I recently learned about a cool problem that involves the four binary operations.
“My life is all arithmetic,” the young businesswoman explained. “I try to add to my income, subtract from my weight, divide my time, and avoid multiplying.”
A fifth-grader teacher, who is spending a week at NCTM Headquarters for the Illuminations Summer Institute, shared the problem with me. He uses it to practice basic operations and to develop an understanding of place value with his students.
Choose any positive integer as the starting value, and choose a different positive integer as the ending value. Then, perform any of the following moves on the starting number:
- Add 1.
- Subtract 1.
- Multiply by 10.
- Divide by 10.
Continue to perform moves until you’ve reached the ending number.
For example, you can get from 8 to 71 with the following sequence of moves:
- Subtract 1: 8 – 1 = 7
- Multiply by 10: 7 × 10 = 70
- Add 1: 70 + 1 = 71
For any given starting and ending number, what is the fewest number of moves required?
In general, can you find an algorithm to predict the minimum number of moves for any given starting and ending numbers?
The teacher who shared the problem with me said that students have fun with 777 as the starting number and 888 as the ending number. It is a fine problem for kids to explore, but it proved to be a red herring for me — it led me to make false assumptions for determining a general solution.