“If our forces are ten to the enemy's one, surround him; if five to one, attack him; if twice as numerous, divide our army into two.”
Force application depends on the odds; the same move is right or wrong by a factor of two.
Chapter 3 turns from aims to arithmetic: ten to one, surround; five to one, attack; twice as numerous, divide your army. The ratios shock modern readers — overwhelming superiority before engaging — because Sun Tzu prices risk at full cost: a battle at par is a coin flip that spends your force either way. The instruction is to let the ratio choose the verb.
A firm with one dominant product and triple the competitor’s resources in a category has a 5:1 edge — the verb is "attack": commit marketing, distribution, and pricing to take the category outright. The same firm entering an adjacent category at rough parity has a different verb: divide — probe with a narrow wedge, learn, and never commit the main force to a coin flip. Most strategy failures are verbs applied at the wrong ratio.
The numbers are ancient field arithmetic, not modern doctrine — translate the principle, not the ratios. The lasting instruction is that advantage is method-specific: the same edge that justifies one move forbids another. And the ladder runs downward too: below parity, the correct verb — as the next verses say — is avoid or leave.