Oh Dear...

Well that attempt to start a novel new distraction flopped a bit. To be fair the question was going to be almost impossible even if you knew exactly what it was you were talking about.

the calculation (5×4×3)×(26³)×(26!/6!10!2¹⁰)= was first used in the 1930's to describe what possibility?

Well let me transport you back 90 years or so to the dawn of Arthur Scherbius's new invention. An electro-mechanical rotor cipher that came to be known as.... ENIGMA.

a military issue ENIGMA machine, with plugboard

It was a wonder in its complexity. Used in the twenties by business's looking to keep their internal communications private, it was adopted by many countries like Germany and Poland. By the Thirties the German military had improved immeasurably on the original making a coding machine that was believed to be unbreakable. Typing in a letter sent an electrical signal through three of five possible rotors (of which the first moves every time a letter was plugged in, the second every 26 times the first moves and the third every 26 times the second moves), then back through them the other way to light up a new letter that the coder can write into an incoherent and completely random order. If that wasn't hard to follow, the military issue device also had a plug board that took the final letter and swapped it with another letter, of which their were ten possible pairs and six left over that didn't swap with anything...

Here is where the calculation fits in. It states the chance of working out the possible set up position of the ENIGMA machine on any ONE day! Here is the break down.

(WARNING: This next section contains some pretty heavy maths, of which I fail alot at. I have tried to explain it in as Leymanish a way as possible, but if fiddly numbers give you a headache look away now.)

OK, here goes...

(5×4×3) Was the chance that any three of the five rotors were used, as once one is fitted their is one less to chose from each time.

(26³) Since each of the three rotors had 26 connectors it could have possibly been routed through, their is 26 possibilities for each rotors start position. Hence their are 26×26×26 possible setups for the rotors, or 26³.

(26!/6!×10!×2¹⁰) This is the tricky one depicting the switchboard. So their are 26 letters in the alphabet. How many ways can you arrange them? Well 26 ways×25 ways×24 ways×23 ways×22.... you get the picture. This is written as 26 factorial, or 26!.

Now because we only need 10 pairs of these letters we don't need to use them all. So we take the 6 possible letter combinations that will not swap. Since their are less each time we remove one this is once more a factorial number, or 6!.

Then we want to pair up the rest of the numbers, not caring what the two pairs are just that they are paired. Again the number dwindles after each one is set so this is 10!.

When two letters are in a pair, they are always going to swap, so they form a stable set. hence 2¹⁰ represents the 10 pairs of 2. Lastly we divide all these possible switchboard combinations with the original 26! to get the possible switch board combos for one days setup...

So what was the possible number of setup options per day you may ask, well. it was a whopping

158,962,555,217,826,360,000

Over one hundred and fifty eight quin-trillion ways it could be set!

Does that not blow your mind! it does not even factor in the three more rotors that the German navy where issued with for their ENIGMA's. How the Polish cryptologists Marian Rejewski, Jerzy Różycki and Henryk Zygalski managed to break the computation and reverse engineer a rudimentary computer to work out possible combinations is far beyond me, and my hat therefore goes off to them.

Maybe this weeks question should be easier. we will see.