A Howling Good Halloween Puzzle

Many of you may have already seen a version of October’s Problem to Ponder, a number puzzle that determines your age on the basis of the number of times that you want to eat chocolate per week. The October Problem to Ponder includes an additional challenge: Why does the puzzle work?

• How many times a week do you want to eat chocolate? (Pick more than one but less than 10 times).
• Multiply that number by 2.
• Add 5.
• Multiply that result by 50.
• If your birthday has occurred this year, add 1760 to that result. If your birthday is still to come this year, add 1759.
  (UPDATED for 2013: If your birthday has occurred this year, add 1763 to that result. If your birthday is still to come this year, add 1762.) 
• Subtract the year (all four digits) in which you were born.
• The result will be a three-digit number. The first digit will be the number of times a week that you want to eat chocolate, and the last two digits will be your age.

Extension: Suppose you wanted chocolate 10 or more times a week, or no chocolate at all. Could you still make the puzzle work?

Note to teachers: You can base the puzzle on anything that your students might want to do a number of times a week—for example, play video games, talk with or text friends, or read a book.

Some Approaches:  

Last month, I had the opportunity to visit an eighth-grade algebra class at Haynes Academy in New Orleans as they discussed this problem. (Thank you again for the visit, Michelle and class!)

Why does this work?

After working through the process for their own chocolate number, the students gathered in groups to try to generalize the problem in some way, using a variable to represent the number of times that they wanted chocolate, along with the year that they were born. Students then shared this reasoning:

“If N is the number of times a week that you want to eat chocolate, then follow the steps, and you go from N —> 2N —> 2N + 5 —> 50 x (2N + 5).  Then you either add 1760 or 1759, depending on whether you’ve had your birthday this year yet or not; for example,

50 x (2N + 5) + (1760) = 100N + (250 + 1760) = 100N + 2010

Now subtract your birth year, and you get (100N + 2010) – (year of birth), which will give you N in the hundreds place, and your age in the 10s and 1s place.”

The Haynes students also decided that the problem would work even if someone didn’t want any chocolate, because then there wouldn’t be a hundreds place in the result, e.g. N = 0 in the hundreds place. And they decided that if a person were more than 100 years old, the process wouldn’t work because the person’s birth year would “mess up the 100s place in the result.”

Great job, Haynes Academy!