My question is whether or not cycles can occur in the abstract game, Clearcut.



A few smart guys (and me) have been trying to either find a cycle or prove that one can't happen. Ten days have passed without success. Super hard problem.

Clearcut was designed by me. The only benefit I will derive from a proof is to be able to say with certainty that Clearcut is finite.

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    $\begingroup$ All relevant details should be presented directly, not in a linked pdf. $\endgroup$ Jul 25, 2023 at 20:11
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    $\begingroup$ Ok. I added a JPG of the rule sheet. $\endgroup$ Jul 25, 2023 at 23:29
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    $\begingroup$ Can you clarify what you mean by “cycle”? $\endgroup$ Jul 26, 2023 at 1:15
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    $\begingroup$ You kill some of mine, I kill some of yours, you kill some of mine... Endless cycle of turns. Game never finishes. $\endgroup$ Jul 26, 2023 at 2:58
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    $\begingroup$ Would it be possible to transcribe the text somehow? That would make the post more readable for people with poor eyesight (zooming in is easier with text than an image, and screen-readers can't read the image), and make the puzzle more searchable. $\endgroup$
    – bobble
    Aug 5, 2023 at 19:05

1 Answer 1


I believe this is a cycle.

Blue and red both take two stones on turn 3.
Each following cycle of 4 turns each they take 2 stones on turn 1 and 3.
(each time with 3 vs 2 extended crossovers.)
enter image description here
Clarification: Starting with dark blue and red placed, some color starts, e.g. blue.
round 1: blue plays blue 1, red plays red 1
round 2: blue: blue 2, red: red 2
round 3: blue: blue 3 and takes red 1+2, red: red 3 and takes blue 1+2
round 4: blue: blue 4, red: red 4
round 5: blue: blue 1 and takes red 3+4, red: red 1, and takes the blue 3+4
round 6: the situation is identical to before round 2, repeat round 2....

  • $\begingroup$ Hope this helps. $\endgroup$
    – Retudin
    Jul 27, 2023 at 15:26
  • $\begingroup$ (Note that it is 1 picture with 2 unconnected groups) If blue 2 is played, red 3 is not there (removed by blue the round before (blue 1), so there is no crosscut. $\endgroup$
    – Retudin
    Jul 27, 2023 at 15:35
  • $\begingroup$ Your new picture makes it very clear. It is much simpler than I thought it would be. $\endgroup$ Jul 27, 2023 at 15:59

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