Still open: the rank of 3 × 3 matrix multiplication

Multiplying two 2 × 2 matrices the schoolbook way takes 8 multiplications. In 1969 Volker Strassen did it with 7, and two years later Winograd proved 7 is the fewest possible. Applied to blocks, over and over, that one saved multiplication cuts an n × n product from n³ to n^2.807, because log₂ 7 = 2.807.

That 7 is a CP rank. The rule for multiplying 2 × 2 matrices is a 4 × 4 × 4 tensor with one 1 for each of the 8 schoolbook products. Each rank-one term of a CP decomposition costs one multiplication, so a rank-R decomposition is an algorithm with R multiplications. Strassen’s algorithm is seven rank-one terms that add up to the tensor exactly.

For 3 × 3 matrices the tensor is 9 × 9 × 9, with 27 ones, and nobody knows its rank. It is at least 19 (Bläser, 2003) and at most 23 (Laderman, 1976). Rank 21 would beat Strassen (n^2.771); rank 22 would not (n^2.814). Deep dive 20 turns the search into a game.