Old challenge (Aubrey Dunne, Michael Marcotty, and Dave Rossum)
Census taker: How old are your three daughters?
Mrs. S: The product of their ages is 36, and the sum of their ages is the address on our door here.
Census taker: I'm good at math, but I cannot tell.
Mrs. S: My eldest daughter has red hair.
Census taker: Oh thanks, now I know.
Can you figure out how old the three daughters are?
Since the census taker cannot tell their ages from the product and sum, there must be at least two possibilities with product 36 and the same sum. Trial and error yields just 1, 6, 6 and 2, 2, 9, both of which have sum 13 (which must therefore be the address). Reference to an "eldest" daughter rules out the first possibility and means that there are a 9-year-old eldest daughter and 2-year-old twins.
Noelle Matteson (age 11) notes the crucial insight, that the census taker knows the address, even though we do not. James Fahs and Diane Larrabee note that the hair color is a "red hairing."
Kent Wenger, Ron Douglass, James Fahs, Michael Eastep, and Bill Lenard argue that one of two 6-year-olds could be "older" by 11 months, or 11 days if adopted, or even 11 minutes if twins, so you cannot rule out 1, 6, 6 for sure. (Other winners chosen from many correct solutions: Hanna Klieber (age 10), Kurt Anderson, Roy Hathaway, Charles Herrin, Ayesha Irani, Dick Kaehler, Michael Jackson, Steven Miller, Tom Neubecker, and Hal Poret.)
Starting from scratch, what would be the best transportation system of the future for getting everybody to work?
* Send answers, comments, and new questions to:
Math Chat, Math Dept., Williams College,
Williamstown, MA 01267
or by e-mail to Frank.Morgan@williams.edu