The greedy player wins in 8, never 7

For a matrix, greedy peeling is optimal: the SVD takes the best rank-one layer, then the best of what is left. Try the same plan here — always play the move that leaves the fewest nonzero cells.

• Only 8 of the 128,000 first moves lower the count, and they are the tensor’s eight single cells. Another 176 keep it; 127,816 raise it.
• Greedy wins in 8, never in 7: it plays the schoolbook rule. Shown Strassen’s first move, its best reply is to take the move back, and it finishes in 10.
• No order avoids the climb: over all 5,040 orders of Strassen’s seven moves the peak is never below 10, and reaches 14 in some.

Nor is there a theorem to lean on in place of the SVD: finding the rank of a tensor is NP-hard (Håstad, 1990). The result belongs to one yardstick — counting nonzero cells — and other measures might rank moves differently.