But what I held in my hand was a strict and precise solution of my equations!
I ran my eye through page after page of calculations that took my breath away with their originality and sheer beauty. Whoever had done it possessed an immense mathematical knowledge to be envied by the world's foremost mathematicians. Almost all the modern armoury of mathematics had been employed: the theory of linear and non-linear differential and integral equations, the theory of the functions of a complex alternating current, and those of groups, and of plurality, and even such apparently irrelevant systems as topology, number theory and mathematical logic.
I nearly cried out in delight when at the end of a synthesis of countless theorems, intermediate calculations, formulae and equations the final solution emerged-a mathematical formula taking up three whole lines.
And to add a touch of the exquisite, the unknown mathematician had given himself the trouble of resolving the long formula into a simpler one. He had found a brief and precise form containing only the more elementary algebraic and trigonometric expressions.
At the very end, on a small inset, there was a graphic representation of the solution.
I could wish for nothing better. An equation which I thought could not be solved in the final form had been solved.
When I had recovered a little from my initial surprise and admiration I went through the photo copies again. Now I noticed that he who had solved my problem had been writing in great hurry and very closely as though trying to save on every scrap of paper and every second of time. Altogether he had written twenty-eight pages and I pictured mentally what a titanic work that had been! Try and pen a letter of twenty-eight closely-written pages in one day or just copy, without following the meaning, twenty-eight pages out of a book, and you will surely find it a hellish job.
But what I had in front of me was not a letter to a friend or a chapter copied out of a book. It was the solution of a most intricate mathematical problem-done in twenty-four hours.
For several hours I studied the closely-written pages, my surprise mounting with each hour.
Where had Kraftstudt found such a mathematician? On what terms? Who was he? A man of genius nobody knew? Or perhaps one of those wonders of human nature that sometimes occur on the border line between the normal and the abnormal? A rare specimen Kraftstudt had unearthed in the Wise Men's Home?
Cases have been recorded of brilliant mathematicians ending their days in a lunatic asylum. Maybe my mathematician was one of those?
These questions plagued me for the rest of that day.
But one thing was clear: the problem had been solved not by a machine, but by a man, a mathematical wizard the world knew nothing about.
The next day, a little calmer, I re-read the whole solution for the sheer pleasure of it this time, just as one will listen again and again to a piece of music one loves. It was so precise, so limpid, so beautiful that I decided to repeat the experiment. I decided to give Kraftstudt Co. one more problem to solve.
That was easy, for I was never short of challenging problems, and I chose an equation which I had always thought impossible to break down so that it could be handled by a computer, let alone be finally solved.
This equation, too, dealt with radio-wave propagation, but it was a specific and very complex case. It was an equation of the type that theoretical physicists evolve for the fun of it and soon forget all about because they are much too complex and therefore of no use to anybody.
I was met by the same young man blinking in the daylight. He gave me a reluctant smile.
"I have another problem-" I began.
Nodding briefly he again led me all the way through the dark corridors to the bleak reception hall.
Knowing the drill now, I went up to the window and handed him my equation.
"So it's not computers that do these things here?"
"As you see," he said without looking up from my equation.
"Whoever solved my first problem is quite a gifted mathematician," I said.
The young man did not say a word, deep as he was in my equation.
"Is he the only one in your employ or have you several?" I asked.
"What has that to do with your requirements? The firm guarantees-"
He had no time to finish, for at that moment the deep silence of the place was shattered by an inhuman scream. I started and listened. The sound was coming from behind the wall beyond the partition. It was like somebody being tortured. Crumpling the sheets with my problem, the young man, throwing a side glance and seizing me by the hand, dragged me to the exit.
"What was that?" I asked, panting.
"You'll have the solution the day after tomorrow, at twelve. You'll pay the bearer."
With those words he left me by my taxi.
3
It is hardly necessary to say that after this event my peace of mind was completely gone. Not for one moment could I forget that terrible scream which had seemed to shake the very stone vaults sheltering Kraftstudt and Co. Besides I was still under the shock of finding such a complicated problem solved by one man in one day. And finally I was feverishly waiting for the solution of my second problem. If this one too was solved, then…
It was with shaking hands that two days later I received a package from the Kraftstudt's girl. By its bulk I could tell that it must contain the solution to the monstrously complicated piece of mathematics. With something akin to awe I stared at the thin creature in front of me. Then I had an idea.
"Please come in, I'll get the money for you."
"No, it's all right." She seemed frightened and in a hurry. "I'll wait outside…"
"Come on in, no point in freezing outside," I said and all but dragged her into the hall. "I must have a look first to see whether the work's worth paying for."
The girl backed against the door and watched me with wide-open eyes.
"It is forbidden…" she whispered.
"What is?"
"To enter clients' flats… Those are the instructions, sir…"
"Never mind the instructions. I'm the master of this house and nobody will ever know you've entered."
"Oh, sir, but they will, and then…"
"What then?" I said, coming nearer.
"Oh. it's so horrible…"
Her head drooped suddenly and she sobbed.
I put a hand on her shoulder but she recoiled.
"Give me the seven hundred marks at once and I will go."
I held out the money, she snatched it and was gone.
Opening the package I nearly cried out with astonishment. For several minutes I stood there staring at the sheaf of photo paper unable to believe my own eyes. The calculations were done in a different hand.
Another mathematical genius! And of greater calibre than the first. The equations he had solved in an analytical form on fifty-three pages were incomparably more complicated than the ones I had handed in die first time. As I peered at the integrals, sums, variations and other symbols of the highest realms of mathematics I had a sudden feeling of having been transferred into a strange mathematical world where difficulty had no meaning. It just didn't exist.
That mathematician, it seemed, had no more difficulty in solving my problem than we have in adding or subtracting two-digit numbers.
Several times I tore myself away from the manuscript to look up a thing in a mathematical manual or reference book. I was amazed by his skill in using the most complex theorems and proofs. His mathematical logic and methods were irreproachable. I did not doubt that had the best mathematicians of all nations and ages, such as Newton, Leibnitz, Gauss, Euler, Lobachevsky, Weierstrass and Hilbert, seen the way my problem had been solved they would have been no less surprised.