Twenty-two years ago, I "published" a newsletter that was shared with middle school mathematics teachers. Below are the first three problems (i.e. old favorites) I included in the first issue.
Problem 1: Lester has a code where letters are used for digits. Each letter stands for exactly one of these digits: 2, 6, 7, 8. What digit does each letter represent if PQ + PQ + PQ = RRS?
Note: Does the problem have more solutions if all of the digits 0-9 are allowed?
Problem 2: Gina and Tom raise cats and birds. They counted all the heads and got 10. They counted all the feet and got 34. How many cats and birds do they have?
Note: Do you have to assume that they had at least 1 of each animal?
Problem 3: A rancher divided his herd of cows among his four children. One son got one-half the herd; his oldest daughter got one-fourth the herd; another son got one-fifth the herd; and his youngest daughter got the remaining 48 cows. How many cows were in the rancher's original herd?
Try to solve all three problems without using formal algebraic equations.
Hint: For all three problems, try to guess-and-adjust.
When solved, ask is there another way I could do this (especially problems #2 and #3) by drawing a picture?
Solution Commentary: No solutions provided...though there are many different solution techniques.