gavinksong Posted March 2, 2015 Report Share Posted March 2, 2015 Can you place eight 2x2 tiles on a standard 8x8 checkerboard such that there is no room for a ninth? There is an elegant solution to this problem. Quote Link to comment Share on other sites More sharing options...
0 gavinksong Posted March 4, 2015 Author Report Share Posted March 4, 2015 eight 2x2.jpg No room for a fifth. Board.png Consider the nine areas marked in red. No 2x2 piece can touch more than one of these areas. Hence with only 8 pieces placed, one of the areas must be available for a ninth piece. Quote Link to comment Share on other sites More sharing options...
0 Rainman Posted March 3, 2015 Report Share Posted March 3, 2015 Consider the nine areas marked in red. No 2x2 piece can touch more than one of these areas. Hence with only 8 pieces placed, one of the areas must be available for a ninth piece. 1 Quote Link to comment Share on other sites More sharing options...
1 bonanova Posted March 2, 2015 Report Share Posted March 2, 2015 No room for a fifth. 1 Quote Link to comment Share on other sites More sharing options...
0 gavinksong Posted March 3, 2015 Author Report Share Posted March 3, 2015 eight 2x2.jpg No room for a fifth. Bonanova! You just blew my mind. That simple thought never even occurred to me. Just because of that, I might consider it an alternative answer. However, I meant that the 2x2 tiles should be in line with the grid. I will not mark it solved until someone comes up with an elegant solution to this (clarified) problem. Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted March 3, 2015 Report Share Posted March 3, 2015 Yeah, what Rainman said. I figured my solution was illegal, and OP says something elegant is needed. But without going off-grid or rotating slightly, which gavinksong has now made explicitly unacceptable, I think it can't be done. I agree with Rainman. Nice proof. Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted March 4, 2015 Report Share Posted March 4, 2015 OK so I totally misread the OP, and Rainman provided the requested elegance. I read it as a challenge rather than as a question. Quote Link to comment Share on other sites More sharing options...
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gavinksong
Can you place eight 2x2 tiles on a standard 8x8 checkerboard such that there is no room for a ninth?
There is an elegant solution to this problem.
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