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Previously asked:

A square table has a coin at each corner. Design an execution sequence, each of whose steps consists of one of the following operations:

ONE (O): The operation (randomly) chooses a coin and flips it. SIDE (S): The operation (randomly) chooses a side of the table and flips the two coins along that side. DIAG (D): The operation (randomly) chooses a diagonal of the table and flips the two coins along that diagonal.

such that at some point during the execution (not necessarily at the end), a state where all coins are turned the same way (all heads or all tails) obtains.

The desired answer is O, D, S, D, O, D, S, D.

Is there a non-technical proof of this answer, and how one may arrive at it?

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  • $\begingroup$ What would you consider to be a "non-technical proof"? $\endgroup$ Apr 29 '15 at 11:10
  • $\begingroup$ An answer that a curious (curious enough to try this puzzle) would have a fair chance to understand. $\endgroup$
    – blackened
    Apr 29 '15 at 11:36
  • $\begingroup$ Have you seen kedargodbole.blogspot.com.au/2008/07/… ? $\endgroup$ Apr 29 '15 at 11:40
  • $\begingroup$ I did. It may be my incompetence, but I don't see it self-evident as a proof. $\endgroup$
    – blackened
    Apr 29 '15 at 11:50
  • $\begingroup$ A variation of the problem is discussed, and some references given, at gurmeet.net/puzzles/tumblers-on-a-rotating-table $\endgroup$ Apr 29 '15 at 11:59
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You already have the answer, so a non-technical proof might be simply checking the answer.

First write down every possible starting condition on a piece of paper. Then for each starting position write down every possible result of the first operation on another piece of paper. Then for each result on that piece of paper, write down every possible result of the second operation on yet another piece of paper. Do this for each step, and check that every end result is the desired outcome.

It is simple, but it may take a long time and use a lot of paper. Perhaps in the process you will notice a pattern that helps you understand the more technical proofs.

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