The old challenge leads to a widely forgotten but handy method for checking addition called "casting out nines."
Old challenge (Joe Herman)
Why does adding a column of three 3-digit hymn numbers yield the same answer whether you add frontwards, backwards, or even sideways?
George Dillard's computer experiments show that it works about 80 percent of the time for three hymns from 1 to 429, and here is why. If there were no carries, you would always just get the sum of all the digits. Every time there is a carry, a 10 becomes a 1, and the sum is reduced by 9. If there happen to be the same number of carries in all four directions, the answers will be the same. In general, they can differ by multiples of 9. Ron Douglass wonders how to predict how often it works.
Joseph Amaral, Harlan Durand, Bill Hasek, and Joe Herman's twin brother note that if you continue the process, ever adding up the digits of your answers, you will always come up with the same final answer. For example, 125 + 236 + 347 = 708; the digits of 708 sum to 15; the digits of 15 sum to 6. On the other hand, backwards, 521 + 632 + 743 = 1,896; the digits of 1,896 sum to 24 (which differs from 15 by 9); but the digits of 24 also sum to 6.
Mary Lou Holmes and Jean Hunter also observe that this is essentially the method of "casting out nines" used to check arithmetic. (So above, for example, the digits of 24 sum to 6, casting off 18, or two 9s.) To check a lengthy sum, just add up all the digits, then add the digits of your answer, and continue until you are down to one digit. The same process applied to your original total should yield the same digit.
For example, let's check the above sum 125 + 236 +347 = 708. The sum of the digits is 1+2+5+2+3+6+3+4+7 = 33; the sum of its digits is 3 + 3 = 6. Applying the same process to the original total 708 yields 7+0+8 = 15 and then 1 + 5 = 6, which checks. This method will find most errors (unless the error happens to be a multiple of 9).
'Good Will Hunting'
The movie about a prize-winning MIT mathematician and a young prodigy lists Harvard physicist Sheldon Glashow and MIT mathematician Daniel Kleitman in the credits. According to Professor Kleitman: "The original story had the hero a brilliant physicist; they contacted Glashow, who said that it didn't ring true as a physicist; he suggested they make him a mathematician and suggested they talk to me. They told me they wanted to hear mathematicians talk.... I muttered a few words, got hold of a post doc, Tom Bohman, and had a discussion with him of various results and topics and things. They busily took notes; they seemed quite alert and intelligent, though mathematically illiterate. After a while (an hour? two hours? who can remember) they left, thanking us profusely for our help." Kleitman appears briefly in the movie, outside a window while the stars are talking at the counter.
New challenge (Michael Marcotty)
Take an old-fashioned bottle of nonhomogenized milk, with the cream risen to occupy the narrow neck at the top, shake it to mix in the cream, and put it back down. Is the pressure of the milk on the bottom of the bottle the same as before mixing, or greater, or less?
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