Problem 256
Tatami-Free Rooms

Tatami are rectangular mats, used to completely cover the floor of a room, without overlap.

Assuming that the only type of available tatami has dimensions 1 2, there are obviously some limitations for the shape and size of the rooms that can be covered.

For this problem, we consider only rectangular rooms with integer dimensions a, b and even size s = a·b.
We use the term 'size' to denote the floor surface area of the room, and — without loss of generality — we add the condition a b.

There is one rule to follow when laying out tatami: there must be no points where corners of four different mats meet.
For example, consider the two arrangements below for a 4 4 room: The arrangement on the left is acceptable, whereas the one on the right is not: a red "X" in the middle, marks the point where four tatami meet.

Because of this rule, certain even-sized rooms cannot be covered with tatami: we call them tatami-free rooms.
Further, we define T(s) as the number of tatami-free rooms of size s.

The smallest tatami-free room has size s = 70 and dimensions 7 10.
All the other rooms of size s = 70 can be covered with tatami; they are: 1 70, 2 35 and 5 14.
Hence, T(70) = 1.

Similarly, we can verify that T(1320) = 5 because there are exactly 5 tatami-free rooms of size s = 1320:
20 66, 22 60, 24 55, 30 44 and 33 40.
In fact, s = 1320 is the smallest room-size s for which T(s) = 5.

Find the smallest room-size s for which T(s) = 200.

These problems are part of Project Euler and are licensed under CC BY-NC-SA 2.0 UK