Triangle cutting problem

Anything goes, but keep it seemly...

Triangle cutting problem

Postby udosuk » Thu Apr 27, 2006 6:20 pm

Could anyone help me on this? I suppose it could be worked out using pen & paper...

A 30-60-90 triangle can be cut into 2 such triangles in only one way:
Image

To cut it into 3 such triangles, there are 3 different ways:
Image

So how many ways in total to cut it into 4 such triangles? Their sizes can be either same or different to each other...
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Re: Triangle cutting problem

Postby Cec » Thu Apr 27, 2006 9:54 pm

udosuk wrote:"So how many ways in total to cut it into 4 such triangles? Their sizes can be either same or different to each other...

Four?
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Postby Hud » Fri Apr 28, 2006 12:04 am

Just one question: does each cut have to go completely through the large triangle? In other words, can a cut butt against another cut inside the large triangle? I' not sure it matters, but it may?
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Postby ronk » Fri Apr 28, 2006 12:16 am

Probably more, but my answer is five.
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Postby ab » Fri Apr 28, 2006 12:20 am

You can convert each 3 into a 4 by halving a green, yellow or red triangle, which gives 9 options, but you can also convert each 2 into a 4 by making one of the two triangles into a 3 which gives rise to 2 more (fingers crossed) so I think 11.
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Postby Ruud » Fri Apr 28, 2006 12:29 am

This is what I could find (so far):

My sail designs for 2006:D

Image

My improved answer is 10.

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Postby ronk » Fri Apr 28, 2006 12:42 am

ab wrote:You can convert each 3 into a 4 by halving a green, yellow or red triangle, which gives 9 options
I totally missed those.

ab wrote:... but you can also convert each 2 into a 4 by making one of the two triangles into a 3 which gives rise to 2 more
But you can do that to each triangle two different ways. And you can make two triangles out of each of the two triangles. Which is what I counted to arrive at my (crappy) answer of 5.

But (I think) each of those 5 is identical to 1 of your 9.

[edit: And Ruud finds a 10th by thinking "outside the triangle":) ... and dividing the 90 degree angle by 3. Cool!]
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Postby udosuk » Fri Apr 28, 2006 5:43 am

Actually the answer is more than 11 (by quite a few in fact)...

But ab & Ruud have done a good job so far (especially Ruud for the nice graphics...)

And Hud, I think the answer to your question is yes (although I might not have fully understood your words)...

You definitely have to "think outside the triangle" to find the remaining ones...:D
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Postby Ruud » Fri Apr 28, 2006 12:17 pm

udosuk wrote:Actually the answer is more than 11 (by quite a few in fact)...

Help! I'm only at 14 now:(

Image

Tell me when to stop.

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Postby ronk » Fri Apr 28, 2006 3:46 pm

Ruud wrote:Help! I'm only at 14 now:(
Use other diagonal of 1st figure.

P.S. My screen's barely wide enough to see all at once.:(
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Postby Ruud » Fri Apr 28, 2006 4:26 pm

Ron wrote:Use other diagonal of 1st figure.

Thanks, we're at 15 now:)

My screen's barely wide enough to see all at once.:(

Image

Ruud.

16, after ab's addition.:)
Last edited by Ruud on Fri Apr 28, 2006 12:53 pm, edited 1 time in total.
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Postby ab » Fri Apr 28, 2006 4:40 pm

Ruud in your first downward pointing triangle, you can leave the red triangle alone and flip the other three around their line of symmetry to get the blue triangle at the thin end:)
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Postby Ruud » Fri Apr 28, 2006 4:55 pm

Hi ab.

Good addition. I don't know if Ron's screen is high enough for another picture, so I updated the previous post:D

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Postby ronk » Fri Apr 28, 2006 5:13 pm

Ruud wrote:Good addition. I don't know if Ron's screen is high enough for another picture, so I updated the previous post:D

LOL (lots of laughs)! Yes, it's plenty high ... perhaps I exaggerated.

I know this will sound like I'm complaining, but with half the triangles upside down, it's really tough to compare and "see" what we've already got. I'll volunteer to flip half of them over, if you'd rather not do it. (My only tool is MS Paint, tho.)

Ron
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Postby Ruud » Fri Apr 28, 2006 5:48 pm

Well, I've had a few laughs too.

Sorry Ron. No need for you to go flippin' I still have the original.:D

Image

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