25 dots are drawn as a 5x5 regular square grid. Can you draw 3 triangles that pass through every dot? The corners of the triangles must lie on the dots, ie., they cannot lie outside the grid.
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asked Jun 28, 2021 at 4:51
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3$\begingroup$ i wonder what is the fewest number of triangles needed if right triangles are forbidden? $\endgroup$– jwezorekCommented Jun 28, 2021 at 22:30
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2$\begingroup$ @jwezorek great idea! I've made this into a new puzzle: puzzling.stackexchange.com/questions/110702/… $\endgroup$– Dmitry KamenetskyCommented Jun 29, 2021 at 2:05
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3 Answers
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I think this is the only solution.
(Edit: It isn't. See teedyay's answer for another solution.)
I found this solution mostly by trial and error, but I did keep in mind that
apart from the 9 triangle vertices there are 25-9=16 other vertices that need to lie on the 9 sides of the triangles, so on average the sides need to visit almost 2 extra vertices each. That makes it almost inevitable that all triangles have two axis-aligned sides and a third 45-degree side, and that one triangle covers two whole sides of the grid.
answered Jun 28, 2021 at 5:12
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1$\begingroup$ Correct and well done! I will post a harder version of this puzzle soon. $\endgroup$ Commented Jun 28, 2021 at 5:17
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Here is an alternative solution to the one already found:
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4$\begingroup$ Oh very nice! I didn't realise that there was another solution. $\endgroup$ Commented Jun 28, 2021 at 14:43
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3$\begingroup$ After seeing the solution for the 7x7 version of the problem I should have realised that trick might apply here as well. Very well done for finding this! $\endgroup$ Commented Jun 28, 2021 at 15:22
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Another symmetric solution:
This was found by noticing that @teedyay's solution has quite a bit of slack in it.
In fact, there are two more similar solutions:
answered Jun 29, 2021 at 4:05