Te Kete Ipurangi Navigation:

Te Kete Ipurangi
Communities
Schools

Te Kete Ipurangi user options:

Frogs: Student Notes

Introduction

These notes are designed to provide you with hints to help you while you work towards solving this problem and also to help you see a much larger picture of both the problem and of mathematics itself. The activity is available online at:

 
The notes below are split into four phases:
  1. Understanding and solving specific problems
  2. Generalising
  3. Proving
  4. Extending
The four phases are the same as those used by mathematicians when they are working on research problems.  
 
Below we have provided a number of hints for you to use if you need help. Try to do as much of the problem as you can yourself, and only use the hints if you get stuck.  

Phase 1: Understanding and solving specific problems

This activity asks you to help three brown frogs and three green frogs swap places. Once you have solved this problem you are challenged to help four and then five frogs of each colour. You are then asked to generalise how many jumps it would take for n frogs of each colour.
The rules of the Frog Problem are as follows:
  • Each frog can only move towards the other end of the lily pads (no going back).
  • A frog can hop onto the next empty lily pad.
  • A frog can jump over a neighbouring frog onto an empty lily pad.
  • Only one frog can occupy a single lily pad.
This phase is set out as a series of hints. Try to solve the problem without using the hints, but if you get stuck, hopefully a hint or two will help you.

Hint 1

If you are not sure how to solve the problem, the first thing you should try is just playing with the frogs. Notice that frogs can only move forwards.

Hint 2

You have probably already noticed that frogs can either take a small hop forward into an empty space immediately in front of them or a big jump forward over another frog into an empty space. 

Hint 3

If you move the frogs of one colour too far forward you will make a blockage that the other colour frogs can’t get past.

Hint 4

When you are moving frogs past each other they should be alternating colours (one brown, then one green, then one brown…). If you get two of the same colour next to each other they might create a block that the other colour can’t get past.

Phase 2: Generalising

In this phase we investigate how many moves it will take n frogs of each colour to move past each other.

Hint 5

To work out the number of moves it takes for each number of frogs, start with the smaller numbers. How many moves does it take for one of each colour? (Note that the computer version will not help you do this but you should be able to work it out easily enough). How many moves for two and then three of each colour?

Hint 6

At this stage it is probably easiest if you make a table of the number of frogs of each colour and the number of moves. 

Hint 7

Your table should look something like this (you may have also used extra columns for hops and jumps):

Frogs (of each colour)
Moves
1
3
2
8
3
15
4
 
5
 

Is it worth thinking about hops and jumps too? Why? Why not?

Hint 8

Can you see any patterns in the table? Think about square numbers or relationships between rows.

Hint 9

There are at least three ways of looking at the moves pattern - the number of moves (N) for a given number of frogs of each colour (n) can be written as:
1.       N = (n + 1)2-1
2.       N = n(n + 2)
3.       N = n2 + 2n.
 
Did you get any of these? How? Did you get something else? How?
 

Phase 3: Proving

The hints above will have allowed you to complete the Frogs Bright Sparks activity, but there is more to be done. Next is the most difficult part, proving that your general solution is true. 

Hint 10

How can you get a handle on the formula for N that you have found? Is it easier to break it up into a smaller problem?

Hint 11

How many jumps do there have to be? How many times does every frog of one colour meet every other frog of the other colour? What happens then? Maybe you could add a column for jumps on your table if you don’t already have one?

Hint 12

Think about how far the frogs need to move. How many lily pads does each frog need to move? Can you use this to work out how many moves you need for n frogs? On average, each frog needs to move one space plus the same number of spaces as there are frogs. How many lily pads is this altogether? Is it worth adding another column to your table?

Hint 13

How many lily pads are covered by jumps?
This means that the rest are covered by hops so we should have got n2 + 2n moves in total.
 
Aside: Finally there is a subtle point that you might like to think about. We have assumed in the counting above that it is always possible for the swap to take place. To actually tie this proof down completely we need to show that we can indeed swap the frogs but this requires an argument based on Mathematical Induction. We don’t expect you to be able to complete this step.
 

Phase 4: Extending the problem

 
Now we get back to some fun stuff again after the hard work of proof. One way that mathematics develops is by mathematicians trying to take a problem and extending it in some way. What other similar problems can we build around this one? Come up with your own ideas and see where they lead you. We have given a few examples below to start you off.
  • Different numbers of frogs on each side – for example 3 brown and 4 green frogs.
  • Different sizes of jumps – for example what if the frogs are allowed to jump two other frogs instead of just one.
  • A different number of blank lily pads in the middle.
  • Mix up the coloured frogs on either side before you start.