The Frog Problem

There are 7 lily pads and 6 frogs (3 dark frogs and 3 light frogs). The pads are arranged horizontally as shown above. With the center lily pad unoccupied, each dark frog sits on a lily pad on the left and each light frog sits on the right. Your task is to swap the 3 dark frogs and 3 light frogs to the opposite sides in a minimum number of moves leaving the center lily pad unoccupied. What is the minimum number of moves necessary to swap the two sets of frogs? What is the minimum number of moves if you have 15 frogs on each side? Identify and discuss the patterns observed. Pay particular attention to the number of hops and slides as you swap the two sets of frogs. Find a general rule for finding the minimum number of moves for swapping any number of frogs assuming that there are equal number of frogs on each side in each case. Discuss how you would adapt to help develop algebraic skills among children in your field placement.

Rules:

1. The frogs move by hopping or sliding onto a vacant lily pad. 2. After each move, each frog must be on a lily pad. 3. A frog can only hop over one other frog to land onto a vacant lily pad. 4. A frog may slide onto the vacant lily pad next to it. 5. Frogs cannot hop or slide backwards. If you have blocked your frogs into an impossible position, you will have to start all over again.