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.
(In reply to
re(2): Parity by Brian Smith)
A different parity argument. This time I will use stripes of four types:
ABCDABCDA
ABCDABCDA
ABCDABCDA
ABCDABCDA
ABCDABCDA
ABCDABCDA
ABCDABCDA
ABCDABCDA
ABCDABCDA
There are an equal number of B, C, and D; and an excess of A. Any 1x4 tile can cover four of a single type of square or one of each type of square.
So look at the totals for B, C and D being covered. The relative difference between any two totals for B, C or D must be a multiple of 4. But when all the 1x4 tiles are placed there is only one leftover square. If that square was one of B, C, or D then the difference would be 1, which is not a multiple of 4. So the leftover square cannot be any of B, C, or D types; it must be of type A.
Now rotate the stripes to run horizontally and the same conclusion applies. Then the leftover square must be one that is type A for both parities, which leaves the pattern I put out originally:
A***A***A
*********
*********
*********
A***A***A
*********
*********
*********
A***A***A
re(3): Parity
