The Pied Piper of Hamelin
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
Problem
"The Pied Piper of Hamelin'' is a story you may have come across. This fellow, who is often dressed in very bright colours, drives the many rats out of town by his playing of pipes. I don't think we know how many rats there were - maybe your story books have said how many. We also do not know how many people lived in the town.
Suppose that there were 100 people and 100 rats. Supposing that all the people and rats have the usual number of legs, there will be 600 legs in the town belonging to People and Rats.
But now, what if you were only told that there were 600 legs belonging to people and rats but you did not know how many people/rats there were?
The first part of this month's challenge is to investigate how many people/rats there could be if the number of legs was 600. To start you off, it is not too hard to see that you could have 150 people and 75 rats; you could have had 250 people and 25 rats. See what other numbers you can come up with. Remember that you have to have 600 legs altogether and rats will have 4 legs and people will have 2 legs.
I just chose 600 because that lets you have 100 people and 100 rats. You could now extend this idea by having 120 people and 120 rats (just as an example) using 720 legs and see what numbers you can come up with. Again, to start you off, you could have 270 people and 45 rats; or 320 people with 20 rats. Now you carry on.
You could choose any number you like for the total number of legs; try some out of your own. I had a hidden rule that whatever number I chose it meant that we could have the same number of people and rats. (600 legs 100 people and 100 rats; 720 legs 120 people and 120 rats.)
So you could choose the number of legs for yourself using that same rule. (Allowing you to have the same number of people and rats.)
It's probably time to have a look at all the results that you have got and see what things you notice. You can then tell me about them.
When you look at the examples that I did for you, you might notice:-
| 600 Legs | a) 100 People 100 Rats | |
| b) 150 People 75 Rats | ||
| c) 250 People 25 Rats | ||
| a) Gives equal numbers of People and Rats | ||
| b) Gives twice as many People as Rats | ||
| c) Gives ten times as many People as Rats. |
In the next example:-
| 720 Legs | d) 120 People 120 Rats |
| e) 270 People 45 Rats | |
| f) 320 People 20 Rats | |
| d) Gives equal numbers of People and Rats | |
| e) Gives six times as many People as Rats | |
| f) Gives sixteen times as many People as Rats. |
This seems as if it could be worth looking at more deeply. I guess there are other things which will "pop up'' to explore.
Then there is the chance to put the usual question "I wonder what would happen if ...?''
Good luck, please send solutions in and any further ideas which came from asking "I wonder what would happen if ...?''
Getting Started
If you had one less rat, what could you replace it with to keep the number of legs the same?
How are you keeping track of what you have done?
Student Solutions
Thanks to all who sent in solutions. Here are some from Oakley School and St. George's school. This is what Emily sent us;
If you TAKE 1 rat you have to ADD 2 people
E.G
2 people + 196 rats 98 people + 101 rats
4 people + 195 rats 96 people + 102 rats
6 people + 194 rats
8 people + 193 rats
10 people + 192 rats
etc.
Hope this helps!
Ben, Hannah & Scott sent in these thoughts;
We figured that if you add 1 rat and take 2 people. For example 108 rats and 84 people, 109 rats and 82 people and 110 rats and 80 people.
Otis, Jason & Sophis sent their solutions for 600 legs;
our solutions are:
298 people, 1 rat
298x2 = 596 legs
1 rat = 4 legs
298 people(596 legs) + 1 rat(4 legs)
150x2 = 300 legs
75x4 = 300 legs
150 people(300 legs) + 75 rats(300 legs)
Teachers' Resources
The Pied Piper of Hamelin
"The Pied Piper of Hamelin'' is a story you may have come across. This fellow, who is often dressed in very bright colours, drives the many rats out of town by his playing of pipes. I don't think we know how many rats there were - maybe your story books have said how many. We also do not know how many people lived in the town.
Suppose that there were 100 people and 100 rats. Supposing that all the people and rats have the usual number of legs, there will be 600 legs in the town belonging to People and Rats.
But now, what if you were only told that there were 600 legs belonging to people and rats but you did not know how many people/rats there were?
The first part of this month's challenge is to investigate how many people/rats there could be if the number of legs was 600. To start you off, it is not too hard to see that you could have 150 people and 75 rats; you could have had 250 people and 25 rats. See what other numbers you can come up with. Remember that you have to have 600 legs altogether and rats will have 4 legs and people will have 2 legs.
I just chose 600 because that lets you have 100 people and 100 rats. You could now extend this idea by having 120 people and 120 rats (just as an example) using 720 legs and see what numbers you can come up with. Again, to start you off, you could have 270 people and 45 rats; or 320 people with 20 rats. Now you carry on.
You could choose any number you like for the total number of legs; try some out of your own. I had a hidden rule that whatever number I chose it meant that we could have the same number of people and rats. (600 legs 100 people and 100 rats; 720 legs 120 people and 120 rats.)
So you could choose the number of legs for yourself using that same rule. (Allowing you to have the same number of people and rats.)
It's probably time to have a look at all the results that you have got and see what things you notice. You can then tell me about them.
When you look at the examples that I did for you, you might notice:-
| 600 Legs | a) 100 People 100 Rats | |
| b) 150 People 75 Rats | ||
| c) 250 People 25 Rats | ||
| a) Gives equal numbers of People and Rats | ||
| b) Gives twice as many People as Rats | ||
| c) Gives ten times as many People as Rats. |
In the next example:-
| 720 Legs | d) 120 People 120 Rats |
| e) 270 People 45 Rats | |
| f) 320 People 20 Rats | |
| d) Gives equal numbers of People and Rats | |
| e) Gives six times as many People as Rats | |
| f) Gives sixteen times as many People as Rats. |
This seems as if it could be worth looking at more deeply. I guess there are other things which will "pop up'' to explore.
Then there is the chance to put the usual question "I wonder what would happen if ...?''
Good luck, please send solutions in and any further ideas which came from asking "I wonder what would happen if ...?''
Why do this problem?
Possible approach
Key questions
Possible extension
Possible support