Extending Problems...To Where Students Should Also Go
Problem 1: How can a farmer plant 10 trees so that she has 5 rows with 4 trees in each row?
Problem 2: Stu Dent won some marbles during a game and put them all in one bag. When questioned about the number of marbles he had in the bag, Stu replied, "I do not know exactly, but when I counted them 2 at a time, there was 1 marble left over. Similarily when I counted them 3, 4, 5, and 6 at a time, there was 1 marble left over. But, when I counted them 7 at a time, there were no marbles left over." What is the smallest number of marbles Stu could have had in the bag?
These problems are fun, but it is even more fun to pursue their extensions...
Extension to Problem 1: What happens if you change one or more of the three given numbers? What triples of numbers will produce a solution "similar" (you get to decide what that means) to the solution for Problem 1. Can you generalize the problem further?
Extension to Problem 2: Is there a solution if instead of 2 marbles, there were 2 marbles left each time (or 0 for the case when counting by 2's)? Suppose you keep the remainder of 1 marble constant but alter the 7 by either increasing or decreasing the countingtime when no marbles remain? Finally, suppose the remainder was always 1 less than the the number of marbles you are counting by? Can you generalize?
Hint: For Problem 1, think creatively...does a row always have to be horizontal?
For Problem 2, you can try algebra, but it might be better to try some sample numbers to test out what is happening numerically...plus, the notions of LCM and GCD might prove helpful.
Solution Commentary: Sorry, you are on your own. These two problems are "classics"...and can be found in many different resources.
