TAOCP 7.2.2.1 Exercise 321
A _one-sided tetrabolo_ is a tetromino considered up to translation and rotation, but not reflection.
Section 7.2.2.1: Dancing Links
Exercise 321. [42] (T. H. O'Beirne, 1962.) In how many ways can a square be formed from the eight one-sided tetraboloes and their mirror images?
Verified: yes
Solve time: 1m02s
Setup
A one-sided tetrabolo is a tetromino considered up to translation and rotation, but not reflection. Each such piece covers exactly $4$ unit squares. Its mirror image is a distinct piece of the same area.
The problem asks for the number of ways to form a square using eight one-sided tetraboloes together with their mirror images, with each piece used exactly once in a non-overlapping tiling.
Let the total number of pieces be $8$. Each piece has area $4$, so any admissible tiling covers an area
$$ A = 8 \cdot 4 = 32. $$
A necessary condition for the existence of a square tiling is that $32$ equals $n^2$ for some integer $n$, where $n$ is the side length of the square.
Solution
Assume a square tiling exists. Let the side length be $n$. The square then consists of exactly $n^2$ unit cells, hence
$$ n^2 = 32. $$
The integer squares nearest to $32$ are $25 = 5^2$ and $36 = 6^2$, and no integer $n$ satisfies $n^2 = 32$.
Thus no square has area $32$, so no decomposition of a square into $8$ tetraboloes of area $4$ each can exist.
Since the existence of a tiling is impossible, the number of such tilings is $0$.
Therefore,
$$ \boxed{0}. $$
This completes the proof. ∎
Verification
Each tetrabolo, whether one-sided or mirrored, covers exactly $4$ unit squares. Eight such pieces cover exactly $32$ unit squares, independent of geometry or orientation. Any square region in the integer grid has area $n^2$ for some integer $n$, and $32$ is not a perfect square. Hence no candidate square region can match the required area, so no exact cover solution exists.
Notes
The argument depends only on area divisibility and does not require any structural analysis of the tetromino shapes or a search via exact cover machinery. In TAOCP terms, the corresponding exact cover instance has no feasible solution because the primary constraint set includes a single item representing total area that cannot be matched by any square configuration.