How mathmaticians won the war

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JudgeMental

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No , just for once its NOT my favourite subject Enigma or Ultra.

How a statistical formula won the war
Thursday July 20, 2006
The Guardian

Here is a story about mathematical deduction that I love, mainly because it is said to be true, and because it had an impact (albeit small) on the outcome of the second world war. It is the story of how a simple statistical formula successfully estimated the number of tanks the enemy was producing, at a time when this could not be directly observed by the allied spy network.
By 1941-42, the allies knew that US and even British tanks had been technically superior to German Panzer tanks in combat, but they were worried about the capabilities of the new marks IV and V. More troubling, they had really very little idea of how many tanks the enemy was capable of producing in a year. Without this information, they were unsure whether any invasion of the continent on the western front could succeed.
One solution was to ask intelligence to guess the number by secretly observing the output of German factories, or by trying to count tanks on the battlefield. Both the British and the Americans tried this, but they found that the estimates returned by intelligence were contradictory and unreliable. Therefore they asked statistical intelligence to see whether the accuracy of the estimates could be improved.
The statisticians had one key piece of information, which was the serial numbers on captured mark V tanks. The statisticians believed that the Germans, being Germans, had logically numbered their tanks in the order in which they were produced. And this deduction turned out to be right. It was enough to enable them to make an estimate of the total number of tanks that had been produced up to any given moment.
The basic idea was that the highest serial number among the captured tanks could be used to calculate the overall total. The German tanks were numbered as follows: 1, 2, 3 ... N, where N was the desired total number of tanks produced. Imagine that they had captured five tanks, with serial numbers 20, 31, 43, 78 and 92. They now had a sample of five, with a maximum serial number of 92. Call the sample size S and the maximum serial number M. After some experimentation with other series, the statisticians reckoned that a good estimator of the number of tanks would probably be provided by the simple equation (M-1)(S+1)/S. In the example given, this translates to (92-1)(5+1)/5, which is equal to 109.2. Therefore the estimate of tanks produced at that time would be 109
By using this formula, statisticians reportedly estimated that the Germans produced 246 tanks per month between June 1940 and September 1942. At that time, standard intelligence estimates had believed the number was far, far higher, at around 1,400. After the war, the allies captured German production records, showing that the true number of tanks produced in those three years was 245 per month, almost exactly what the statisticians had calculated, and less than one fifth of what standard intelligence had thought likely.
Emboldened, the allies attacked the western front in 1944 and overcame the Panzers on their way to Berlin. And so it was that statisticians won the war - in their own estimation, at any rate.

So there you have it, maths won the war :D
 

thedonster

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Some other great examples of mathematicians' influence on the war:
1) The breaking of the Japanese naval communications code. Unlike the Enigma saga, the Japanese code (I think it was called "Purple") was broken by pure math analysis. This had a huge influence on the war in the Pacific, including killing Yamamoto.
2) My favourite was the analysis of the V1 bomb. My math prof in first year university taught this one and I still remember it. When the V1 started to hit London their accuracy suggested an advanced guidance system which had the British really worried. They divided a map of London into a grid and plotted the hits. A mathmatical analysis showed the pattern of hits was random, and the rockets were guided by careful fuel measurement only.
3)Then there is the ultimate achievement: the atomic bomb. Couldn't have been done without some very advanced mathematicans' work.
 

Rameusb5

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The Pre-War Polish cracking of Enigma was very much math based. That was one interesting story though.

Wasn't the Enigma device a Polish design (or at least based on a Polish design)?


And it's my opinion that the bean-counters won WWII. For all their technical superiority, the Germans just couldn't compete with the combined resources of the Allies.
 

JudgeMental

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Wasn't the Enigma device a Polish design (or at least based on a Polish design)?

Nope, the Enigma was a German design, the Poles got hold of an Enigma machine from the Germans when it was inadvertantly sent to their Polish embassy not in a diplomatic pouch. The Poles did a brilliant job copying the gubbins of it and getting it back to the Germans without their knowing what was done. Of course all that was a help to decoding Enigma traffic but without the brilliant work of the Polish analysts who very much had their backs to the wall it, would have not have been enough.
 

Zbojnik

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Nope, the Enigma was a German design, the Poles got hold of an Enigma machine from the Germans when it was inadvertantly sent to their Polish embassy not in a diplomatic pouch. The Poles did a brilliant job copying the gubbins of it and getting it back to the Germans without their knowing what was done. Of course all that was a help to decoding Enigma traffic but without the brilliant work of the Polish analysts who very much had their backs to the wall it, would have not have been enough.

They actually cracked the code long before they set there eyes on an Enigma machine.
 

JudgeMental

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They actually cracked the code long before they set there eyes on an Enigma machine.
The Poles aquired an Enigma to copy in 1928 (source Col Tadeusz Lisicki), they DID NOT crack it before then.It was obtained when it was sent to the German Legation in Warsaw. It had been sent via commercial transport instead of 'Diplomatic Bag' It revealed some key ingredients to the poles including the rotor wiring, and the stecker board or plug board which provided extra encipherment.
BS4 did an amazing job, but according to Marian Rejewski in an interveiw given by him in '76 part of their success was down to aquiring a commercial Enigma machine which they modded to military specs. Another key source was the spy codenamed 'Ashe' who gave the French Intelligence service: a manual for Enigma, a codebook and a sample enciphered message with its plain text equivalent. The French passed this information to BS4.
Don't get me wrong Zbojnik, NONE of this takes anything away from the brilliant (and there is no other word for it) work of Marian Rejewski, Jerzy Rozycki and and Henryk Zygalski or the other members of BS4. I am just filling in some of the blanks. It is acknowledged that the Poles had broken Enigma in 1932. One can only sympathise with them, can you imagine what its like knowing whats going to happen to your country and not be able to do anything about it?
The Poles don't often get the credit for the work they did on Enigma, which is a shame as without them WW2 would have taken a different course. Thats no exaggeration, it was their groundbreaking work which gave the Allies a start on Enigma. Sadly many of the Poles did not live to see the fruits of their labours.
 
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Zbojnik

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The Poles aquired an Enigma to copy in 1928 (source Col Tadeusz Lisicki), they DID NOT crack it before then.It was obtained when it was sent to the German Legation in Warsaw. It had been sent via commercial transport instead of 'Diplomatic Bag' It revealed some key ingredients to the poles including the rotor wiring, and the stecker board or plug board which provided extra encipherment.
BS4 did an amazing job, but according to Marian Rejewski in an interveiw given by him in '76 part of their success was down to aquiring a commercial Enigma machine which they modded to military specs. Another key source was the spy codenamed 'Ashe' who gave the French Intelligence service: a manual for Enigma, a codebook and a sample enciphered message with its plain text equivalent. The French passed this information to BS4.
Don't get me wrong Zbojnik, NONE of this takes anything away from the brilliant (and there is no other word for it) work of Marian Rejewski, Jerzy Rozycki and and Henryk Zygalski or the other members of BS4. I am just filling in some of the blanks. It is acknowledged that the Poles had broken Enigma in 1932. One can only sympathise with them, can you imagine what its like knowing whats going to happen to your country and not be able to do anything about it?
The Poles don't often get the credit for the work they did on Enigma, which is a shame as without them WW2 would have taken a different course. Thats no exaggeration, it was their groundbreaking work which gave the Allies a start on Enigma. Sadly many of the Poles did not live to see the fruits of their labours.

Hmm. That's very interesting. Everything I read on the topic points differently. Polish contribution to the war is something that interestes me due to my heritage. Can you send any references about Enigma my way?
 

JudgeMental

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Some of the sources where I read about it:

Enigma: Not the drama book, this one by Hugh Sebag-Montifiore deals mainly with the work done OUTSIDE Bletchley Park. It deals with the seizures of Enigma and materiel by outside agencies.

The Code Book: Simon Singh Deals with codes in general.

The Secret War: Brian Johnson An old book accompanying a cracking series on TV. Makes up for the fact that it was written earlier by dealing with the people who were DIRECTLY involved in the work rather than descendents.

Like I said, mate, NONE of this in any way diminishes the work done by the Poles. The unsung heroes of Enigma.In many ways it makes it all the more poignant.

I PM'd you my msn if you want to ask anything about it.
 
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DrFubar

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another great source for the story of the enigma and the role the poles had in breaking it is: Enigma How The Poles Broke The Nazi Code by Wladyslaw Kozaczuk and Jerzy Straszak.
 

Seth_Soldier

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For the dday(or was it for the lend lease), the mathematician have create a new math sector : the optimization of the ressources.
Based on the ressources and the need they have set the priority to have the best optimization.

:D at least it is what i've heard when i've learned AMPL.
 

Colt .45 killer

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neet find, but until a matamatician picks up a gun and shoots somebody he hasent really won the war, he contributed to the wars being won.
 

Fu. Svedberg

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Don't forget all the ballistics tables that where calculated for different kinds of ammunition, guns, ambient temperatures and gun temperatures all done by hand :eek:
 

Colt .45 killer

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IM not taking anaything away from all those who supported the war. but saying that mathamaticians won the war is not true, that title belongs the the soldiers who fought, died and killed to survive and win the war; they should atleast be respected for that, because they arent respected for anaything else in our modern society. those mathamaticians could have predicted everything, found ways to break every communitcation and calculate every shells path to the milimeter, but without a sailor to fire the gun, and radio operators to intercept messages, tankers to benefit from knowing they outnumber the enemy .. and basic frontline information, all you have is a pile of paper with cryptic math on it that isnt good for anaything else then fire starter.

a more proper title to this thread could have been " how mathamaticians contributed to the winning of ww2", the winning of any war has to be handed to those who died fighting it.
 

JudgeMental

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The title was taken from the original article in the newspaper. No slur against squaddies was intended. 24561433 is not a particular maths number, its my Army number. I was a squaddie for 22 years and as such fully respect them and what they do. I was only in 2 little wars, and don't envy the people who were involved in the big ones. Chill Colt ;)
 

Nestor Makhno

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I used my fingers to win my math exam.

What did you have to do to the examiner with those fingers? ;) :p

On the statistical analysis of the V1 landings I heard another story relating to the accuracy of the guidance and aiming of the V1. This is probably about 3rd hand hearsay so don't take this as 'historical source info'.

The first V1s to land on London were pretty much centred bang on the centre of London where, obviously, they did a lot of damage. British miliitary intelligence had pretty much rolled back all major Nazi spy networks in the British mainland at a fairly early stage, either executing them, turning them or deliberately feeding them misinformation.

They let information filter back to Germany that the majority of the V-1's were landing about 15km too far north. The reaction of the Germans was to adjust for this, which meant that the V-1's were now centred on the town of Croydon (where I live) which received a terrible pasting as a consequence.

Other explanations for Croydon getting particularly hammered are:

a) prevailing winds or,

b) V-1s launched from France running out of fuel too soon or, my favourite theory...

c) someone from the German strategic command had actually been there pre-war and decided to take the place out as a favour to humanity.

Certainly NATO predictions of WP nuclear strike patterns showed that the largest nuke in most simulated attacks would hit Croydon. 100 megatons would probably be just enough to remove that amount of piss-stained concrete and shopping centres.
 
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