A recurring version of the 11 x 11 puzzle asks whether an 11 x 11 space can be divided into smaller integer squares where no two squares share the same dimension. This question sits within the famous mathematical challenge known as "squaring the square."
In the 1930s, mathematicians Leonard Brooks, Cedric Smith, Arthur Stone, and William Tutte translated this geometric dilemma into graph theory. They represented square tilings as electrical networks: each horizontal boundary line between squares becomes a node, and each square acts as a resistor carrying current equal to its side length.
Using this electrical analogy alongside automated proof checking, researchers proved that a "simple perfect squared square", a square divided into smaller squares of all different sizes, with no sub-rectangle formed by a subset of them, cannot exist in small dimensions.
The lowest possible order for a simple perfect squared square was discovered by A. J. W. Duijvestijn in 1978. It requires 21 distinct smaller squares fitted inside an outer square measuring 112 x 112. An 11 x 11 boundary is far too constrained. The total area of 121 cannot be written as the sum of distinct integer squares:
$$s1^2 + s2^2 + \dots + s_k^2 = 121$$
Because any distinct square set containing integers greater than 1 quickly overshoots 121 (for instance, $7^2 + 6^2 + 5^2 = 49 + 36 + 25 = 110$, leaving only 11 units, which cannot be formed by remaining distinct squares), the partition fails instantly. The 11 x 11 grid admits uniform tilings (121 squares of size 1, or non-distinct subsets), but the claims of unique, elegant non-uniform packing circulating online are mathematically impossible.
