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To be explicit, the shapes pictured below, with reflections permitted.

The 20 shapes to be arranged

Can these be packed into a rectangle?

This puzzle arose from discussion on r/mathmemes. No solution was posted (and I don't know the solution, either).

Some preliminary analysis:

  1. Adding the single square makes this impossible. The total area of the pictured shapes is 88, so adding one gives a prime number.
  2. There are no parity concerns, as the grid will require even parity (either 4x22 or 8x11) and the sum of piece parity is also even.
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  • $\begingroup$ There are no parity concerns whatever parity your selected grid requires, because there are odd-sized pieces involved, any one of which can be used to adjust parity. $\endgroup$
    – theonetruepath
    Commented Jul 1 at 8:16

3 Answers 3

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The answer is

enter image description here

Apologies for reusing OP's colour scheme. The big violet blob consists of the Z pentomino and the T tetris piece.

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    $\begingroup$ One solution out of many $\endgroup$
    – Retudin
    Commented Jul 1 at 6:26
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Here's a solution to the 4x22. It's made up of a 4x(2,3,4,6,7). 4x(1,2,3,4,5,7) and 4x(1,2,3,4,6,6) are both impossible.

enter image description here

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    $\begingroup$ The subrectangles also cleanly rearrange into a large number of 8×11 solutions. $\endgroup$
    – AxiomaticSystem
    Commented Jul 1 at 13:11
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A simpler method to see that 4x22 is possible:

First, it is known that there are many ways to pack 12 pentominoes into 4x15.

Then it suffices to show that we can pack 5 tetrominoes, 2 trominoes and 1 domino into 4x7.

While we cannot pack 5 tetrominoes into 4x5 (parity problem with T), there are many ways to pack 4x7 with the help of smaller pieces, including but not limited to:

TTTIIII
LTrr..i
LSSrOOi
LLSSOOi

OOiiiT.
OOSSTT.
LLLSSTr
LIIIIrr

ILL..OO
ILiiiOO
ILTSSrr
ITTTSSr

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