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Xindaris
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icon 3 Geometry Puzzles (+2)  
So, I was very bored today, with nothing but a pencil and paper to entertain myself. I wound up making up a few geometry puzzles based on isosceles triangles. They were developed independently, but there's a pretty good chance someone else already came up with them. Oh well. Anyway, if someone's interested in trying them out, here they are.

Puzzle 1:

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In this figure,
-The 3 blue lines are equal in length and all parallel.
-The 2 orange lines are equal in length.
-The 2 green lines are equal in length, and perpendicular to the blue ones.
-The triangle's area is 12 square units.
...How long is an orange line?


Puzzle 2:

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In this figure,
-The 3 blue lines are equal in length, parallel and 1 blue line is perpendicular to the green lines.
-The 2 green lines are equal in length, and their sum is equal to that of the 3 blue lines combined blue line.
-The 2 orange lines are equal in length, and that length is 1 unit.
...What is the area of the triangle minus the circle?

(WARNING: This one's answer is kind of a nasty number; I couldn't figure out how to make both the answer and the question have neat, simple numbers for this.)


Puzzle 3:

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In this figure,
-The two things that look like squares are indeed squares.
-The 2 green lines are equal in length.
-The 2 red lines are equal in length.
...The inner square's area is what number times the outer square's area? The figure itself suggests a number, but how can you be sure?

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[Last edited by Xindaris at 12-13-2019 02:00 AM]
12-12-2019 at 11:26 PM
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Xindaris
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icon Re: 3 Geometry Puzzles (0)  
I just want to note to anyone who's already trying these somehow that I just realized I forgot one assumption in puzzle 2. It's been edited in now.

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12-12-2019 at 11:37 PM
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TripleM
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icon Re: 3 Geometry Puzzles (+1)  
Puzzle 2 is impossible.

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12-13-2019 at 12:17 AM
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Dischorran
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icon Re: 3 Geometry Puzzles (0)  
Meanwhile, my totally non-mathematician (see post above mine for proof) work gets me to
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12-13-2019 at 12:34 AM
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Nuntar
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File: Problem 3.bmp (111.9 KB)
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icon Re: 3 Geometry Puzzles (+1)  
#3
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[Last edited by Nuntar at 12-13-2019 12:54 AM]
12-13-2019 at 12:53 AM
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Xindaris
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icon Re: 3 Geometry Puzzles (0)  
TripleM:
Obviously I didn't have a way to check geometry while making these, so it's believable I came up with an impossible geometric situation for #2. And yet I did get a valid real number as a solution anyway...
I don't think my solution actually uses the "3 blue lines are the same", so maybe if we take that part out it should be a geometrically possible thing?

Nuntar:
Nice.
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[Last edited by Xindaris at 12-13-2019 01:59 AM]
12-13-2019 at 01:54 AM
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Nuntar
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icon Re: 3 Geometry Puzzles (0)  
I don't think that's a rigorous proof.
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12-13-2019 at 02:15 AM
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Xindaris
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icon Re: 3 Geometry Puzzles (0)  
Reply to Nuntar below:
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[Last edited by Xindaris at 12-13-2019 05:50 AM]
12-13-2019 at 05:46 AM
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Dying Flutchman
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icon Re: 3 Geometry Puzzles (+1)  
quote:
Nuntar wrote:
I don't think that's a rigorous proof.
Click here to view the secret text



There's an even easier way. After splitting the setup along AE, we have a diagonal line running SW-NE. That's the side of the triangle. Now, because all is E/W symmetric, and both squares thus share the same center, the corner of the small square must be on the diagonal SE-NW (not drawn in the picture). So the corner can only be exactly in the center of the 'halved setup'. Hence, it is exactly a quarter of the size of the big square.

Yeah, I know. It's not as convincing without a picture. But trust me. It's easy :)

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12-13-2019 at 08:08 AM
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Dying Flutchman
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icon Re: 3 Geometry Puzzles (+1)  
Number 1 has a particularly nice setup, I think.

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[Last edited by Dying Flutchman at 12-13-2019 08:58 AM]
12-13-2019 at 08:37 AM
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Nuntar
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icon Re: 3 Geometry Puzzles (0)  
quote:
Dying Flutchman wrote: Yeah, I know. It's not as convincing without a picture. But trust me. It's easy :)


I know it's easy; my point is just that it's important to distinguish between what you can prove and what you "know" from looking at the diagram.

In your proof, it seems to me like you are assuming that the other diagonal (BE) passes through the top-left corner of the large square. In an accurately drawn diagram it will, of course. But that doesn't mean it's safe to assume it.

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12-13-2019 at 02:19 PM
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Dying Flutchman
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icon Re: 3 Geometry Puzzles (+1)  
quote:
Nuntar wrote:
quote:
Dying Flutchman wrote: Yeah, I know. It's not as convincing without a picture. But trust me. It's easy :)


I know it's easy; my point is just that it's important to distinguish between what you can prove and what you "know" from looking at the diagram.

In your proof, it seems to me like you are assuming that the other diagonal (BE) passes through the top-left corner of the large square. In an accurately drawn diagram it will, of course. But that doesn't mean it's safe to assume it.


Oh, but it is safe to assume that, because the small square is placed symmetrically in the middle of the large square.

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12-13-2019 at 02:43 PM
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Xindaris
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icon Re: 3 Geometry Puzzles (+1)  
Okay, I'm pretty sure puzzle 1 is just plain broken as written. If--and I haven't looked it up but I'm just trusting that folks know what they're talking about--the triangle set up that way has to be equilateral then it's not the one I had in mind. Besides that, I think I wrote that problem somehow assuming more knowledge than I actually gave because I originally "wrote it backwards", starting from what is now the answer and trying to work out what that would make the area. So here's what I thought was going on with that one:
Click here to view the secret text

Wow, really boring right? And the circle is irrelevant. This is what happens when I try to make up puzzles tired, I guess--at least for the first try.

I think I like DF's answer better.

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[Last edited by Xindaris at 12-16-2019 04:33 AM]
12-16-2019 at 04:33 AM
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Dying Flutchman
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icon Re: 3 Geometry Puzzles (+1)  
Well, out turns out again that making puzzles is often harder that solving them. All drod architects may recognize that feeling :)

Meanwhile, you had many of us nicely nerd sniped. So it's been worth it.

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12-16-2019 at 07:20 AM
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