6
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This puzzle is part of the Monthly Topic Challenge #11: Now in 3D.


There are nine distinct known three-cell Garden of Eden patterns in the Life Without Death cellular automata, including the two well-known triominoes. The nine pieces are as follows:

* . *
. . .
* . .
* * .
. . .
* . .
* * .
. . .
. . *
* . .
. . *
* . .
* . .
. . *
. * .
. * .
* . *
* . .
. * *
* .
* *
* * *

Note that some of these pieces are disconnected, but you can't dismantle them. The cells inside each piece must retain their relative positions.


The task is, can these nine pieces (treated as 1-cell high prisms) be fit perfectly into a 3x3x3 cube?

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1
  • 1
    $\begingroup$ You can rotate or flip the puzzle pieces, but you can't dismantle them - though many of them were actually disconnected. $\endgroup$ Jun 5, 2023 at 2:05

1 Answer 1

4
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Is it possible?

Yes!

Labelling the Gardens of Eden for reference:

 R . R
 . . .
 R . .

 S S .
 . . .
 S . .

 T T .
 . . .
 . . T

 U . .
 . . U
 U . .

 V . .
 . . V
 . V .

 . W .
 W . W

 X . .
 . X X

 Y .
 Y Y

 Z Z Z


The following packing works:

 W V W    R W R    S S U
 T Y V    Z Z Z    U T T
 V Y X    R Y X    S X U

This was pretty much all manual trial and error. A trivial checkerboard colouring has the R and W patterns both mono-coloured (everything else is 2:1 or 1:2), so I tried keeping them on opposite colours for balance, which fairly quickly lead to the above solution. I suspect there are others.

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    $\begingroup$ I did a quick check by computer, and there are 44751 solutions (plus their rotations and reflections), so once you get over the confusing nature of disconnected pieces, it shouldn't be too hard to solve it by hand. $\endgroup$ Jun 5, 2023 at 13:51

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