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gavinksong

Checkerboard Challenge

Question

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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attachicon.gifeight 2x2.jpg

No room for a fifth.

 

 

attachicon.gifBoard.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.

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post-52528-0-12956500-1425421701_thumb.p

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.

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attachicon.gifeight 2x2.jpg

No room for a fifth.

 

Bonanova! You just blew my mind. That simple thought never even occurred to me.  :o

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.

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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.

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OK so I totally misread the OP, and Rainman provided the requested elegance. I read it as a challenge rather than as a question. :duh:

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