A reader wrote in to the newspaper's puzzle column with a complaint. She had been given a list of warehouse totals and asked which of them could be shared out evenly among nine people. She had done nine long divisions and it had taken her the whole evening.
The columnist wrote back that there was a test for it, and that it took one line. He added that he was not being clever: the test is older than the newspaper, and every accountant of her grandmother's generation had known it by heart.
Add up the digits of the number. If that sum can be divided by nine, so can the number itself. If it cannot, neither can the number.
Take four thousand seven hundred and twenty-five. Its digits add to eighteen, and eighteen is nine twos. So the number divides evenly, and indeed it is nine times five hundred and twenty-five. Take three thousand one hundred and forty-two. Its digits add to ten, which is not a multiple of nine, so no amount of long division will make it come out even.
What the columnist wanted his reader to notice was not the trick itself but its shape. The question "does this divide evenly?" and the question "what is the answer?" are different questions, and the first one is far cheaper than the second.
"You do not need to open the box to know it is empty," he finished. "Sometimes you can weigh it."