Fill a 5x5x5 box with 25 identical pieces, each made of five unit cubes: a piece of two units glued to a...

geometry/tiling/pentomino.cube New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

Fill a 5x5x5 box with 25 identical pieces, each made of five unit cubes:
a piece of two units glued to a piece of three units, so that the
resulting piece is four units long and two units high (an
N-pentomino).

Show solution

Here are the 3 solutions I found to filling a 5x5x5 cube with 25
N-pentominoes.

AAABB JBBBL MNOPL RSTLL XXXLY
CCAAD JCCCG MNOPP RSTUV RSXXY
EEEDD JJEEG MNOOP MNTUV RSTYY
FFFDG KJFFG KQQOP MNUUV RSTYV
HIIDG HHIII KHQQQ KHUWW KWWWV

The top two layers (pieces denoted ABCDEFGIJ) form a symmetrical block
so could be replaced by its mirror image, to get a second solution.

Another solution is:

AACCC JCCKL JJRKS MJUKK MJVVK
BAAAD MNEOL MQROS MTUOO VVVWO
BBEDD FNELL FQRRS FTUUS XXXWW
FBEDH FNNLH GQQRH GTTUS YYXXW
GBEDI GPNII GPQIH PPTIH PYYYW

There is apparently another solution, but I don't know it.

-- Jaap Scherphuis <jaap@ermine.ox.ac.uk>

[ Editor: the reply "Wooden puzzle - complete solution" (January 1997)
to Jaap Scherphuis's posting above. ]

Here are the 4 unique solutions (i.e. after discarding all rotations
and reflections).  Three of them match the solutions listed above after
doing the necessary rotations/reflections and relabelling.

AAALL   BLLLX   CMMMX   DEEEX   EEOOO
FFAAW   BFFFY   CNNMM   DHHHX   DOOVX
GGGWW   BBGGY   CCNNN   HHPPP   DPPVV
IIIWY   JBIIY   JCQQQ   JQQRR   DRRRV
JSSWY   JKSSS   KKTTT   KTTUU   KUUUV

AAAKK   BKKKU   CLQTU   DMRUU   EEEUW
FFAAV   BFFFX   CLQTT   DMRSY   DMEEW
GGGVV   BBGGX   CLQQT   CLRSY   DMRWW
HHHVX   IBHHX   INNQT   CLSSY   DMRWY
JOOVX   JJOOO   IJNNN   IJSPP   IPPPY

AAAUV   BKRUU   CKRRU   DLLRU   EEERX
FGAAV   BBSTY   CKMMM   DKLLL   DKEEX
FGSVV   FBSTY   CMMNN   CNNNX   DOOOX
GGSVY   FBTTY   HHHPP   CPPPX   DQQOO
GISWY   FITWW   IIHHW   IJJJW   JJQQQ

AAAJJ   BJJJT   BBPTT   CBPTW   DBPTX
EGAAV   EHPQY   EEPQW   CEQQW   DDQUX
FGKVV   FHKRY   FKKRW   CKRRX   CDRUX
GGLVY   HHLSY   FLLSW   FLSSX   CDSUU
GMMVY   HIMMM   IINNN   INNOO   IOOOU

-- Ilan Mayer <ilan@isgtec.com>, January 1997