Questions 26-50
Square Two
In which we put pentominoes into boxes in different ways
26: Can you assemble the 12 pentominoes into a rectangular box?
Yes, you can. In fact, when I was a kid, I got a set of pentomino pieces made out of plastic that came in a rectangular box. This was something of a puzzle to figure out how to get them back into the box.
27: What size boxes might 12 pentominoes fit into?
We can figure out that there will be 60 squares in our box (12 pentominoes, 5 squares each, 12 x 5 = 60) for the pentominoes to fit without any space left over. This gives us a few possibilities:
1 x 60
2 x 30
3 x 20
4 x 15
5 x 12
6 x 10
And that's all the sized boxes that can fit 60 squares. But that doesn't tell us if they can actually fit.
28: Can a set of 12 pentominoes fit into a 1 x 60 box?
No. If you're using all 12 different pentominoes, it's easy to see that only one of them can fit into a very narrow box that's one square wide. If you wanted 12 of the 1x5 "O" pentomino, then yes. But if you're using any of the other pentominoes, they're just too wide to fit.
29: Can a set of 12 pentominoes fit into a 2 x 30 box?
No. The same reasoning as above applies - some of the pentominoes would fit, but several cannot.
30: Can a set of 12 pentominoes fit into a 3 x 20 box?
Yes! You were getting impatient, weren't you? Here's one solution:
31: Can a set of 12 pentominoes fit into a 4 x 15 box?
Yes! Here's one solution:
32: Can a set of 12 pentominoes fit into a 5 x 12 box?
Yes! Here's one solution:
32: Can a set of 12 pentominoes fit into a 6 x 10 box?
Yes! Here's one solution:
33: Could a set of 12 pentominoes fit into a square box?
The smallest square box that would fit the 60 individual little squares that would make up the 12 pieces would be a box that was eight squares on a side, giving us 8 x 8 = 64 squares. That leaves 4 extra squares left empty.
34: Could a set of 12 pentominoes fit into a square box with the four corner squares removed?
Yes!
35: Could a set of 12 pentominoes fit into a square box with the four center squares removed?
Yes!
36: Could a set of 12 pentominoes fit into a square box with four edge squares removed?
Yes! TODO Insert picture.
37: Could a set of 12 pentominoes fit into a 5 x 12 box with the long, straight pentomino at the end?
Yes!
38: Could a set of 12 pentominoes fit into a 5 x 12 box with the long, straight pentomino in column two?
No. If the long, straight pentomino was in column two, that would mean that all of column one would have to be a single pentomino, and that pentomino would be a duplicate of the long, straight pentomino which we are not allowing.
39: Could a set of 12 pentominoes fit into a 5 x 12 box with the long, straight pentomino in column three?
No. If the long, straight pentomino was in column three, that would mean that columns one and two together were formed by two pentominoes. If you imagine marking off the space that the first pentomino used, the unused space would be symmetric and therefore those two pentominoes would be duplicates of each other, which we are not allowing.
40: Could a set of 12 pentominoes fit into a 5 x 12 box with the long, straight pentomino in column four?
No
41: Could a set of 12 pentominoes fit into a 5 x 12 box with the long, straight pentomino in column five?
No
42: Could a set of 12 pentominoes fit into a 5 x 12 box with the long, straight pentomino in column six?
No
43: Could a set of 12 pentominoes fit into a 5 x 12 box with the long, straight pentomino in column seven through twelve?
By symmetry, a long straight pentomino in column seven is the same question as if it was in column six, eight is the same as five, and so on.
44: Could a set of 12 pentominoes fit into a 5 x 12 box with the long, straight pentomino in row 2 near the center?
Maybe? TODO
45: Could a set of 12 pentominoes fit into a 5 x 12 box with the long, straight pentomino in row 2 near the end?
Maybe? TODO
46: Could a set of 12 pentominoes fit into a 5 x 12 box with the long, straight pentomino in row 1 near the center?
Maybe? TODO
47: Could a set of 12 pentominoes fit into a 5 x 12 box with the long, straight pentomino in row 1 near the end?
Maybe? TODO
48: Could a set of 12 pentominoes fit into a 3 x 20 box with the long, straight pentomino in row 1 at the end?
No - this would make two gaps of 1x5 that could only be filled by a 1x5 pentomino, which would be duplicates of each other, as well as duplicating the long, straight pentomino we started with.
49: Could a set of 12 pentominoes fit into a 3 x 20 box with the long, straight pentomino in row 1 somewhere that's not at the end?
Maybe? TODO
50: Could a set of 12 pentominoes fit into a 3 x 20 box with a cut across the box, so it's a 3x5 + 3x15 or a 3x10 + 3x10?
Probably not - there are only two 3x20 solutions, according to wikipedia, and they sound like they don't have cuts across the box.
TODO demonstrate 3x5 solutions
demonstrate 3x15 solutions
???: are there rectangular sub-boxes? 5x3 seems like it ought to be a sub-box (see above). I think 4x5 is probably a sub-box, leading to some rotations within solutions.
Consider my favorite layout, that's two 6x5 boxes
5x3 + 15x3 ???
10x3 + 10x3 ???
5x1 + 5x11 YES (trivially) (see Q37)
5x2 + 5x10 NO - 5x2 can't exist by yin/yang monad-y construction.
5x3 + 5x9 ???
5x4 + 5x8 ???
5x5 + 5x7 ???
5x6 + 5x6 YES
5x3 embedded within a larger rect ???
5x4 embedded within a larger rect ???
5x5 embedded within a larger rect ???
5x6 embedded within a larger rect? (5x6+5x6 works, 5x6 inside a 6x10 probably not)
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