# Three triangles passing through every dot of a 5x5 grid

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.

• i wonder what is the fewest number of triangles needed if right triangles are forbidden? Commented Jun 28, 2021 at 22:30
• @jwezorek great idea! I've made this into a new puzzle: puzzling.stackexchange.com/questions/110702/… Commented Jun 29, 2021 at 2:05

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.

• Correct and well done! I will post a harder version of this puzzle soon. Commented Jun 28, 2021 at 5:17

Here is an alternative solution to the one already found:

• Oh very nice! I didn't realise that there was another solution. Commented Jun 28, 2021 at 14:43
• 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! Commented Jun 28, 2021 at 15:22

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: