A square with side length 49 is completely tiled with the following non-overlapping shapes: 600 1×4 rectangles and a unit square. Find the number of possible locations of the unit square.
The important part about the size of the square it is of the form 4x+1: 49 = 4*12+1. I will illustrate using 9=4*2+1 but the argument will be the same.
Apply a four color stripe parity on the diagonal.
ABCDABCDA
BCDABCDAB
CDABCDABC
DABCDABCD
ABCDABCDA
BCDABCDAB
CDABCDABC
DABCDABCD
ABCDABCDA
There is an equal number of B, C, and D each; and one more A than the others. So the unit square is one of the A's.
Now apply the parity going along the other diagonal. This creates a different set of A's; and the unit square must be common to both.
That leaves this pattern.
A***A***A
*********
*********
*********
A***A***A
*********
*********
*********
A***A***A
For the size 9 (9=4*2+1) case as illustrated then there is (2+1)^2=9 possible places for the unit square. Then for the size 49 (49=4*12+1) there will be (12+1)^2=169 possible places for the unit square.
Parity
