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.
无法铺满榻榻米的房间
榻榻米是一种长方形的垫子，常常用来不重叠地铺满房间的地板。
假定现在只有唯一一种形状为1×2的榻榻米，显然这对于能够铺满榻榻米的房间的形状和大小都将有所限制。
在这个问题中，我们考虑的房间为长方形，相邻两边长为a和b，且面积s=a·b是偶数。
我们用“大小”这个词来表示房间的地板面积，此外，不失一般性，我们假定a ≤ b。
在铺榻榻米的时候必须遵循一个规则：不能有四个榻榻米共用一个顶点。
例如，考虑下面两种铺满4×4房间的铺法：
左边的铺法是可行的，而右边的则不是：有一个红色的”X“出现在当中，标记出被四个榻榻米共用的顶点。
因为这个原因，有些面积为偶数的房间依然不能铺满榻榻米：我们称之为无法铺满榻榻米的房间。
进一步地，我们定义T(s)是面积为s且无法铺满榻榻米的房间种数。
最小的无法铺满榻榻米的房间面积为s = 70，两边长为7×10。
其它所有面积为70的房间都能铺满榻榻米；这些房间的大小分别是：1×70， 2×35和5×14。
因此，T(70) = 1。
同样地，我们可以验证T(1320) = 5，因为当房间面积为s = 1320时，恰好有5种房间无法铺满榻榻米：
20×66，22×60，24×55，30×44和33×40。
事实上，s=1320是使得T(s) = 5的最小房间面积s。
求出使得T(s) = 200的最小房间面积s。