なお、Peter Winkler氏は時枝記事にも登場した人だ>>86 Sergui Hart氏は、>>263のPUZZLESのページで、”Choice Games”のＰＤＦを投稿した人だ 0510１３２人目の素数さん2016/09/10(土) 14:10:21.11ID:1RTeFNgE いやお前もう黙ってろよ 0511１３２人目の素数さん2016/09/10(土) 14:10:54.54ID:q7Skbg74>>507 つづき 英 stackexchange http://math.stackexchange.com/questions/371184/predicting-real-numbers Predicting Real Numbers edited May 15 '13 Jared Mathematics Stack Exchange (抜粋) Here is an astounding riddle that at first seems impossible to solve. I'm certain the axiom of choice is required in any solution, and I have an outline of one possible solution, but would like to see how others might think about it.
100 rooms each contain countably many boxes labeled with the natural numbers. Inside of each box is a real number. For any natural number n, all 100 boxes labeled n (one in each room) contain the same real number. In other words, the 100 rooms are identical with respect to the boxes and real numbers.
Knowing the rooms are identical, 100 mathematicians play a game. After a time for discussing strategy, the mathematicians will simultaneously be sent to different rooms, not to communicate with one another again. While in the rooms, each mathematician may open up boxes (perhaps countably many) to see the real numbers contained within. Then each mathematician must guess the real number that is contained in a particular unopened box of his choosing. Notice this requires that each leaves at least one box unopened.
99 out of 100 mathematicians must correctly guess their real number for them to (collectively) win the game. What is a winning strategy? （抜粋おわり） 0512１３２人目の素数さん2016/09/10(土) 14:12:51.62ID:q7Skbg74>>511 つづき 英 mathoverflowは参考になる http://mathoverflow.net/questions/151286/probabilities-in-a-riddle-involving-axiom-of-choice Probabilities in a riddle involving axiom of choice - MathOverflow: edited Dec 9 '13 Denis (抜粋) The question is about a modification of the following riddle (you can think about it before reading the answer if you like riddles, but that's not the point of my question):
The Riddle: We assume there is an infinite sequence of boxes, numbered 0,1,2,…
. Each box contains a real number. No hypothesis is made on how the real numbers are chosen. You are a team of 100 mathematicians, and the challenge is the following: each mathematician can open as many boxes as he wants, even infinitely many, but then he has to guess the content of a box he has not opened. Then all boxes are closed, and the next mathematician can play. There is no communication between mathematicians after the game has started, but they can agree on a strategy beforehand.
You have to devise a strategy such that at most one mathematician fails. Axiom of choice is allowed. （この後＜11＞でAlexander Prussによる確率分布の議論があるよ） （抜粋おわり） 0513１３２人目の素数さん2016/09/10(土) 14:13:47.09ID:q7Skbg74>>512 英 mathoverflowで>>512関連 http://mathoverflow.net/questions/152787/can-an-infinite-number-of-mathematicians-guess-the-number-in-a-box-with-only-one Can an infinite number of mathematicians guess the number in a box with only one error? - MathOverflow edited Dec 26 '13 user44653 (抜粋) In this question＊） the following observation was made: ＊）上記 Probabilities in a riddle involving axiom of choice - MathOverflow: edited Dec 9 '13 Denis mathoverflow にリンクされている
Consider a sequence of boxes numbered 0, 1, ... each containing one real number. The real number cannot be seen unless the box is opened. Define a play to be a series of steps followed by a guess. A step opens a set of boxes. A guess guesses the contents of an unopened box. A strategy is a rule that determines the steps and guess in a play, where each step or guess depends only on the values of the previously opened boxes of that play. Then for every positive integer k , there is a set S of k strategies such that, for any sequence of (closed) boxes, there is at at most one strategy in S that guesses incorrectly.
My question is this: Can k be countably infinite (instead of a positive integer)? If not, is there a proof?
[Edit: the original question also asked whether k can be uncountable; this was answered by Dan Turetsky in the negative in comments]. （抜粋おわり） 0514１３２人目の素数さん2016/09/10(土) 14:15:47.79ID:q7Skbg74>>513 つづき これは内容的には無視していいかもしれんが、mathoverflowより時期が早いよね http://brainden.com/forum/topic/16510-100-mathematicians-100-rooms-and-a-sequence-of-real-numbers/ 100 mathematicians, 100 rooms, and a sequence of real numbers Asked by Jrthedawg, July 22, 2013 New Logic/Math Puzzles - BrainDen.com - Brain Teasers (抜粋) Question I am a collector of math and logic puzzles, and this must be the best I've ever seen.
100 rooms each contain countably many boxes labeled with the natural numbers. Inside of each box is a real number. For any natural number n, all 100 boxes labeled n (one in each room) contain the same real number. In other words, the 100 rooms are identical with respect to the boxes and real numbers. Knowing the rooms are identical, 100 mathematicians play a game. After a time for discussing strategy, the mathematicians will simultaneously be sent to different rooms, not to communicate with one another again. While in the rooms, each mathematician may open up boxes (perhaps countably many) to see the real numbers contained within. Then each mathematician must guess the real number that is contained in a particular unopened box of his choosing. Notice this requires that each leaves at least one box unopened. 99 out of 100 mathematicians must correctly guess their real number for them to (collectively) win the game. What is a winning strategy? （抜粋おわり）
