Frogs

Introduction

There are two blue frogs and two red frogs (as in the diagram above).

A frog can jump over one other frog onto an empty lily pad or it can slide onto an empty lily pad which is immediately next to it.  

Only one frog, at a time, is allowed on each lily pad. 

Now the idea is for the blue frogs and red frogs to change places.  So, the red frogs will end up on the side where the blue frogs started and the blue frogs will end up where the red frogs began.

The challenge is to do this in as few slides and jumps as possible. 

What is the smallest possible number of slides and jumps?  How do you know you have found the smallest possible number?

Support

Use this to help you out!

Step 1: predict what the next patterns may be.  

• You may want to start small with 1 red and 1 blue frog instead before moving onto 2 red and 2 blue frogs etc.

• Do you always have to make a particular move or are there choices to be made, more than one of which will lead you to the answer? How do you decide which option to choose?

Step 2: organise your findings in a systematic way.  

• What is some important information to record?  Number of blue and red frogs?  Number of slides and jumps?  Empty lily pads at any particular point?  

• What information is required to explain exactly what you have done? How can you record what you have done?

• What is the most efficient way of recording the sequence of moves? Hint: it can be done very minimally!

• Hint: as a minimum, you should have two rows: the number of frogs per colour and the number of moves.

Step 3: come up with some general rules.

• What patterns are there in the sequence of moves?

• Does it matter which coloured frog starts? 

• Is there a connection between which frog makes the first move and which makes the last, and does this change as you change the number of frogs?

• What happens if there are a different number of frogs per colour? What is the connection between the number of frogs in each colour and the number of moves it takes?

• What patterns can you spot in the sequences of moves, e.g. which coloured frog moves, where is the empty lily pad at any point, how many jumps and slides are there?

• What happens if the number of red and blue frogs are different e.g. 2 red and 3 blue frogs etc.? Can you develop a rule that allows you to calculate the number of moves needed based on how many frogs there are of each colour?

Step 4: verify (a.k.a. check) that your rules work.  This can be achieved by comparing your drawings with the answers found using your rules.

Step 5: justify (a.k.a explain) why the rules work.

For further support, look at this page on nrich.org

Extension

Is there a pattern that links the number of frogs per colour with the number of moves required? 

Can you generalise these results into a two-way table? Hint: rows are the number of red frogs and columns the number for the number of blue frogs. 

Can you spot further patterns?

The above pattern is related to the position-to-term rule for quadratic sequences.  This is beyond what you are required to learn, but if you are interested, take a look at this video and at this website.