One of the major myths about war is that it is a field where human cleverness is routinely exercised, where generals deploy troops like chess pieces and outsmart each other with gambits. Certainly there have been a few instances of clever strategic and tactical choices throughout history; my favorite is the ancient general who decoyed the enemy's cavalry off the field, then posted a line of elephants to frighten the horses so that they would not come back. (This has been called the only successful use of the otherwise unmanageable elephant in warfare.)
However, close analysis of the average war--certainly of modern wars-- indicates that it involves not calculus at all but simple addition and subtraction. While there are many famous stories about World War II strategy and tactics, any close study of that war inevitably reveals the truth: the Allies won because Hitler took on too many adversaries at once; the winner, as in most wars, was the side which could lose the most men and machines and keep fighting. In the end, it was the immense supply of young American men and the factories back home producing guns and aircraft that won World War II, and not any clever decisions made by Eisenhower or Patton.
A good place to start any study of World War II is with the paratroops--begin your study where the myth is strongest. The paratroopers have the reputation of being the bravest men in the war, and indeed they were. Anyone who would strap so many pounds of equipment and supplies to his body, jump out of an airplane and descend helplessly and slowly through withering enemy fire, is either a hero or an idiot (Plato pointed out many centuries ago that the distinction is not always clear.) But the corresponding myth--that the paratroopers were the fine scalpel of sophisticated tactics--dissolves upon close examination. In fact, the whole concept of paratroops becomes ridiculous in light of what actually happened. A parachutist has limited control over his descent, especially if the pilot has dumped him in the wrong place, and there is little guarantee that the members of a unit will land together. For an excruciating long time as he descends he is a plainly visible and helpless target for any enemy in the vicinity, and the moment he lands he must struggle to free himself from his parachute. If he lands in water he is completely helpless and will probably drown immediately. Parachutes may be a good way to sneak a few commandos into enemy territory, but are a really foolish way to deploy large groups of men. Hitler, more pragmatic than we were, never used his paratroops again after Crete, where they were slaughtered in large numbers. Read any book on the Normandy invasion, or watch the movie The Longest Day, and you will discover that pilots panicked and dropped their loads miles from the drop zone, that gliders crashed and split open, and that some of the drop zones turned out to be marshes in which the heavily laden troops sank and drowned. In fact, no unit landed intact with its leaders and equipment, and was able to carry out its original mission; the story of the airborne units at Normandy is the story of isolated men hooking up with men from other units and even with troops of other nationalities, moving in an approximate direction towards the often distant goal, and improvising missions to carry out. While most of the troops seem to have fought extremely bravely, they certainly had good reasons in light of what happened to think of their leaders as idiots; someone sitting comfortably in front of a map in England had imagined that it would be possible to land intact units right on the drop zones in relative safety. If the infantry hadn't broken through from the beaches, of course, the Normandy airborne experience would simply have been another horrifying failure like the later Operation Market Garden, where the tanks weren't able to get through Belgium in time to support the airborne units dropped into Holland. The most powerful images in The Longest Day, nicely representative of the horror and helplessness of airborne deployment, are of the paratrooper hanging from a church steeple, watching as one of his colleagues falls into the flames of a burning house and another goes down a well.
There is no dispute possible about the fact that we won in Normandy because we had more young and fresh troops, more guns, tanks, and airplanes than the enemy, who had no planes or tanks in that part of France and whose troops were largely old men and young boys after years of war and a disastrous invasion of Russia. War is not chess; it is an exercise in brute strength, won by the side which can afford to waste more lives and material than the other.
For some related thoughts, see Nuclear Strategy is an Oxymoron.