/sci/ - Science & Math

>>12609665
>>12613031
Alright, here's my proof of why it's impossible:

Lemma 0: The congruent parts cannot be a 2x4 rectangle.
The proof of this is trivial.

Lemma 1: The congruent parts cannot have a dimension of 5 (as in, a part either occupies both columns A and E or occupies both rows 0 and 4).
The proof of this is as follows: suppose the congruent parts have a dimension of 5. Without loss of generality, suppose one of the congruent parts occupied both columns A and E. Then the other two congruent parts cannot occupy both rows 0 and 4, as they would have to cross the first part, so they too must occupy both columns A and E. When the grid is cut, it is clear that there is a top part, a middle part, and a bottom part. Without loss of generality, suppose the removed square is below the middle part. If any of the middle part occupied row 4, it would be impossible for the top part to be above the middle part, so row 4 must be completely occupied by the top part. Then, in order to be congruent, the middle part occupies a complete row by itself. Call this row X. The space between row 4 and row X must be occupied by the three squares of the top part not in row 4 and, in order to maintain congruence between the top and middle part, either zero or three squares of the middle part not in row X. So the space must be occupied by either three or six squares. However, the space between row 4 and row X must be a multiple of five. Then it is a contradiction, and the congruent parts cannot have a dimension of 5.