## Minimum number of 45-60-75 triangles in a square

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### Minimum number of 45-60-75 triangles in a square

Here's another puzzle:

What is the minimum number of 45-60-75 triangles that a square can be divided into?

Here is an image of a flawed answer:

Any ideas?
bjnoel
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### Re: Minimum number of 45-60-75 triangles in a square

Okay, since there have been no takers:

The first thing I notice is that the image is not just flawed but a little misleading, because 45, 60 and 75 are a Pythagorean triplet. So, we can make a rectangle out of two such triangles (my images aren't quite in proportion):

The next thing is to find the lowest common denominator of 45 and 60, which is 180. So, our answer looks like this:

24 triangles.

Lomax

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### Re: Minimum number of 45-60-75 triangles in a square

Lomax,

That's a neat interpretation, that this question might be word play!

I took a "45-60-75 triangle" to mean a triangle with angles of 45 degrees, 60 degrees, and 75 degrees. This interpretation is reinforced by the image which appears to depict triangles with such angles. But perhaps this is misdirection, as in a riddle?

Honestly, I didn't even consider that a Math-related puzzle could have such an element to it.
Natural ChemE
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### Re: Minimum number of 45-60-75 triangles in a square

Oops, my bad. The bit where they added up to 180 should have been a giveaway for me haha. I have absolutely no idea how to approach the question then :P

Lomax

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### Re: Minimum number of 45-60-75 triangles in a square

Lomax,

I don't know! It's rather suspicious that the numbers selected were such that
452 + 602 = 752
and
45 + 60 + 75 = 180.
These numbers, since they're unspecified, could be taken to mean either angles or sides. And the OP did state that the picture showed a "flawed" solution; it could've been intentional misdirection. While I'm not sure if it was his intent, I'd argue that your answer is a valid interpretation.
Natural ChemE
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### I N C E P T I O N

First of all: Your solution is awesome, Lomax.

Second of all: This doesn't exactly count as a solution, but here's one with an infinite number of triangles (with angles $45^\circ$, $75^\circ$, $60^\circ$) --

xcthulhu
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### Re: Minimum number of 45-60-75 triangles in a square

Nice one Lomax! No, it wasn't intended to be solved that way, very creative. The numbers do indeed represent the angles rather than the sides.

Good one xcthulu, but is it the minimum?
bjnoel
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### Re: Minimum number of 45-60-75 triangles in a square

What is the minimum number of 45-60-75 triangles that a square can be divided into?
Hang on divided into! Not bye
Using xcthulhu’s diagram there is a square inside! 12 triangles
Ummm maybe not you could just use 4 triangles if that were the case
In any case I really like xcthulhu’s diagram as a design solution for areas

Fuqin
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### Re: Minimum number of 45-60-75 triangles in a square

Hey,

I can't figure out a solution. Here is another attempt; it's not correct. I haven't illustrated this but I feel this leads to an infinite series of smaller and smaller triangles just as my first attempt.

~XCT
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xcthulhu
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### Re: Minimum number of 45-60-75 triangles in a square

I kinda see what XCT is getting at. Unless you can find a way to fit in 100% of the square with finitely many triangles, then the answer must be infinite.

You could try a proof by contradiction?

Assume we've fit one with $k$ such triangles. Now consider that for each such triangle, we can break this triangle into two right angle 60-30-90 and 45-45-90 triangles. This implies we've filled the square with 2k of these two distinct right angle triangles. Is this possible?

jshort
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### Re: Minimum number of 45-60-75 triangles in a square

xcthulhu wrote:Hey,

I can't figure out a solution. Here is another attempt; it's not correct. I haven't illustrated this but I feel this leads to an infinite series of smaller and smaller triangles just as my first attempt.

~XCT

Yep. There's something about the differences in angles that prevents a finite solution, but I can't wrap my mind around it. If you want to be pedantic (sometimes I do), the OP did not specify Euclidean, so you can always tile a spherical rectangle (Ouch!).