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So start with an upper case H, and then draw $3$ straight lines. What is the greatest number of closed triangles that you can form? For example:

an example

Note that triangles inside of triangles only count once (e.g. 5 & 6 in the image don't count): however, triangles inside of other triangles do not count

And you aren't allowed to extend the cross line of the H (e.g. 5 in the image doesn't count) Example of what's not allowed

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4 Answers 4

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Here's a solution for 7 triangles:

enter image description here

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    $\begingroup$ You can make $8$ from that picture. See if you can figure it out ;) $\endgroup$
    – Mr Pie
    Mar 1, 2019 at 13:48
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    $\begingroup$ No, you cannot. $\endgroup$
    – Bass
    Mar 1, 2019 at 15:10
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    $\begingroup$ Decrease the angle between three red lines and rotate them ccw slightly. Now two rightmost red lines in the original form will make another triangle with the right pillar of H. According to OP only extending the cross line is not allowed. $\endgroup$
    – no more sigsegv
    Mar 1, 2019 at 17:13
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    $\begingroup$ @btw please draw a picture. The two rightmost red lines (the ones that go through the right edge of the picture) already make a triangle with the right pillar of the H, and hopefully you aren't suggesting that three straight lines could somehow make more than one triangle. $\endgroup$
    – Bass
    Mar 1, 2019 at 18:21
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    $\begingroup$ I was mistaken! After lots of analysis from drawing at home and all, methinks $7$ triangles is, in actual fact, the maximum amount that can be made from $3$ straight lines and with the OP's rules. My apologies! But, to your answer, my compliments! :D $\endgroup$
    – Mr Pie
    Mar 2, 2019 at 6:40
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Here's one with six triangles (7 if you count triangles outside of triangles, which you don't):

H with lines

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    $\begingroup$ I suspect that this is the highest that can be gotten, but I'm not sure. $\endgroup$
    – Brandon_J
    Mar 1, 2019 at 1:52
  • $\begingroup$ My friend alleges that he can get 7, but won't show me how. Considering writing a python script to help prove that 6 is the max. Also I feel like there may be a mathematical proof of this with analytical geometry $\endgroup$
    – Curtis
    Mar 1, 2019 at 5:07
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    $\begingroup$ If you just moved all of your blue lines up or down a bit you’d get 7 won’t you. $\endgroup$
    – tyobrien
    Mar 1, 2019 at 15:38
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    $\begingroup$ @tyobrien you'd get one more triangle in the middle, but the two triangles right next to the new one would become quadrangles, right? $\endgroup$
    – Bass
    Mar 1, 2019 at 15:55
  • $\begingroup$ Oh yes that would be correct $\endgroup$
    – tyobrien
    Mar 1, 2019 at 15:56
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Does this count as 8 triangles?

enter image description here

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    $\begingroup$ The question says "Note that triangles inside of triangles only count once", which (I think) is a more complicated way of saying "the triangles are not allowed to overlap". $\endgroup$
    – Bass
    Mar 1, 2019 at 9:32
  • $\begingroup$ @Bass Precisely! $\endgroup$
    – Curtis
    Mar 1, 2019 at 14:12
  • $\begingroup$ @Bass it is possible to create two triangles that overlap, but neither is inside the other, so they're not just different ways of saying the same thing. $\endgroup$
    – Monty Harder
    Mar 1, 2019 at 15:31
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Please have a look at this pic using 3 red lines on letter H. I count 10 triangles

total triangles 9

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    $\begingroup$ Overlapping triangles are not counted, so there are only 5 in your arrangement. $\endgroup$
    – Jaap Scherphuis
    Mar 1, 2019 at 13:41
  • $\begingroup$ Yeah, I agree with @JaapScherphuis $\endgroup$
    – Brandon_J
    Mar 1, 2019 at 18:41
  • $\begingroup$ @JaapScherphuis Event with overlapping I count 6. How are there 10 here? $\endgroup$
    – Hazel へいぜる
    Mar 1, 2019 at 22:33
  • $\begingroup$ @PerpetualJ I count 10 with overlapping rot13(gbc bs gb u gb zvqqyr yvar gvzrf gjb naq gbc bs u gb obggbz erq yvar gvzrf gjb. ) not sure if that helps $\endgroup$
    – Yout Ried
    Mar 1, 2019 at 23:50

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