# The Payphone Problem

Extracts from this document...

Introduction

A.M.D.G MATHS COURSEWORK – PAYPHONE PROBLEM 01-May-07

This coursework is about finding all the possible combinations for putting in to

Payphones various different coins and using those results to try to find a Formula that

Works so you would successfully be able to predict how many coins you would have to Put in the payphone for the next total without having to go through all the listings. I

Have tried to set all the possible listings into an easy to read and an easy to follow

Pattern so that if I have made any mistakes they are easy to see. There are three parts

To this coursework, the first 2 parts are an investigation into specific coins used and

After the first 2 investigations there is a formula that works for those coins. The third

Investigation is a more general case, showing shortcuts and also the relevance of prime Numbers to the formula’s from the first 2 cases.

Investigation 1

This investigation is to try and find a formula for putting in 10p and 20p coins into a

Payphone.

Middle

1

20p

2

30p

3

40p

5

50p

8

60p

13

Investigation 2

This is an investigation to show the different comibations of putting in a 10p and 50p

into a pay phone and seeing if there is any pattern that forms from the results. From

this pattern I will try and find a formula. In this investigation I have started the call cost from 40p as I assume that if I started with a 10p, the first three results would all be the same, as the 50p would be redundant in any call less than 50p; therefore the data that I accumulated for the first three results would be useless and the formula would be incorrect.

These are all the combinations for 40p

10 10 10 10

There is 1 combination for 40p

These are all the combinations for 50p

50

10 10 10 10 10

There are 2 combinations for 50p

These are the combinations for 60p

50 10

10 10 10 10 10 10

10 50

There are 3 combinations for 60p

These are the combinations for 70p

10 10 10 10 10 10 10

50 10 10

10 50 10

10 10 50

There are 4 combinations for 70p

These are the combinations for 80p

50 10 10 10

10 50 10 10

10 10 50 10

10 10 10 50

10 10 10 10 10 10 10 10

There are 5 combinations for 80p

These are the combinations for 90p

10 10 10 10 10 10 10 10 10

10 10 10 10 50

10 10 10 50 10

10 10 50 10 10

10 50 10 10 10

50 10 10 10 10

Conclusion

Tn = Tn -1 + Tn -5

-1 = 10p coin

-5 = 50p coin

so for any formula using coins you could use the formula

Tn = Tn - X + Tn -Y

X and Y being the coins used in the formula.

This is also true of the first formula using the 10p and 20p coins:

-1 = 10p coin

-2 = 20p coin

Further investigation shows that this theory can be disproved to an extent. The theory

will only work if both the coin values used are prime numbers. For example if I used a 10p coin and a £1.00 coin the formula:

Tn = Tn -1 + Tn -10

would not work because 10 is not a prime number. As with the X and Y formula,

unless both X and Y are prime numbers it will not work.

Conclusion

I have found the formulas for the payphone problem and I have investigated further,

And found that for all the pay phone problem formulas, prime numbers are very

Important. If I had more time to investigate I would of tried all the possible coins and

Found their formula and seen if prime numbers were important in those to, for example 20p coin and a 50p coin or even tried using a pound coin.

L.D.S

BY: ALEX WHICHELO

This student written piece of work is one of many that can be found in our GCSE Pay Phone Problem section.

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