superprismatic Posted September 3, 2011 Report Share Posted September 3, 2011 Here's a 4×5 grid made up of squares: -------------------------- | | | | | | -------------------------- | | | | | | -------------------------- | | | | | | -------------------------- | | | | | | -------------------------- [/code] And here are 5 shapes, each made out of 4 squares: [code] Shape A: ------ | | ---------------- | | | | ---------------- Shape B: ----------- | | | ---------------- | | | ----------- Shape C: --------------------- | | | | | --------------------- Shape D: ------ | | ---------------- | | | | ---------------- Shape E: ----------- | | | ----------- | | | ----------- Can you fill the 4×5 grid with these five shapes? You are allowed to flip them and/or rotate any or all of them. Quote Link to comment Share on other sites More sharing options...

0 bushindo Posted September 4, 2011 Report Share Posted September 4, 2011 Here's a 4×5 grid made up of squares: -------------------------- | | | | | | -------------------------- | | | | | | -------------------------- | | | | | | -------------------------- | | | | | | -------------------------- And here are 5 shapes, each made out of 4 squares: Shape A: ------ | | ---------------- | | | | ---------------- Shape B: ----------- | | | ---------------- | | | ----------- Shape C: --------------------- | | | | | --------------------- Shape D: ------ | | ---------------- | | | | ---------------- Shape E: ----------- | | | ----------- | | | ----------- Can you fill the 4×5 grid with these five shapes? You are allowed to flip them and/or rotate any or all of them. Some thoughts, mostly 90% hand waving and 10% proof by exhaustion I don't think it is possible to perform this task. I wish I had an elegant proof, but unfortunately I don't. We'll just have to make do with proof by exhaustion. The first thing we need to do is reason where shape C would go. Shape C could be placed horizontally or vertically. So let's examine the two cases 1) If C is placed vertically, then it will take 1 entire column of the 4x5 grid. The optimal placement in this case would be at either end, and the problem reduces to placing shape A, B, D, and E on a 4x4 grid.a) Now the problem is where to place shape E on this 4x4 matrix. Because of rotations, there are really only 3 choices b) In each of the 3 cases above, we can easily work through the remaining 3 pieces and show that they don't fit. 2) If C is placed horizontally, then it must be placed at either the topmost of bottom-most row. It is easy to see that placing C in row 2 or 3 would fail to lead to a solution. So, let's say that we place C at the bottom row. There are 3 choices from herea) It is easy to work through the remaining cases and show by exhaustion that the task required by the OP is not possible. Quote Link to comment Share on other sites More sharing options...

0 witzar Posted September 4, 2011 Report Share Posted September 4, 2011 It is not possible. Think about 5x4 grid as a chessboard with 10 black squares and 10 white squares. Now try to paint with black and white tiles A-E. Each tile except D will have 2 white squares and 2 black squares, no matter how you paint it. But tile D will have 3 black and 1 white or 3 white and 1 black. Therefore it is not possible to cover with tiles A-E an area with the same number of white and black squares. Quote Link to comment Share on other sites More sharing options...

0 bushindo Posted September 4, 2011 Report Share Posted September 4, 2011 It is not possible. Think about 5x4 grid as a chessboard with 10 black squares and 10 white squares. Now try to paint with black and white tiles A-E. Each tile except D will have 2 white squares and 2 black squares, no matter how you paint it. But tile D will have 3 black and 1 white or 3 white and 1 black. Therefore it is not possible to cover with tiles A-E an area with the same number of white and black squares. That's brilliant, witzar. Good work! Quote Link to comment Share on other sites More sharing options...

## Question

## superprismatic

Here's a 4×5 grid made up of squares:

Can you fill the 4×5 grid with these

five shapes? You are allowed to flip

them and/or rotate any or all of them.

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## 3 answers to this question

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