## Tatami-Free Rooms

### Problem 256

Published on Saturday, 19th September 2009, 01:00 am; Solved by 494; Difficulty rating: 80%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.