32 1. Trisecting Angles of their contemporaries, only to find limited success as an adult. But Gauss was unbelievably precocious as a child he was even more spectacular as an adult. At the age of 2 Gauss taught himself to read. At the age of 3 he observed his father working on the figures from a payroll for some fellow workers. When his father had finished, Gauss who had been quietly observing him from the ground, said, “Father, the reckoning is wrong, the figure should be . . . ,” and quoted a number. After checking his numbers, Gauss’s father found the mistake. The correct answer was the one Gauss had given. A certain Mr. J. G. B¨ uttner had the misfortune of being a rather dull man who was also Gauss’s first teacher in elementary school. He also has the reputation of having been rather cruel and a heavy user of the cane he constantly kept at hand. But canning was standard practice at the time (the good old days), and it isn’t certain whether B¨ uttner was unusually cruel. But he did assign arithmetic problems that befuddled his students and had little pedagogical value (perhaps a more severe form of cruelty). When Gauss was 10, B¨ uttner told the pupils to add all the whole num- bers between 1 and 100. In a few seconds Gauss marched to B¨ uttner’s table and laid his slate down. When the others had finished, B¨ uttner turned over Gauss’s slate to see the single number 5050 — the correct answer. Perplexed (and, some say, disappointed — in the retelling of this story one wonders if we see the effect of having had a pedantic math teacher on the part of the tellers, an experience just about everyone has had at least once), B¨ uttner asked Gauss to explain himself. Gauss wrote the sum out twice, once forward and once backward like this: 1 + 2 + 3 + 4 + · · · + 100 100 + 99 + 98 + 97 + · · · + 1 Gauss explained that the two sums together are twice what is desired. But if the numbers are added vertically, you always get 101. Since there are 100 such terms and their sum is twice what is wanted, the correct answer is 50 · 101 = 5050. The method was known to mathematicians, but not to B¨ uttner. Gauss had discovered this method for himself. This event was a watershed in Gauss’s life. To his credit, and perhaps in- dicating that he was another overly criticized mathematics teacher, B¨uttner realized he had seen an exceptional talent and ordered a special arithmetic book for his star pupil. Also, B¨ uttner’s assistant, 18-year-old Martin Bartels, later a professor of mathematics, began to help Gauss. Gauss’s career as a student continued with similar success. To pay for his education, Duke Carl Wilhelm Ferdinand of Braunschweig became his

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