The Brown-Freedman-Halbeisen-Hungerbühler-Pirillo-Varricchio problem asks, is there an infinite word over a finite subset of N, the non-negative integers, containing no two consecutive blocks of the same length and the same sum?

The question was apparently first raised by Brown and Freedman in a 1987 paper, then independently by Pirillo and Varricchio in a 1994 paper, and by Halbeisen and Hungerbühler in 2000.

It follows from results of Dekking that such a word exists avoiding four consecutive blocks.

Recent results of Cassaigne, Currie, Schaeffer, and Shallit (2011) show that such a word exists avoiding three consecutive blocks.

-- JeffreyShallit - 13 Jul 2011