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Similar to this question. 5-color coloring game.

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

They now play a game such that:

Player $$ picks a region and player $$ 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 $$ have a winning strategy for every possible map?

David Raveh
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blademan9999
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    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) – Calvin Lin Jun 25 '23 at 16:25
  • I'm not even convinced by the answer to the linked question which claims that A has a winning strategy with 5 colors. – Misha Lavrov Jun 25 '23 at 19:42

1 Answers1

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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.

Snared
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  • Map must be Euclidean – blademan9999 Jun 25 '23 at 17:44
  • 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 – Snared Jun 26 '23 at 02:12
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    @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".) – Misha Lavrov Jun 26 '23 at 03:12
  • 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. – Snared Jun 27 '23 at 04:03
  • 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? – Snared Jun 27 '23 at 06:14
  • I'm not going to ask a new question just because I didn't perfectly clarify this one, an edit is more appropriate. – blademan9999 Jul 19 '23 at 11:35
  • @Snared This is why there are policies on Math SE which implore askers to provide details, and potential answerers to ask questions before posting answers. Generally speaking, if you can answer a question here in a one liner, there is a good chance that the asker left off some detail. Use the comments to seek clarification before answering. – Xander Henderson Aug 05 '23 at 18:15
  • That being said, @blademan9999, it is not appropriate to edit a question in a way which invalidates an existing answer. While I do not like the "gotcha!" aspect of this question, it is a valid answer to the question you originally asked, and editing your question to invalidate the answer is not okay. In such a case, it is better to ask a new question with the appropriate details. – Xander Henderson Aug 05 '23 at 18:16
  • I edited it to clarify and add a detail which I though was clearly implied by the question. – blademan9999 Aug 06 '23 at 16:23
  • @blademan9999 you realize the Earth is mapped yet is a sphere? I'm not sure why you thought that a map implies that the underlying space is of dimension 2 specifically. We map tori, $n$-spheres, even non-orientable spaces using maps. What you did it add the word "Euclidean", which is still wrong, since tori can be embedded in R^3 and inherit the subspace topology metric, that being Euclidean. So either way the answer stands, if you really wanted to edit the question to be a "Gotcha" you would have specify the map further than simply being Euclidean. – Snared Aug 06 '23 at 17:52
  • The whole 4 colour theorem, planar, most maps, planar, have you ever seen a toroidal map. Plus a spherical map can easily be turned to planar one, and vice Versa. Posting a second near indentic question instead of just a doting would likely be against the rules – blademan9999 Aug 07 '23 at 03:09
  • a spherical map cannot be turned into a planar one, since euler characteristic is invariant under isomorphisms, yet the sphere and plane have euler characteristics 1 and 2, so what you said cannot be true.. And one question is general, the other would be specific to planar maps if that is what you wish... – Snared Aug 09 '23 at 16:34
  • And yes, maps between tori are extremely natural - they will appear many times in most mathematician careers.. Tori are the product of two circles, so any pair of maps from two circles to one would produce a natural torus function. If you were playing a game where horizontal sides and vertical sides are identified... There are so many examples that I could go on and on. Anyway, I hope you can find someone to answer the more specific version of your original question, or maybe you plan to self-answer the question.. sorry for the misunderstanding, I wish I knew you meant planar in the beginning. – Snared Aug 09 '23 at 16:45