems in biochemistry is the protein structure prediction problem [ 4 ] [ 5 ] – how to predict a protein 's structure from its sequence.
Leo Gerville-Reache, Why do we change whatever amount we found in the first envelope: the Wikipedia two envelopes problem commented., arXiv:1402.3311 Panagiotis Tsikogiannopoulos, Variations on the Two Envelopes Problem , Hellenic Mathematical Society, Mathematical Review 77-78: 3-25.
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Can 3SUM be solved in strongly sub-quadratic time, that is, in time O(n 2−ϵ) for some ϵ>0? Can the edit distance between two strings of length n be computed in strongly sub-quadratic time? (This is only possible if the strong exponential time hypothesis is false.) Can X + Y sorting be done in o(n 2 log n) time?
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