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To solve this problem, first we re-read the rules to make sure we understood them. Then we got five chairs and four children to act out the roles of the frogs and toads. The toads had jumpers on, while the frogs wore t-shirts. It took a while of jumping and sliding around but we finally found a solution. The grid showing frog and toad slides and jumps helped when we realised that you needed 1 in each to start or you would get stuck.
We found out that if two frogs or two toads are next to each other, you could not move again. On most turns there were two possible moves. We had to think ahead to make sure we weren’t going to end up with two creatures together.
After finding a solution with 2 frogs and 2 toads, we changed the number of frogs and toads and solved the problem again. We made a note of how many moves it took in total to get all the frogs and toads to the other side.
Next, we put all the results into a chart and looked for a pattern.
Kate noticed that with an even amount of frogs and toads on both sides (e.g. 4 frogs and 4 toads) you always got an even number of moves. With an odd number on either side, you always had an odd total of moves.
We started looking at other patterns where the number of frogs and toads was equal on each side. We listed the number of toads and frogs on one side of a table and the total number of moves on the other. We worked out what you needed to multiply the number of each creature by to make the total number of moves and recorded this in the table. We noticed that you need to multiply by one more each time you add a creature.
We also noticed that to find the number you multiply by, you add 2 to the number of creatures on each side. For example if you had 24 frogs and 24 toads, you would multiply by 26. From this we worked out a formula.
n stood for number of frogs/toads on each side.
m stood for the total number of moves
(n + 2) x n = m
This works for odd and even numbers of frogs, so long as there is an equal number of frogs and toads on each side.
What if there were lots of frogs? If there were 2672 toads on one side and 2672 frogs on the other, you would calculate:
2672 x (2672 + 2) = 2672 x 2674
You would need to complete 7,139,584 moves. If each move took you just one second, this would take 118,993 minutes, or 1983 hours, or 82.6 days. That’s without stopping for a second to eat, sleep, drink or go to the toilet!