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I am playing with non flat pentacubes (i.e. 5-cube non-flat puzzle pieces), trying to fill all possible volumes of 60 cubes (then using 12 over the 17 possible pieces).
Up to now, I made it for all the volumes (meaning: 6x5x2, 10x3x2, 5x4x3) except the 2x2x15 one!

Does anyone know if at least one solution exists?

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  • $\begingroup$ Welcome to PSE! Just wanted to let you know that it is customary (not essential) to accept an answer that you find most useful. You can do this by clicking on the check/tick mark under the vote count around the top left of an answer. (It turns green after you click it.) $\endgroup$ Commented 19 hours ago

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You can tile a 2 × 2 × 5 grid as follows.

tiling 2 × 2 × 5 grid with four non-flat pentacubes

Stacking three of these vertically gives the desired tiling.

Update: the OP clarified in the comments below that all 12 pentacubes must be different. Here’s one way to do this. First, I’ll borrow @PebNischl’s answer to make a 2 × 2 × 5

one third of the full tiling

Next, I can make the following shape.

one sixth of the full tiling

This can be combined with its mirror image (which has J2 and N1) to make another 2 × 2 × 5. Finally, I can make the last shape above in a different way.

another sixth of the fill tiling

Once again, this can be combined with its mirror image (which has J4 and V1) to make another 2 × 2 × 5. Combining all the shapes gives a 2 × 2 × 15 with twelve different pentacubes.

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  • $\begingroup$ Many thanks for your answer! It is fine, but actually, I wasn't accurate enough: I meant with 12 "different" pieces ... $\endgroup$ Commented 23 hours ago
  • $\begingroup$ @ImageCreatorMaster. No worries. Please update the question with the above additional information. Other users shouldn’t have to read the comments under an answer to get the full question. $\endgroup$ Commented 23 hours ago
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    $\begingroup$ @ImageCreatorMaster. See my updated answer. $\endgroup$ Commented 22 hours ago
  • $\begingroup$ What software you used for this images? $\endgroup$ Commented 12 hours ago
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    $\begingroup$ @justhalf. Yeah I have to order the images manually but it’s easy to do that on Freeform. It also has edge-alignment, centre-alignment, spacing, etc., which make it easier to align the cubes. It also has shadow feature for any image, which creates a mild illusion that the cubes cast some shadow. $\endgroup$ Commented 12 hours ago
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Edit: This should be it:

CompleteSolution Essentially, you can build multiple 2x2x5-towers from unique cubes and stack those up.

Original post:

To get a solution from unique pentacubes, the solution from @Pranay can be modified slightly (if I understand the edit correctly):

One cube from the second and third pentacube need to be swapped around. Second row is rotated by 90° to see the arrangement better. Solution

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    $\begingroup$ +1. But I think the OP wants all 12 to be different. $\endgroup$ Commented 23 hours ago
  • $\begingroup$ Yes, I misunderstood the question. The new solution should be what was intended, but I see that you just beat me to it $\endgroup$ Commented 22 hours ago
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    $\begingroup$ Cool. We came up with the same solution. I updated my answer too. $\endgroup$ Commented 22 hours ago
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    $\begingroup$ Whaooo. Looks like we have serious guys in this forum. Many thanks to you both! 👍👏 $\endgroup$ Commented 17 hours ago

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