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Covering a chessboard (Posted on 2005-11-12) Difficulty: 2 of 5
You are given 21 3x1 rectangular pieces to cover an 8x8 chessboard. Since the board has 64 squares, which square on the chessboard must you cut out so that the 21 given pieces exactly cover the remaining 63 squares? Or it is impossible, no matter which square you remove?

See The Solution Submitted by pcbouhid    
Rating: 3.6667 (6 votes)

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Solution Count the Ways | Comment 13 of 15 |

There are 178 basic arrangements of the tiles, so that together with rotations and reflections there are 8 times this, or 1424. The "basic" ones considered, have the empty square in the third column of the third row, and have the piece that is covering the outer corner nearest this empty space placed horizontally. The reflection about the main diagonal would place that piece vertically. Rotations of 0, 90, 180 and 270 degrees clockwise constitute the four possible rotations for each of the two reflections (that is original plus its reflection).

aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd 
eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd
gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd
ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk
ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm
nnnooopq nnnopppq nnnopqqq nnnopqrs nooopppq nooopqqq nooopqrs nopppqqq nopppqrs nopqqqrs
rrrssspq rrrosssq rrropsss tttopqrs nrrrsssq nrrrpsss ntttpqrs norrrsss notttqrs noptttrs
tttuuupq tttouuuq tttopuuu uuuopqrs ntttuuuq ntttpuuu nuuupqrs notttuuu nouuuqrs nopuuurs
aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd 
eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd
gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd
ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk
ghlllmmm ghlllmmm ghlllmmm ghlllmno ghlllmno ghlllmno ghlllmno ghlmmmno ghlmnnno ghlmnooo
nopqrrrs nopqrsss nopqrstu pppqrmno pqqqrmno pqrrrmno pqrstmno pqlrrrno pqlmrrro pqlmnrrr
nopqttts nopqrttt nopqrstu sssqrmno psssrmno pqsssmno pqrstmno pqlsssno pqlmssso pqlmnsss
nopquuus nopqruuu nopqrstu tttqruuu ptttruuu pqtttuuu pqrstuuu pqtttuuu pqtttuuu pqtttuuu
aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd 
eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd eeefffcd
gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd gh iiicd
ghjjjkkk ghjjjkkk ghjjjklm ghjjjklm ghjjjklm ghjjjklm ghjjjklm ghjkkklm ghjklllm ghjklmmm
ghlmnooo ghlmnopq ghnnnklm ghnnnklm ghnnnklm ghnnnklm ghnopklm ghjnnnlm ghjknnnm ghjklnnn
pqlmnrst rslmnopq ooopqklm opppqklm opqqqklm opqrsklm qrnopklm opjqqqlm opjkqqqm opjklqqq
pqlmnrst rslmnopq rrrpqsss orrrqsss oprrrsss opqrsttt qrnopsss oprrrsss oprrrsss oprrrsss
pquuurst rstttuuu tttpquuu otttquuu optttuuu opqrsuuu qrtttuuu optttuuu optttuuu optttuuu
aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabbbcd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd 
eeefffcd eeefffcd eeefffcd eeefffcd eeefghcd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd
gh iiicd gh iiicd gh iiicd gh ijkcd ij fghcd gh biiid gh biiid gh biiid gh biiid gh biiid
ghjklmmm ghjklmmm ghjklmno ghlijkmn ijkfghlm ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk
ghjklnnn ghjklnop ghjklmno ghlijkmn ijknnnlm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm
opjklqrs qrjklnop pqjklmno oplqqqmn opkqqqlm nnnooopq nnnopppq nnnopqqq nnnopqrs nooopppq
optttqrs qrsssnop pqrrrsss oprrrsss oprrrsss rrrssspq rrrosssq rrropsss tttopqrs nrrrsssq
opuuuqrs qrtttuuu pqtttuuu optttuuu optttuuu tttuuupq tttouuuq tttopuuu uuuopqrs ntttuuuq
aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd 
eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd
gh biiid gh biiid gh biiid gh biiid gh biiid gh biiid gh biiid gh biiid gh biiid gh biiid
ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk
ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmno ghlllmno
nooopqqq nooopqrs nopppqqq nopppqrs nopqqqrs nopqrrrs nopqrsss nopqrstu pppqrmno pqqqrmno
nrrrpsss ntttpqrs norrrsss notttqrs noptttrs nopqttts nopqrttt nopqrstu sssqrmno psssrmno
ntttpuuu nuuupqrs notttuuu nouuuqrs nopuuurs nopquuus nopqruuu nopqrstu tttqruuu ptttruuu
aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd 
eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd
gh biiid gh biiid gh biiid gh biiid gh biiid gh biiid gh biiid gh biiid gh biiid gh biiid
ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjklm ghjjjklm ghjjjklm
ghlllmno ghlllmno ghlmmmno ghlmnnno ghlmnooo ghlmnooo ghlmnopq ghnnnklm ghnnnklm ghnnnklm
pqrrrmno pqrstmno pqlrrrno pqlmrrro pqlmnrrr pqlmnrst rslmnopq ooopqklm opppqklm opqqqklm
pqsssmno pqrstmno pqlsssno pqlmssso pqlmnsss pqlmnrst rslmnopq rrrpqsss orrrqsss oprrrsss
pqtttuuu pqrstuuu pqtttuuu pqtttuuu pqtttuuu pquuurst rstttuuu tttpquuu otttquuu optttuuu
aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd aaabcccd 
eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfffd eeebfghd
gh biiid gh biiid gh biiid gh biiid gh biiid gh biiid gh biiid gh biiid gh bijkd ij bfghd
ghjjjklm ghjjjklm ghjkkklm ghjklllm ghjklmmm ghjklmmm ghjklmmm ghjklmno ghlmijkn ijklfghm
ghnnnklm ghnopklm ghjnnnlm ghjknnnm ghjklnnn ghjklnnn ghjklnop ghjklmno ghlmijkn ijklnnnm
opqrsklm qrnopklm opjqqqlm opjkqqqm opjklqqq opjklqrs qrjklnop pqjklmno oplmqqqn opklqqqm
opqrsttt qrnopsss oprrrsss oprrrsss oprrrsss optttqrs qrsssnop pqrrrsss oprrrsss oprrrsss
opqrsuuu qrtttuuu optttuuu optttuuu optttuuu opuuuqrs qrtttuuu pqtttuuu optttuuu optttuuu
aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd 
eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff
gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii
ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk
ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm ghlllmmm
nnnooopq nnnopppq nnnopqqq nnnopqrs nooopppq nooopqqq nooopqrs nopppqqq nopppqrs nopqqqrs
rrrssspq rrrosssq rrropsss tttopqrs nrrrsssq nrrrpsss ntttpqrs norrrsss notttqrs noptttrs
tttuuupq tttouuuq tttopuuu uuuopqrs ntttuuuq ntttpuuu nuuupqrs notttuuu nouuuqrs nopuuurs
aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd 
eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff
gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii
ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk ghjjjkkk
ghlllmmm ghlllmmm ghlllmmm ghlllmno ghlllmno ghlllmno ghlllmno ghlmmmno ghlmnnno ghlmnooo
nopqrrrs nopqrsss nopqrstu pppqrmno pqqqrmno pqrrrmno pqrstmno pqlrrrno pqlmrrro pqlmnrrr
nopqttts nopqrttt nopqrstu sssqrmno psssrmno pqsssmno pqrstmno pqlsssno pqlmssso pqlmnsss
nopquuus nopqruuu nopqrstu tttqruuu ptttruuu pqtttuuu pqrstuuu pqtttuuu pqtttuuu pqtttuuu
aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd 
eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff
gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii gh bciii
ghjjjkkk ghjjjkkk ghjjjklm ghjjjklm ghjjjklm ghjjjklm ghjjjklm ghjkkklm ghjklllm ghjklmmm
ghlmnooo ghlmnopq ghnnnklm ghnnnklm ghnnnklm ghnnnklm ghnopklm ghjnnnlm ghjknnnm ghjklnnn
pqlmnrst rslmnopq ooopqklm opppqklm opqqqklm opqrsklm qrnopklm opjqqqlm opjkqqqm opjklqqq
pqlmnrst rslmnopq rrrpqsss orrrqsss oprrrsss opqrsttt qrnopsss oprrrsss oprrrsss oprrrsss
pquuurst rstttuuu tttpquuu otttquuu optttuuu opqrsuuu qrtttuuu optttuuu optttuuu optttuuu
aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd 
eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff
gh bciii gh bciii gh bciii gh bcijk gh bcijk gh bcijk gh bcijk gh bcijk gh bcijk gh bcijk
ghjklmmm ghjklmmm ghjklmno ghlllijk ghlllijk ghlllijk ghlllijk ghlllijk ghlllijk ghlllijk
ghjklnnn ghjklnop ghjklmno ghmmmijk ghmmmijk ghmmmijk ghmmmijk ghmmmijk ghmmmijk ghmmmijk
opjklqrs qrjklnop pqjklmno nnnooopq nnnopppq nnnopqqq nnnopqrs nooopppq nooopqqq nooopqrs
optttqrs qrsssnop pqrrrsss rrrssspq rrrosssq rrropsss tttopqrs nrrrsssq nrrrpsss ntttpqrs
opuuuqrs qrtttuuu pqtttuuu tttuuupq tttouuuq tttopuuu uuuopqrs ntttuuuq ntttpuuu nuuupqrs
aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd 
eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff eeebcfff
gh bcijk gh bcijk gh bcijk gh bcijk gh bcijk gh bcijk gh bcijk gh bcijk gh bcijk gh bcijk
ghlllijk ghlllijk ghlllijk ghlllijk ghlllijk ghlllijk ghlllijk ghlllijk ghlmnijk ghlmnijk
ghmmmijk ghmmmijk ghmmmijk ghmmmijk ghmmmijk ghmmmijk ghmnoijk ghmnoijk ghlmnijk ghlmnijk
nopppqqq nopppqrs nopqqqrs nopqrrrs nopqrsss nopqrstu pqmnorrr pqmnorst oplmnqqq oplmnqrs
norrrsss notttqrs noptttrs nopqttts nopqrttt nopqrstu pqmnosss pqmnorst oprrrsss optttqrs
notttuuu nouuuqrs nopuuurs nopquuus nopqruuu nopqrstu pqtttuuu pquuurst optttuuu opuuuqrs
aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd 
eeebcfgh eeebcfgh eeebcfgh eeebcfgh eeebcfgh eeebcfgh eeebcfgh eeebcfgh eeebcfgh eeebcfgh
ij bcfgh ij bcfgh ij bcfgh ij bcfgh ij bcfgh ij bcfgh ij bcfgh ij bcfgh ij bcfgh ij bcfgh
ijkkkfgh ijkkkfgh ijkkkfgh ijkkkfgh ijkkkfgh ijkkkfgh ijkkkfgh ijkkkfgh ijkkkfgh ijkkkfgh
ijlllmmm ijlllmmm ijlllmmm ijlllmmm ijlllmmm ijlllmmm ijlllmmm ijlllmmm ijlllmmm ijlllmmm
nnnooopq nnnopppq nnnopqqq nnnopqrs nooopppq nooopqqq nooopqrs nopppqqq nopppqrs nopqqqrs
rrrssspq rrrosssq rrropsss tttopqrs nrrrsssq nrrrpsss ntttpqrs norrrsss notttqrs noptttrs
tttuuupq tttouuuq tttopuuu uuuopqrs ntttuuuq ntttpuuu nuuupqrs notttuuu nouuuqrs nopuuurs
aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd 
eeebcfgh eeebcfgh eeebcfgh eeebcfgh eeebcfgh eeebcfgh eeebcfgh eeebcfgh eeebcfgh eeebcfgh
ij bcfgh ij bcfgh ij bcfgh ij bcfgh ij bcfgh ij bcfgh ij bcfgh ij bcfgh ij bcfgh ij bcfgh
ijkkkfgh ijkkkfgh ijkkkfgh ijkkkfgh ijkkkfgh ijkkkfgh ijkkkfgh ijkkkfgh ijkkkfgh ijkkkfgh
ijlllmmm ijlllmmm ijlllmmm ijlllmno ijlllmno ijlllmno ijlllmno ijlmmmno ijlmnnno ijlmnooo
nopqrrrs nopqrsss nopqrstu pppqrmno pqqqrmno pqrrrmno pqrstmno pqlrrrno pqlmrrro pqlmnrrr
nopqttts nopqrttt nopqrstu sssqrmno psssrmno pqsssmno pqrstmno pqlsssno pqlmssso pqlmnsss
nopquuus nopqruuu nopqrstu tttqruuu ptttruuu pqtttuuu pqrstuuu pqtttuuu pqtttuuu pqtttuuu
aaabcddd aaabcddd aaabcddd aaabcddd aaabcddd aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef 
eeebcfgh eeebcfgh eeebcfgh eeebcfgh eeebcfgh gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef
ij bcfgh ij bcfgh ij bcfgh ij bcfgh ij bcfgh hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef
ijkkkfgh ijkkkfgh ijklmfgh ijklmfgh ijklmfgh hijjjkkk hijjjkkk hijjjkkk hijjjkkk hijjjkkk
ijlmnooo ijlmnopq ijklmnnn ijklmnnn ijklmnop hilllmmm hilllmmm hilllmmm hilllmmm hilllmmm
pqlmnrst rslmnopq opklmqqq opklmqrs qrklmnop nnnooopq nnnopppq nnnopqqq nnnopqrs nooopppq
pqlmnrst rslmnopq oprrrsss optttqrs qrsssnop rrrssspq rrrosssq rrropsss tttopqrs nrrrsssq
pquuurst rstttuuu optttuuu opuuuqrs qrtttuuu tttuuupq tttouuuq tttopuuu uuuopqrs ntttuuuq
aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef 
gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef
hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef
hijjjkkk hijjjkkk hijjjkkk hijjjkkk hijjjkkk hijjjkkk hijjjkkk hijjjkkk hijjjkkk hijjjkkk
hilllmmm hilllmmm hilllmmm hilllmmm hilllmmm hilllmmm hilllmmm hilllmmm hilllmno hilllmno
nooopqqq nooopqrs nopppqqq nopppqrs nopqqqrs nopqrrrs nopqrsss nopqrstu pppqrmno pqqqrmno
nrrrpsss ntttpqrs norrrsss notttqrs noptttrs nopqttts nopqrttt nopqrstu sssqrmno psssrmno
ntttpuuu nuuupqrs notttuuu nouuuqrs nopuuurs nopquuus nopqruuu nopqrstu tttqruuu ptttruuu
aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef 
gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef
hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef
hijjjkkk hijjjkkk hijjjkkk hijjjkkk hijjjkkk hijjjkkk hijjjkkk hijjjklm hijjjklm hijjjklm
hilllmno hilllmno hilmmmno hilmnnno hilmnooo hilmnooo hilmnopq hinnnklm hinnnklm hinnnklm
pqrrrmno pqrstmno pqlrrrno pqlmrrro pqlmnrrr pqlmnrst rslmnopq ooopqklm opppqklm opqqqklm
pqsssmno pqrstmno pqlsssno pqlmssso pqlmnsss pqlmnrst rslmnopq rrrpqsss orrrqsss oprrrsss
pqtttuuu pqrstuuu pqtttuuu pqtttuuu pqtttuuu pquuurst rstttuuu tttpquuu otttquuu optttuuu
aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef aaabcdef   
gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef gggbcdef  
hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef hi bcdef  
hijjjklm hijjjklm hijkkklm hijklllm hijklmmm hijklmmm hijklmmm hijklmno  
hinnnklm hinopklm hijnnnlm hijknnnm hijklnnn hijklnnn hijklnop hijklmno  
opqrsklm qrnopklm opjqqqlm opjkqqqm opjklqqq opjklqrs qrjklnop pqjklmno  
opqrsttt qrnopsss oprrrsss oprrrsss oprrrsss optttqrs qrsssnop pqrrrsss  
opqrsuuu qrtttuuu optttuuu optttuuu optttuuu opuuuqrs qrtttuuu pqtttuuu  

 


  Posted by Charlie on 2005-11-13 19:18:14
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