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Similar to this question. 5-color coloring game.

Let there be two players, $𝐴$ and $𝐵$, and a map.

They now play a game such that:

Player $𝐴$ picks a region and player $𝐵$ colors it such that the region is a different color than all adjacent regions. Player $B$ wins if at the end of the game, the map is colored such that no two adjacent regions are the same color. Player $A$ wins if at any point in time that becomes impossible. If there are four available colors, then does player $𝐴$ have a winning strategy for every possible map?

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    $\begingroup$ I believe that player A doesn't have a winning strategy for some map, and that's partly reflected in the difficulty of the 4-color theorem (where you can't induct your way through it) $\endgroup$
    – Calvin Lin
    Jun 25, 2023 at 16:25
  • $\begingroup$ I'm not even convinced by the answer to the linked question which claims that A has a winning strategy with 5 colors. $\endgroup$ Jun 25, 2023 at 19:42

1 Answer 1

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A does not always have a winning strategy. Suppose the map is of a Torus. Since it is well known that 7 colors are needed to color arbitrary partitions of a Torus, B can sometimes have a completely winning strategy. It all depends on the euler characteristic of the surface of the map.

Edit: Let $E = \mathbb R^3$ be endowed with the Euclidean metric. Embed a torus $T \subset \mathbb E$ and consider it with the topology given by the Euclidean metric on $E$. Hence, we are dealing with an Euclidean map, yet there are partitions of the underlying space for which 7 colors are needed so that no two touching partitions are the same color.

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  • $\begingroup$ Map must be Euclidean $\endgroup$ Jun 25, 2023 at 17:44
  • $\begingroup$ Seriously? You changed your question immediately after my correct answer was posted. If you wanted to restrict to Euclidean maps you should have opened a new question. Now my answer, which was correct but is now incorrect after you edited your question, is getting downvoted into oblivion and my rating is suffering unjustly. Next time just post a new question $\endgroup$
    – Snared
    Jun 26, 2023 at 2:12
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    $\begingroup$ @Snared Drawing a map in the plane is a default assumption. Your answer, to begin with, is nothing more than a "gotcha" like saying "well, actually, 2+2=7 can be true if we're working mod 3". If you seriously thought that toroidal maps might be relevant, you should have left a comment to check, first. (Also, a single downvote - not from me - is hardly "oblivion".) $\endgroup$ Jun 26, 2023 at 3:12
  • $\begingroup$ Obviously there are geometries for which this game is impossible and that is the answer to the question. Then they changed the question to specify a geometry and ask. Now nobody even cares about the question as its relevance just immediately dropped. Yet you thought it was natural. That's crazy. $\endgroup$
    – Snared
    Jun 27, 2023 at 4:03
  • $\begingroup$ It's three downvotes not a single one @MishaLavrov all of which were added after the OP changed his question to invalidate the answer, by the way. Makes no sense. Was I supposed to delete my answer after that point? $\endgroup$
    – Snared
    Jun 27, 2023 at 6:14

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