$3 \times 3 \times 3$ grid with the center cube filled. There are 26 cubes remaining. How many $2 \times 2 \times 1$ tiles could we place without overlapping?

Consider a
4
5
6
7

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3d model of the solution: https://skfb.ly/ES6B

Oras Phong
- 5 years, 12 months ago

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[This was a wrong solution.]

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Okay, maybe I'm just not reading this question correctly. If we eliminate 2 opposite corner cubes from the remaining 26 cubes, we can array three 2x2x1 tiles around one missing cube, and three more 2x2x1 tiles around the other.

What is the one-line solution, because maybe it'll help me find the solution to that other problem.

Michael Mendrin
- 6 years ago

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Yea, you're right, we can put 6. Darn it. i miscounted in my argument, and this is now less interesting with the answer of 6.

But that's exciting! In relation to the inspiration problem:

If we can fit 4 cubes on a face with 2 (adjacent) edges restricted off, then we can fit 6*4 of them! Of course, we can't do 4 on a complete restriction, but it seems likely that we could do a "loose" restriction where we allow slight overlap that is accounted for in the packing.

We also have the opposite corner cubes to play with!

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Okay, you can always delete this post, but I'd still like to hear that one-line solution. I like leaving no stone unturned. Even mistakes can be valuable.

Michael Mendrin
- 6 years ago

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@Michael Mendrin – My faulty solution was:

Each $2*2*1$ square must cover 2 "edge" cubes (IE not corner, not middle). However, i miscounted and thought that there were only 8 edge cubes, but there are actually 12.

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@Calvin Lin – Unfortunately, I've got to go now, gone for a couple of days. As I said in my reply to you in that other problem, I do think it's possible for 8 red cubes to touch one of the faces of the blue cube, but the solution is a mess and will take some time to forward an accurate graphic of how it's done. Unfortunately, even that will not get us any closer to a 24 solution.

Michael Mendrin
- 6 years ago

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Let A, B, C, D, E and F be the tiles and x be the filled center cube. Here we go: