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This is an unusual ask, but bear with me... One night back in the 1970's. I was playing Othello with a friend when I realised that, not only were there no more moves with four empty spaces remaining but the board was diagonally symmetrical. Its symmetry was one where the white counter on the one half was mirrored by an reverse counter on the other side. In my opinion creating an immense depth of complexity. We quickly drew a grid on the Othello box and copied the pattern.There was no intent by either of us to create this situation by a planned strategy of symmetrical positioning or a tit for tat, like for like play. We were utterly bewildered how this happened and that it was at all possible. The trillions of possible positions and our non intentional game play make a repetition of this possible but not very likely in the short term. Can anyone place pieces on an Othello board to reproduce this situation, not by playing the game just placing the pieces, and not necessarily the same pattern but one with a symmetry that has the play locked? Kudos to anyone that can!

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  • $\begingroup$ A diagonal symmetry with white on one side and black on the other side would require some very special pieces on the main diagonal that serves as the symmetry axis. $\endgroup$ – Bass Dec 27 '19 at 9:15
  • $\begingroup$ Presumably, the axis of symmetry is allowed free choice of counters. $\endgroup$ – Lawrence Dec 27 '19 at 14:53
  • $\begingroup$ @Lawrence possibly that, or maybe OP actually meant some other kind of symmetry. I'll be waiting for the clarification. $\endgroup$ – Bass Dec 27 '19 at 16:36
  • $\begingroup$ @Bass Hmm. You have unearthed a fundamental flaw in my recollection. I have now to consider where unless I can give fairly exacting details of its symmetry my puzzle seems pointless........On reflection, it must have been symmetrical on another axis as symmetrical it most definitely was. $\endgroup$ – Hobittual Dec 27 '19 at 22:05
  • $\begingroup$ Possibly related: boardgames.stackexchange.com/questions/20539/… $\endgroup$ – Dmitry Kamenetsky Dec 29 '19 at 13:49
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The diagonal symmetry with opposing colours on opposing sides of the axis isn't really possible because there are pieces on the symmetry axis too.

Since OP agreed that this observation presents a problem, here's the "most symmetrical" finished position with 4 empty spots I could find.

enter image description here

The checkered spots can be any colour you want (as long as they are all filled, and the symmetries are preserved), since the important thing is that

any straight line ending in one of the empty spots must have pieces of one colour only.

This pattern has

  • point symmetry through the center
  • 90 degree rotational anti-symmetry (closest thing to a diagonal anti-symmetry I could think of)
  • equal number of points for both (naturally)
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  • $\begingroup$ Ah you beat me. I was going to use exactly the same solution. $\endgroup$ – Dmitry Kamenetsky Dec 29 '19 at 13:31
  • $\begingroup$ I appreciate the suggestion, the illustration obviously bears out the conditions I vaguely recalled. Whilst being correct, the solution does not look as though it was derived by the game play of Othello, to me, I could be wrong but the central transparent sections are far too simplistic to have appeared at random with both players attempting to thwart the other. The four empty corner squares were definitely occupied. $\endgroup$ – Hobittual Feb 3 at 23:09

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