Madame L spent about an hour while she was trying to go to sleep last night, trying to figure out the answer to this riddle. Then her sister, Ellen, sent her the answer.
Ellen had started a project of writing about a number of items she's grateful for, as many items each day in the month of November as that day of the month. So, for instance, on the seventh day of the month she wrote about seven things she's thankful for. She kept it up through the 16th, Saturday, at which point she decided that the lists were "getting longer and longer" while not serving any valuable purpose. Not that she wasn't thankful, and not that she couldn't think of many more things she was thankful for, but she was going to go on to actually thank the people she's thankful for. Great idea!
But it made Madame L think of a problem she'd heard of recently. So she mentioned it to Ellen, who had already heard of it, and knew the solution. Here's what Ellen wrote:
"I remember Daddy telling the riddle of the first-grade teacher who was tired and wanted to give the children a task to do on their own so she could rest for a few minutes. She told them to add all the numbers from 1 to 100. She thought she'd have some peace and quiet for at least an hour, but only a couple of minutes later, much to her dismay, a little boy came to her desk with the answer. She asked him how he did it so quickly, and he explained how simple it was. If you start at each end and add the numbers--100 + 1--and work your way in toward the middle--99 + 2, then 98 + 3, and 97 + 4, etc.--then each answer is 101. By the time you get to the middle, you would have done it half as many times as the number of numbers you are adding, which in his case would be 50, so 101 times 50 is the answer. So with my 30 days of gratitude, then it would be 31 x 15 which is, as you said, 465."
What an elegant solution to the problem! Thanks, Ellen, and thanks for all you do.