The Payphone Problem Essay, Research Paper

The Pay Phone Problem Introduction This coursework is about finding all the possibe combinations for putting in to

payphones various diffrent coins and using those results to try to find a Formula that

works so you would succsefully 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 courswork , 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 genral case, showing shortcuts and also the relevnce 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. The formula will be used to predict the next number in the sequence without

having to do all the listings. Below are all the listings up to 70p. These are all the combinations for 10p. 10p There is only 1 combination for 10p.

These are all the combinations for 20p. 10 10

20 There are 2 combinations for 20p

These are all the combinations for 30p 10 10 10

20 10

10 20 There are 3 combinations for 30p

These are all the combinations for 40p 10 10 10 10

20 20

10 10 20

10 20 10

20 10 10 There are 5 combinations for 40p

These are all the combinations for 50p

10 10 10 10 10

20 10 10 10

10 20 10 10

10 10 20 10

10 10 10 20

20 20 10

20 10 20

10 20 20 There are 8 combinations for 50p

These are all the combinations for 60p 10 10 10 10 10 10

20 10 10 10 10

10 20 10 10 10

10 10 20 10 10

10 10 10 20 10

10 10 10 10 20

10 10 20 20

10 20 10 20

20 10 10 20

20 10 20 10

20 20 10 10

10 20 20 10

20 20 20 There are 13 combinations for 60p

These are all the combinations for 70p 10 10 10 10 10 10 10

10 10 10 10 10 20

10 10 10 10 20 10

10 10 10 20 10 10

10 10 20 10 10 10

10 20 10 10 10 10

20 10 10 10 10 10

20 20 10 10 10

20 10 20 10 10

20 10 10 20 10

20 10 10 10 20

10 20 10 10 20

10 10 20 10 20

10 10 10 20 20

10 10 20 20 10

10 20 20 10 10

10 20 20 20

20 10 20 20

20 20 10 20

20 20 20 10 There are 21 combinations for 70p

Now I have collected all my results I shall make a results table. Results Table

N T

Amount No. of Ways

10 1

20 2

30 3

40 5

50 8

60 13

70 21

The sequence goes up in a regular pattern – this formular shows this pattern and makes

it easy to predict the next value. To get the next value you have to add the previous

terms together to get the Nth term.

Therefore if the amount was 80 and you had to find out how many ways there are, you

have to take the previous two terms and add them together – so you would add

T6 + T7 together.

Therefore

13 + 21 =34.

34 = T8

To test out this theroy I have done a list for 80p to check that my formular works,

according to the pattern 80p should have 34 squences in it. 80p 10 10 10 10 10 10 10 10

10 10 10 10 10 10 20

10 10 10 10 10 20 10

10 10 10 10 20 10 10

10 10 10 20 10 10 10

10 10 20 10 10 10 10

10 20 10 10 10 10 10

20 10 10 10 10 10 10

20 10 10 10 10 20

20 10 10 10 20 10

20 10 10 20 10 10

20 10 20 10 10 10

20 20 10 10 10 10

10 20 20 10 10 10

10 10 20 20 10 10

10 10 10 20 20 10

10 10 10 10 20 20

10 10 20 10 20 10

10 20 10 10 20 10

20 10 10 10 20 10

20 10 10 20 10 10

10 20 10 20 10 10

20 10 20 10 10 10

10 10 20 20 20

10 20 10 20 20

20 10 10 20 20

20 10 20 10 20

20 20 10 10 20

20 20 10 20 10

20 20 20 10 10

20 10 20 20 10

10 20 20 20 10

20 20 20 This list proves that my theory works because there are 34 sequences like I predicted

using the formular. 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 combinations 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 There are 6 combinations for 90p

These are the combinations for £1.00 50 50

10 10 10 10 10 10 10 10 10 10

50 10 10 10 10 10

10 50 10 10 10 10

10 10 50 10 10 10

10 10 10 50 10 10

10 10 10 10 50 10

10 10 10 10 10 50 There are 8 combinations for £1.00

These are the combinations for £1.10 50 50 10

50 10 50

10 50 50

10 10 10 10 10 10 10 10 10 10 10

10 10 10 10 10 10 50

10 10 10 10 10 50 10

10 10 10 10 50 10 10

10 10 10 50 10 10 10

10 10 50 10 10 10 10

10 50 10 10 10 10 10

50 10 10 10 10 10 10 There are 11 combinations for £1.10 These are the combinations for £1.20 10 10 10 10 10 10 10 10 10 10 10 10

50 50 10 10

50 10 50 10

50 10 10 50

10 50 10 50

10 10 50 50

10 50 50 10

10 10 10 10 10 10 10 50

10 10 10 10 10 10 50 10

10 10 10 10 10 50 10 10

10 10 10 10 50 10 10 10

10 10 10 50 10 10 10 10

10 10 50 10 10 10 10 10

10 50 10 10 10 10 10 10

50 10 10 10 10 10 10 10 There are 15 combinations for £1.20 Amount No. of ways

40 1

50 2

60 3

70 4

80 5

90 6

100 8

110 11

120 15

130 20

Formula = Tn = Tn -1 + Tn -5

From this formula I can predict that the next number in the sequenc for £1.40 should

be 26. I can predict this because Tn-1 = 20 and Tn-5 = 6 so 20 + 6 = 26.

To prove this, here are all the listings for £1.40 10 10 10 10 10 10 10 10 10 10 10 10 10 10

10 10 10 10 10 10 10 10 10 50

10 10 10 10 10 10 10 10 50 10

10 10 10 10 10 10 10 50 10 10

10 10 10 10 10 10 50 10 10 10

10 10 10 10 10 50 10 10 10 10

10 10 10 10 50 10 10 10 10 10

10 10 10 50 10 10 10 10 10 10

10 10 50 10 10 10 10 10 10 10

50 10 10 10 10 10 10 10 10 10

50 10 10 10 10 50

10 50 10 10 10 50

10 10 50 10 10 50

10 10 10 50 10 50

10 10 10 10 50 50

10 10 10 50 50 10

10 10 50 50 10 10

10 50 50 10 10 10

50 50 10 10 10 10

50 10 50 10 10 10

50 10 10 50 10 10

50 10 10 10 50 10

10 50 10 10 50 10

10 10 50 10 50 10

10 50 10 50 10 10 There are 26 possible sequences in £1.40, which proves my formula. There are also several other formulars that I have found which are : Tn = ( Tn – 1 + Tn – 2 ) – 3

Tn = ( Tn – 1 + Tn – 3 ) – 2

Tn = ( Tn – 1 + Tn – 4 ) – 1 All of these formulars work. As I found out , the last two parts of the equasion have to

add up to 5 . eg. Tn – 4 + – 1 = 5 There should be another formula.

Tn = ( Tn – 1 + Tn – 1 ) – 4. This formula will not work because through research I

have found that unless all the numbers in the formula are prime numbers it will not

work and 4 is not a prime number. General Investigation. I have found that my formulars relate to the coins used. Tn = Tn -1 + Tn -5

-1 = 10p coin

-5 = 50p coin So for any formular using coins you could use the formular

Tn = Tn – X + Tn -Y

X and Y being the coins used in the formular. This is also true of the first formular 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 formular :

Tn = Tn -1 + Tn -10

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

unless both X and Y are prime numbers it will not work. I have noticed that for both the 2 investigations there is a shortcut for one of the

answers. In the first formular using 10p and 20p you can predict the total of the coins.

In this example it would be 30p. This is also true of the experiment using the 10p and

50p. If you found out the combination for the 10p and 50p then you could give the

answer for 60p (the sum total of 10p and 50p). Conclusion I have found the formula´s for the payphone problem and I have investigated further,

and found that for all the pay phone problem formula´s, 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.