It is the rule in war, if ten times the enemy's strength, surround them; if five times, attack them; if double, be able to divide them; if equal, engage them; if fewer, defend against them; if weaker, be able to avoid them.
Therefore the skillful leader subdues the enemy's troops without any fighting; he captures their cities without laying siege to them; he overthrows their kingdom without lengthy operations in the field.
So there are five ways of knowing who will win. Those who know when to fight and when not to fight are victorious. Those who discern when to use many or few troops are victorious. Those whose upper and lower ranks have the same desire are victorious.
Now the reason the enlightened prince and the wise general conquer the enemy whenever they move and their achievements surpass those of ordinary men is foreknowledge.
Humble words and increased preparations are signs that the enemy is about to advance. Violent language and driving forward as if to the attack are signs that he will retreat.
When the leader is morally weak and his discipline not strict, when his instructions and guidance are not enlightened, when there are no consistent rules, neighboring rulers will take advantage of this.
An army may be likened to water, for just as flowing water avoids the heights and hastens to the lowlands, so an army avoids strength and strikes weakness.
So the principles of warfare are: Do not depend on the enemy not coming, but depend on our readiness against him. Do not depend on the enemy not attacking, but depend on our position that cannot be attacked.
When he utilizes combined energy, his fighting men become, as it were, like rolling logs or stones... The energy developed by good fighting men is as the momentum of a round stone rolled down a mountain thousands of feet in height.