I am trying to compile a list of mind-boggling/curious historical facts in mathematics that will inspire and attract young people (9–11 years old) to the discipline of Mathematics. Do you have one fact to share? Multidisciplinary historical facts are welcome!
I recall being inspired, at about the age of 12, by the widely-known anecdote of how the mathematical talent of the famous mathematician Carl Friedrich Gauss was first recognized at the age of nine. His teacher put his students to work on summing the first one hundred integers, and young Gauss came up with the answer almost immediately: 5050.
Our math teacher provided some historical context: Gauss had been born into a working-class family in Brunswick in Northern Germany in 1777. He was the only child of his father's second marriage. His father worked as a bricklayer and gardener, and while he could read, write, and perform simple arithmetic, Gauss's mother was functionally illiterate. From the age of seven Gauss attended a public school: A one-room school house with a schoolmaster in charge of one hundred students. Corporal punishment was common. Students used slate tablets as the use of paper was too expensive (the use of slate tablets remained common in German elementary schools until after World War II).
Our teacher did not tell us right away how Gauss had figured out the result almost instantaneously; instead he challenged us to ponder how he might have done it. I spent an entire afternoon trying to figure it out, and I seem to recall I worked out a formula based on averaging. The math teacher revealed Gauss's (supposed) technique the next day: One can form pairs $100+1$, $99+2$, etc and in this way generate fifty pairs each summing to $101$, thus the final sum is $50 \times 101 = 5050$.
It should be noted that the specifics of this computation seem to be a bit of a modern invention. The earliest biography of Gauss does include this anecdote and claims Gauss himself as the source, but it just mentions that Gauss's teacher tasked his students with summing an unspecified arithmetic series during instruction in arithmetic. It further states that it was the custom at that school that when finished with a task students would drop off their slate tablet at the teacher's desk, one on top of the other. Gauss turned in his slate almost immediately with the words "There it lies" and then sat quietly until the end of class, when the teacher flipped the stack of slate tables around, inspected the top-most slate and to his astonishment found that Gauss had written down the correct answer and nothing else:
Wolfgang Sartorius von Waltershausen, Gauss zum Gedächtniss. Leipzig: S. Hirzel 1856 (online)
A recent publication has tried to trace how details were added to this story over time as it was retold numerous times:
Brian Hayes, "Gauss's Day of Reckoning." American Scientist, Vol. 94, No. 3, May 2006, pp. 200-205 (online)
The mathematician Hardy told the following story when he went to visit the mathematician Ramanujan: "I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." (Indeed: $1729 = 10^3+9^3 = 12^3+1^3$)
It's amazing that Ramanujan was able to come up with this instantly.
I am adding a second story. In 1939, George Bernard Dantzig arrived late for a graduate-level statistics class. There were two problems written on the board. Because he was late he didn't know they were examples of "unsolved" statistics problems. Instead he took them for part of a homework assignment, copied them down, and solved them. He used these solutions as his PhD thesis. I
It's said that if he had known they were unsolvable he wouldn't have tried to solve them.
Factoring numbers is usually taught at around that age, and students are taught about prime and composite numbers and how to find all the divisors of a number. It's not too much additional work to discuss perfect numbers and amicable numbers and to demonstrate why 220 and 284 are an amicable pair.
The curious historical fact is that after centuries of brilliant mathematicians exploring number theory, it wasn't until 1867 that a sixteen year old boy in Italy, Nicolò I. Paganini, discovered the second smallest pair, that 1184 and 1210 are amicable.