# 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.