# Shuffle anti-squares of every even length from 24

Open problem settled, formally verified in Lean 4. Vamshi Jandhyala.

> Shuffle anti-squares exist in every even length from 24, as Grytczuk, Pawlik and Pleszczyński conjectured.

Canonical: https://vamshij.com/research/shuffle-anti-squares
Code and Lean proofs: https://github.com/jvvk/mathematics/tree/main/shuffle-anti-squares
Paper (PDF): https://github.com/jvvk/mathematics/blob/main/shuffle-anti-squares/paper/antisquares.pdf

A binary word is a shuffle square if its letters can be coloured with two colours so that each colour, read in order, spells the same word: `1100` is one (both copies spell `10`), `0110` is not. An even word, with an even number of each letter, is a shuffle anti-square if none of its cyclic rotations is a shuffle square. Grytczuk, Pawlik and Pleszczyński found that the shortest anti-squares have length 24 and conjectured that they exist in every even length from 24 upward ([arXiv:2308.13882](https://arxiv.org/abs/2308.13882), Conjecture 1).

The paper proves the conjecture with two explicit families,

- `U_k = 0^(k+9) 1 0^5 11 0^(k+9) 1 0^2 1 0 111` (length 2k + 34), with a short proof by hand, and
- `W_k = 0^(k+5) 1 0^2 1111 0^(k+4) 111 0 1111` (length 2k + 24),

and extends the first: multiplying every zero run of `U_k` by any λ and replacing every one by a block of μ ones, μ odd, still gives an anti-square (Theorem 4; both conditions are needed). Since one cut can only rotate a word, it follows that some even word of every length from 24 needs two cuts before its pieces can be rearranged into a shuffle square (Corollary 6). Grytczuk, Pawlik and Ruciński had found such a word of length 24 and conjectured that two cuts always suffice ([arXiv:2503.22043](https://arxiv.org/abs/2503.22043), Section 6.3); a computer check confirms this for every anti-square up to length 34.

Preprint, 8 October 2026. The author used AI tools (Claude, Anthropic) in this work, as described in the paper's acknowledgements, and takes full responsibility for its content.