# Pairs of Pairs of Pentominoes

Split the 12 pentominoes into three sets of four. Can you pair up pentominoes so that each set makes two of the same shape?

For instance, one of your three sets could look like this:

That uses the L, P, F, and U pentominoes, meaning you'd no longer be able to use those in any other sets.

• For those that don't know them all offhand. Dec 23, 2016 at 20:29
• I need more "P"s! Dec 23, 2016 at 21:10
• Are we allowed to have holes in the middle of the shape? Dec 24, 2016 at 5:14
• @wildBillMunson: Sure, if you want!
– Deusovi
Dec 24, 2016 at 6:08
• Haha as it turned out, I didn't need it! GREAT puzzle!! :D Dec 24, 2016 at 6:34

Finally arrived at this solution after playing around on https://www.scholastic.com/blueballiett/games/pentominoes_game.htm for way longer than I care to admit!!!! :)

First pair (UI-TF)

Second pair (WX-PY)

Third pair (VZ-LN)

• That was my solution too!
– Deusovi
Dec 24, 2016 at 6:46
• Awesome! You really hooked me with this puzzle! It took me hours to find this!!!! And now I look at it, and it seems so simple and elegant!! :D Dec 24, 2016 at 6:48

If you want solutions without holes, then wildBillMunson's answer of UITF/WXPY/VZLN is the only partition possible of the pentominoes into sets of 4 that form two of the same shape, though you actually have four options for the LZVN set:

If you allow holes, however, there are actually three more partitions of pentominoes into 3 sets that work: FLXY/IUTZ/VWPV, FTPZ/ILNW/VXUY, and FVNW/IUTZ/LYPX:

Some of those sets have more than one possible way to split them into pairs, but interestingly enough, LN-VZ is the only set of two pairs that can produce more than one final shape! All others have a unique solution.

Here's my Python code, if anyone is interested in how I got this. I'm not 100% convinced it's bug-free, so there might be extra configurations beyond the ones shown here.

Partial,

Every set I found needs the U.. quite frustrating. Here is what I got so far, maybe it can be useful to someone until I find more.
u+y & v+x, p+t & f+u, v+y & u+i, u+x & p+f

• I don't know how long you've been at it, but I finally found a solution (just posted it) after hours and hours of moving pieces around randomly on scholastic.com/blueballiett/games/pentominoes_game.htm. Wow I probably shouldn't have spent so much time on this! Dec 24, 2016 at 6:31