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


Consider the following pentacube (made from 5 unit cubes):

Pentacube with base 1x2x2 block with one unit cube above the base

It is possible to pack four of these pentacubes to form a 2x2x5 rectangular block. And from that packing it is possible to pack any rectangular block whose three side lengths are 2i, 2j and 5k where i, j and k are arbitrary positive integers.

By volume considerations, for any packing of a rectangular block with these pentacubes, there must be a side length that is a multiple of 5. But is it possible that one or both of the other side lengths is an odd number? For example, is it possible to pack a 5x5x5 cube with these pentacubes?

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  • $\begingroup$ I'm having trouble even visualizing the 2x2x5, lol, it's not that trivial I guess. (I got it alrd now from the first answer) $\endgroup$
    – justhalf
    Commented Jun 30, 2023 at 5:42
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    $\begingroup$ If you add the three-dimensional tag and edit in the template header this could be a last-ditch entry for this month's MTC... $\endgroup$
    – Stiv
    Commented Jun 30, 2023 at 6:12
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    $\begingroup$ On Torsten Sillke's site there are lots of results for polycube packings including a page with results for this pentacube. $\endgroup$ Commented Jun 30, 2023 at 9:09

3 Answers 3

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Yes, for instance we can make a 2x7x15 block.

Put together two pentacubes to make a P pentomino two layers deep:
enter image description here

Then, arrange 21 P pentominoes in 2D to make the 7x15 rectangle below.

enter image description here

Image from https://polyominoes.org/data/5P

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    $\begingroup$ I like this solution because it solves a 3-dimensional puzzle using a 2-dimensional technique. $\endgroup$ Commented Jun 30, 2023 at 22:58
  • $\begingroup$ @Bass’ edit makes @xnor’s answer even better because it shows how the 2 level P pentomino is constructed. Also when you tap/click on the pentomino diagram you get a larger version of it (great for people with reduced vision). $\endgroup$ Commented Jul 1, 2023 at 20:27
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5x5x5 block:

is impossible: Consider the 27 gold cubelets in the following image:
5x5x5 with all cubes with only odd coordinates marked
Each pentacube can only cover 1 of these cubelets, so you need at least 27 pentacubes. However, those 27 pentacubes will have 135 cubelets in total, and there are only 125 cubelets of space in the 5x5x5 cube.

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    $\begingroup$ Great use of minecraft $\endgroup$
    – justhalf
    Commented Jul 1, 2023 at 11:36
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Here is a small one with sides of 2x3x5:

enter image description here

However, I don't believe that a 5x5x5 is possible. I do not have a proof, but the fact that it is mostly 2x2, which is hard to even make a base of 5x5, makes it not probable.

5 times the 2x3x5 together makes 5x5x6, which is the closest I get to 5x5x5. To get that, use the following base:

enter image description here

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  • $\begingroup$ Nice diagrams, what did you use to make those? $\endgroup$
    – xnor
    Commented Jun 30, 2023 at 20:42
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    $\begingroup$ I just googled "online 3d block builder", and this was the first site to come up :) cbc.ca/kids/games/play/3d-block-builder $\endgroup$
    – Lezzup
    Commented Jun 30, 2023 at 20:45

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