[superprismatic]

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.

[bushindo]

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 here

a) It is easy to work through the remaining cases and show by exhaustion that the task required by the OP is not possible.

[witzar]

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.

[bushindo]

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!