The Professor and his protégé

Herr Schmidt was tired. Tired of the cold, tired of the class of forty-seven pre-teen children that sat before him and, he surprised himself to think, tired of his protégé. In fact, the more he pondered, the more he realised it was his protégé that was the root of all his problems. Life had been simple before young Carl was delivered to his door to be a deserving recipient of Herr Schmidt’s careful tutelage.

“Ah, the days before Carl” mused the moustachioed Schmidt to himself. Life was simple then. His forty-six eager students would listen attentively, with awe even, to his every word, and then he would pick up his chalk, write a few tricky sums on the blackboard, sit back, open his newspaper and enjoy a peaceful twenty minutes as the nearly noble young of the Duchy of Brunswick set about their arithmetic, struggling in silent, scholarly endeavour.

“Herr Schmidt, Herr Schmidt. I am finished.”  The professor’s reverie was once again broken, mere moments after it had begun. However, his immediate irritability gave way to a small, barely discernible, smile. An idea had entered the maestro’s mind, and not just any idea, a brilliant idea. “This will keep him busy for a fair while” he thought to himself. “Perhaps until luncheon – possibly beyond” and his mind turned to thoughts of wurst and kraut.

“Gauss,” he boomed a little too loudly, “Gauss, you are a talented scholar, of that there is no doubt. I have a task for you that is worthy of your consideration, but be warned, it may be a little too much for you just yet. You will need to focus and concentrate; it is not a simple problem and will take you some time. Do you think you can accept such a challenge?”

Young Carl Friedrich trembled with excitement. He knew Herr Schmidt was a good man, the best, most dedicated teacher in the Duchy, and he had a special problem for him. “Of course, sir. If you think I am worthy of such a challenge, I will dedicate my mind to it.”

“Very well, my boy” responded Herr Schmidt. “You are to sum all the counting numbers between one and one hundred. As I say, not a simple or swift task. Now go, sit, sit, sit.”

As the young protégé scuttled back to his desk, the professor picked up his newspaper, emitted a soft sigh of relief and began to think again of his luncheon to come, his eyes gently closing …

He was soon rudely shaken from his peace. “Five thousand and fifty, five thousand and fifty” came a tremulous cry from Carl Friedrich Gauss. “What do you mean boy” grumbled the professor, “what do you mean. You have merely made up an answer. Guesswork is not the hallmark of a scholar, young man.  You must re-evaluate your solution, and I must re-evaluate your suitability for this class. Five thousand and fifty indeed. There is no way you can have completed such a sum so swiftly.”

Gauss was confused. He had been set a task, a special task by the magnificent Herr Schmidt and he had solved that task, quickly and eloquently, and yet Herr Schmidt seemed displeased? And then in it dawned on him: it was a test. To gain the professor’s true esteem he must now explain his solution.

“Sir, you are a sage and wise teacher, and I am honoured that you set me such a task, but even more so that you demand I explain my solution.  As you know, sir, the trick is not just keep adding the numbers, one plus two is three, three plus three is six, plus four makes ten.  That’s what a simple Hanoverian would do” he whispered, with a shy smile.

 “No sir, I considered the counting numbers from one to one hundred, and then considered the same numbers but in reverse order, from one hundred descending to one.” The professor wasn’t sure where this was going, but his interest was piqued. “Continue boy, continue” he demanded.

“Well sir, then I paired up the numbers from the two lists: one with one hundred, two with ninety-nine, three with ninety-eight etc.  Each pair will sum to one hundred and one, and there will be one hundred pairs, so I multiplied one hundred and one by one hundred …”

“… giving ten thousand, one hundred” interrupted the professor “which is wrong.”

“But sir, we have two lists of the numbers from one to one hundred, so now we must halve your ten thousand, one hundred to give the answer of five thousand and fifty” announced a triumphant Gauss.

There was a pause. A pause that stretched into a silence. The other boys, sensing something afoot, looked up from their slates. A tension filled the cold, damp classroom. Breathes were held. Jonas Fischer, a particularly nervous boy, slid under his desk. And then the silence shattered.

“Well, what are you waiting for boy? Sit, sit” bellowed Herr Schmidt, pulling up a chair next to his desk, a smile breaking across the corners of his mouth. And together they sat, the professor and his protégé, checking his working, trying other examples and codifying a formula for the general case. For the first time in years, Herr Schmidt was enjoying himself, enjoying mathematics again, so much so, and so engrossed in the task, that he almost forgot to send the boys to lunch. Almost.

He particularly enjoyed his wurst for lunch that day.

Author’s note: this story is fiction. There is an unreliable anecdote that he discovered the formula for an arithmetic progression whilst at primary school, but it makes a great story, one that I often tell my students and, as it is National Story Telling week I thought I would put pen to paper, embellish my tale a little, and write it down.  I hope you enjoyed it, if not the tale, then the simplistic brilliance (not mine) of how to add up the integers from one to one hundred.

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