The 3-1 rule of land combat is a military aphorism rooted in operations research that states that an attacking force should have a 3 to 1 advantage over a defending force in order to succeed. The reason the attacking force must be so large of course, is to account for losses that can occur when assaulting a fortified position. A smaller attacking force often assumes much greater risk by not having enough soldiers to create and exploit a breach; likewise, choice of terrain, cover, level of preparedness, area denial, distance to resupply, distance to reinforcements, and home field advantage (morale) is usually on the side of the defender.
1986 survey of 600 land battles from 1600 to 1973 by the U.S. Army Concepts Analysis Agency.
Statistics above (though arguably vague) suggest that the 3:1 rule is not always necessary for an attacking army to win (the attacker still won close to 50% when facing 1:1 odds). However the 3:1 rule might still be a good rule of thumb when planning offensive operations. When an invading army numbered over 3:1, they succeeded 74% of the time.
My critique and questions for researchers:
How valid is the 3:1 rule in the first place? (ex successful offensive operations led by Nazi Germany, but then their failure against the Soviets).
Does the 3:1 rule apply to all offensive ground operations, or only in cases of frontal assaults when facing a peer competitor?
Is mass and concentration of forces a fitting refutal to the 3:1 rule? Or is it an extension?
How does airpower alter the 3:1 rule?
As a general rule to land combat, what is the most effective force multiplier (tech, terrain, mobility, supply, firepower, coordination of forces, or actual numbers)?
When attempting a breakthrough, should you ever divide your forces? And what are the major risks?