# What is the most triangles you can make from a capital "H" and 3 straight lines?

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:

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

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

Here's a solution for 7 triangles:

• You can make $8$ from that picture. See if you can figure it out ;) Mar 1, 2019 at 13:48
• No, you cannot.
– Bass
Mar 1, 2019 at 15:10
• 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. Mar 1, 2019 at 17:13
• @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.
– Bass
Mar 1, 2019 at 18:21
• 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 Mar 2, 2019 at 6:40

Here's one with six triangles (7 if you count triangles outside of triangles, which you don't):

• I suspect that this is the highest that can be gotten, but I'm not sure. Mar 1, 2019 at 1:52
• 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 Mar 1, 2019 at 5:07
• If you just moved all of your blue lines up or down a bit you’d get 7 won’t you. Mar 1, 2019 at 15:38
• @tyobrien you'd get one more triangle in the middle, but the two triangles right next to the new one would become quadrangles, right?
– Bass
Mar 1, 2019 at 15:55
• Oh yes that would be correct Mar 1, 2019 at 15:56

Does this count as 8 triangles?

• 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".
– Bass
Mar 1, 2019 at 9:32
• @Bass Precisely! Mar 1, 2019 at 14:12
• @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. Mar 1, 2019 at 15:31

Please have a look at this pic using 3 red lines on letter H. I count 10 triangles

• Overlapping triangles are not counted, so there are only 5 in your arrangement. Mar 1, 2019 at 13:41
• Yeah, I agree with @JaapScherphuis Mar 1, 2019 at 18:41
• @JaapScherphuis Event with overlapping I count 6. How are there 10 here? Mar 1, 2019 at 22:33
• @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 Mar 1, 2019 at 23:50