Squares

[bonanova]

How many nonoverlapping unit squares on the plane can be partially overlapped by a single unit square?
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[Zag]

I can get 7. I'm not convinced that's the best possible, but I don't see a way to improve it.

[bonanova]

It's counterintuitive.
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[Trojan Horse]

Does that mean you found a way to get 8, bonanova?

[bonanova]

Yes sir. I didn't discover it, I became aware of it, a while back.
A clue would be to ask what is intuitive about the 7 solution.
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[itisally]

bonanova, such statements make me want me to reply in crazy ways. So the question stated how many on "the" plane, but it doesn’t state that the squares are on that plane. If the plane intersected the squares at a perpendicular then the answer could be an infinite number of squares.
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I could agree with you, but then we would both be wrong.

[bonanova]

I see your point. That's not the intent. The question is meant to be straightforward.

Lay out a bunch of nonoverlapping red unit squares on the plane. Place on top of this assemblage a single blue unit square. How many red squares can the blue square partially overlap? By overlap we mean share portions of the plane that possess nonzero areas.

We rule out the trivial answer of nine, in which eight touch only at their edges. The blue square does not move, once placed. The squares are themselves in the plane. This is a two dimensional question. Except we observe it from the third.

Intuition may not provide a clue. Disregard if you like. I'm in favor of mental health.
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[novice]

Here's the pretty straightforward 7-square solution, at least.

[novice]

[itisally]

So with that in mind I tried spreading out the 7 arrangement to see how much space I could create before things lost contact, then I tried rotating groups to get all the "extra" space on one end.

The other thing that is intuitive to me is the use of a 45 degree rotation. I feel certain that the answer lies somewhere in the ideas of triangles, but it is allusive.
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I could agree with you, but then we would both be wrong.

[Zag]

Yeah, I tried a fair bit more, to no avail.

BTW, your allusion in the last word in your post is eluding me.

[itisally*]

I was really wishing I was at home so that I wasn't trying to move squares around in Word. lol.

Maybe illusive would have been better, spelling is a challenge for me.


Thinking out loud: we use the 45 rotation because it allows the hypotenuse to bisect the inner red square with extra length at the corners. How will this idea help me move forward, or is this the road block to thinking outside the box.

[Suspence]

I thought you were looking for elusive...

I have paper cutouts on my desk, and I'm still not getting anywhere. I was thinking of trying Word. I guess the grass is always greener...
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I hate people who try to write interesting things in their signature.

[Zag]

[itisally]

Zag,
I love that answer!!
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I could agree with you, but then we would both be wrong.

[Elethiomel]

The other thing that might be counter-intuitive is if the answer is asymmetric. But I'm not getting anywhere on this, either.

[bonanova]

I think there is no way to spoiler an image. Is that right? Or where does one upload images now? I'll hold off if people are active.
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_{Vidi, vici, veni.}

[Elethiomel]

You can just post a link to the image instead of including it in an Img tag.

[Trojan Horse]

No matter what happens, this puzzle will belong in the comeuppances thread (in Off Topic).

The only question is, will bonanova get the comeuppance, or will the rest of us?

[bonanova]

[bonanova]

Eight squares.
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_{Vidi, vici, veni.}

[Zag]

[Elethiomel]

Cool. That's a great puzzle.

[itisally]

Oh my goodness! I was sooooooo close. I just couldn't squeeze the one in the middle.
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I could agree with you, but then we would both be wrong.