Different order.
Same quantities.
Hidden structure.
Can a word avoid this forever?
Look at what each half contains
Both halves use the same letters, the same number of times. Only the order changes.
First half
a ×1 b ×1
Second half
a ×1 b ×1
The counts match. The order does not.
Mathematicians call this an abelian square: two adjacent blocks with the same letter counts, in any order.
It happens with longer blocks too.
How long can it be avoided?
Over three letters, a verified finite example reaches 25,379 letters while avoiding every abelian square whose two halves each have length at least two.
The verified words came from V. Keränen and A. Gavrilenko; this project checked them rather than discovered them.
A finite example, however long, is still finite. It does not show that an infinite word exists.
If the rule is weakened so that only abelian squares whose halves have length six or more must be avoided, an infinite ternary word is known.
What remains open is whether an infinite ternary word can avoid them already from half-length two onward — equivalently, whether its only abelian squares can be the single-letter repetitions aa, bb and cc. This question is attributed to Mäkelä in the literature.
ABELISK — Hidden Echoes
A puzzle built on the same mathematics. Symbols repeat in ways you cannot see until you count them — and your task is to break the repetition.
Open means inspectable
Every figure on this site traces to a numbered row in a public claim ledger, with its source and the date it was last checked. Results that turned out to be wrong are not deleted — they stay on the record, with the reason.