September 12th, 973
It was late at night (or, perhaps more precisely, very early in the morning) and Wülfgang was the only one still awake in his restaurant. All his staff had gone upstairs to bed and, truth be told, that was where he belonged as well. But there was something about Pfinder’s signs and countersigns that gnawed and gnawed at him and so he found himself sitting at one of the tables by the light of a lone candle staring at columns of numbers painstakingly copied from Pfinder’s ledger.
11… 47; 34… 100; 22… 64 – the numbers swam about the page in chaos. No order, no meaning. Nothing.
“Syllables!” he exclaimed. “Twen-tee-two goes with six-tee-four! Thir-tee-one goes with eight-y-one!” But, alas, it didn’t hold true for all of them… there was “elev-en and for-ty-sev-en,” for instance. You could make it work with “elev-en-ty-one and for-ty-sev-en” but that was just ridiculous. Everyone knew that “eleventy-one” was just Hobbit rubbish.
In desperation, he tried it again in Dwarvish. “Vein-te-dos y ses-en-ta-cua-tro!” But that was even worse! He buried his head (and whatever part of his voluminous beard would also fit) in his hands. These numbers were all over the place, like the measurements of his mother’s cooking! A pinch of this and a handful of that – no rhyme or reason but somehow, by means of some sort of magic, they just worked.
OK, how about this? His quill scratched out a few sums and differences on his parchment…
8 + 2 = 10
4 + 7 = 11
5 – 2 = 1 + 2
7 – 3 = 3 + 1
5 + 9 = 14
But when he got to 16, the pattern was broken. If there had been a pattern in the first place, that is. Why would you add sometimes and subtract at other times? Well, maybe you’d do it if you were trying to be tricky…
Despite the late/early hour, he had to say that this number game was intriguing and he’d be damned if he’d allow himself to be out-thought by a bunch of half-wit drunkards in some stinking alley in some half-assed quarter of Edicaria. There HAD to be some relationship between the signs and counter-signs and it had to be simple enough that these troglodytes could figure it out with ten fingers or less!
And if they were trying to be tricky troglodytes, they might use more than one rule for calculating a countersign from a sign. Adding sometimes, subtracting sometimes. But probably always multiplying because the countersigns were always larger than the signs. OK then, how many rules might there be?
Let’s see now… a sign of 10 goes with a countersign of 28. How do you get from 10 to 28… could be 10 x 2 + 8 = 28. Could be 3 x 10 – 2 = 28. OK, that’s either 2n + 8 or 3n – 2.
And 11 goes with 47. How do you get from here to there? Well 11 x 4 + 3 = 47. 4n + 3.
And 12 and 25. What’s that all about? How about 12 x 2 + 1? Yeah, that works but it’s 2n + 1.
And 14 and 59. That could be 14 x 4 + 3 or 4n + 3.
Wülfgang sat bolt upright. “Hey, I just saw that! Yeah, there it is – 11 goes to 47 the same way!”
He flipped the parchment over to the other side and copied down all the signs in a vertical column – this time in numerical order. He focused on the signs that ranged from 10 – 19; that made an unbroken series of samples. Next to them he wrote the corresponding countersigns… a countersign of 28 went to the right of a sign of 10. A countersign of 47 went to the right of a sign of 11 and so on.
<<Scritch, scritch, scratch>> went the quill. <<Scratch, scratch, scritch>>. And when he was done, the results took his breath away. There was a regular pattern!
3n-2 then 4n+3 then 2n+1 then repeat! Over and over and over again! The lumpen headed troglodyte bastards were dividing by three, taking the remainder, and using that to determine which formula to apply!
And then, at that moment, and perhaps for the first time in his life, Wülfgang recalled his old math teacher, Mrs. Stonebeard fondly. She was a rusty old pickax but she knew her stuff and she knew exactly how to pound it into the heads of unruly Dwarflings whether they wanted it or not. If not for her ministrations, he might not have cracked this puzzle.
Smiling crookedly, he indulged in a memory of countless hours spent practicing “graphing” with Mrs. Stonebeard. Ah, what the hells. He got up, grabbed his best cleaver from the kitchen for a straightedge and a fresh sheet of parchment and ground out yet one more graph, this time in honor of Mrs. Stonebeard. When he was done, he gazed upon his work in satisfaction and that crooked grin split his beard once again,
Nothing as complicated as syllable counts then! Just simple arithmetic! Although how those drunken oafs managed to multiply 17 by 4 without taking off their boots was beyond him. Ah well, another mystery for another day…