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Mathematical Mysteries in WWII

  During World War II , when Germany launched an offensive against several countries such as France, British Prime Minister Churchill, at the request of France, used more than a dozen aircraft of the air defense team to fight against Germany. These aircraft squadrons must be maintained and operated from airfields on the mainland. The British aircraft suffered heavy losses in the air battle. At the same time, the French Prime Minister asked to continue to send 10 more squadrons of aircraft. Churchill decided to agree to the request. After the cabinet knew about the matter, it hired mathematicians to analyze and predict, and established a regression prediction model based on the statistics of dispatched aircraft and battle-damaged aircraft. After a quick study, it was found that if the loss rate of the replenishment rate remains unchanged, the number of aircraft will decline very quickly. In one sentence, it can be summed up as “if the current loss rate is lost for two weeks, one British “Hurricane” fighter in France will be lost. does not exist anymore”, asking the cabinet to veto the decision. In the end, Churchill agreed to this request, and except for the 3 squadrons remaining in France, all the rest of the planes returned to England to retain their strength for the next British defense.
  
  Ingeniously deal with Japanese bombings In the early days of the Pacific War in
  
  World War II , U.S. ships were repeatedly attacked by Japanese aircraft, and the loss rate was as high as 62%. The U.S. military urgently dispatched a large number of mathematicians to quantitatively analyze 477 battle cases, and came to two conclusions: First, when the Japanese aircraft adopted high-altitude dive bombing, the loss rate of the U.S. ships using the rapid swing evasion tactics was 20%. The loss rate is 100%; the second is that when the Japanese aircraft adopted low-altitude dive bombing, the average loss of the U.S. ships using rapid swing and slow swing was 57%. The U.S. military found the best method based on the principle of maximum and minimum in game theory: when an enemy plane strikes, it adopts a rapid swing evasion tactic. It is estimated that this decision by the US military has reduced the ship loss rate from 62% to 27%.
  
  Accurately estimate the approach route of the Japanese ships
  
  During World War II in New Guinea, the U.S. military received information that the Japanese would send a large escort fleet to Lae, New Guinea, from the port of Rabaul on the east coast of New Britain. The Japanese fleet may take two routes, the voyage is two days. Among them, the northern route is cloudy and foggy, and the visibility is not easy to observe; the southern route has good visibility and is easy to observe. The U.S. military also has two options for action, namely, concentrating the main aviation force on the north-south route for reconnaissance and bombing. If the Japanese army chooses to take the northern route and the US military also chooses the northern route, they will only have two days of bombing time at most, and there may even be no bombing time at all due to the influence of the weather; The line reconnaissance was delayed for a day, and there was only one day of bombing time at most, and there might even be no bombing time at all due to the influence of the weather. If the Japanese army chooses to follow the southern route and the U.S. military chooses the northern route, because the reconnaissance on the northern route is delayed for a day, they will have a day of bombing; Therefore, if the Japanese chose to follow the northern route, they would be bombed for 0 to 3 days; if the Japanese chose to follow the southern route, they would be bombed for 3 days. From this, the U.S. military concluded that the Japanese army would take the northern route. This is indeed the case. The Japanese fleet suffered heavy losses.
  
  Avoid German Submarines
  
  Reasonably Before 1943, Anglo-American convoys in the Atlantic were frequently attacked by German submarines. At that time, the strength of the United Kingdom and the United States was limited, and they were unable to send more escort ships. For a time, the German “submarine warfare” made the Allies devastated. To this end, a U.S. Navy admiral went to consult several mathematicians. Mathematicians, using probability analysis, found that the encounter between the fleet and the enemy submarine was a random event. From a mathematical point of view, this problem has a certain law: the smaller the size of a certain number of ship formations, the greater the number of formations; the greater the number of formations, the greater the probability of encountering the enemy. The U.S. Navy accepted the mathematician’s suggestion and ordered the fleet to gather in the designated sea area, then collectively pass through the dangerous sea area, and then each sailed to the predetermined port. As a result, the probability of the Allied fleet being attacked and sunk was reduced from the original 25% to 1%. Greatly reduced losses.
  
  Calculating the depth of explosion of depth charges During
  
  World War II, British and American transport fleets were often attacked by German submarines while sailing in the Atlantic Ocean. The British Air Force often sent bombers to use depth charges to attack German submarines, but the bombing effect was always unsatisfactory.
  To this end, the British army invited some mathematicians to study this problem. It was found that the submarine only dived 7.6 meters from the time when the British aircraft began to dive to the explosion of the depth charge, while the depth charge of the British aircraft sank to 21 meters and exploded, thus causing low damage to the submarine. After scientific demonstration, the British army resolutely adjusted the fuze of the depth charge, and the explosion depth was adjusted from 21 meters to 9.1 meters. As a result, the bombing effect was increased by 4 times. The German army thought that the British army had some new weapons.
  
  In October 1942, General Patton led more than 40,000 U.S. troops and took 100 warships
  
  to Morocco, which is 4,000 kilometers away from the United States, and landed in the early morning of November 8. On November 4, a strong northwesterly wind suddenly blew on the sea surface, and the stormy waves caused the ship to tilt up to 42°. The weather did not improve until November 6. Headquarters in Washington, fearing that the fleet would be wiped out by high winds, telegraphed Patton’s fleet to land at any other port on the Mediterranean coast. Patton calls back: I’ll go as planned, regardless of the weather.
  At midnight on November 7, the sea suddenly calmed down, and Patton’s army landed successfully as planned. Afterwards, people said that it was a fluke victory, and that the “blood and gallbladder general” bet the life of the soldiers. In fact, General Patton and meteorologists studied in detail the law of wind and wave changes and related parameters in the Moroccan sea area before departure. He knew that although there were strong winds in the sea area from November 4 to 7, according to the usual maximum wave height and wavelength of the sea area and ships The proportional relationship between them is just not enough to capsize the ship, and it will not cause harm to the entire fleet. Instead, November 8 was a good day for landfall. It is precisely by using scientific predictions and reliable parameters that Patton seizes the “terrible opportunity” and suddenly appears in front of the enemy.

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