The sequence goes up in a regular pattern - this formula shows this pattern and makes it easy to predict the next value. To get the next value you have to add the 2 previous terms together to get the Nth term.
Therefore if the amount was 60 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
T4 + T5 together.
Therefore
5 + 8 = 13.
13 = T6
To test out this theory I have done a list for 60p to check that my formula works,
According to the pattern 60p should have 13 sequences in it.
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
This list proves that my theory works because there are 13 sequences like I predicted
Using the formula.
Hear is the new table with the results for 60p
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
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
Results table
Formula = Tn = Tn -1 + Tn -5
From this formula I can predict that the next number in the sequence for £1.40 should
Be 26. I can predict this because Tn-1 = 15 and Tn-5 = 5 so 15 + 5 = 20
To prove this, here are all the listings for £1.30
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 50 10
10 10 10 10 10 10 50 10 10
10 10 10 10 10 50 10 10 10
10 10 10 10 50 10 10 10 10
10 10 10 50 10 10 10 10 10
10 10 50 10 10 10 10 10 10
50 10 10 10 10 10 10 10 10
50 10 10 10 50
10 50 10 10 50
10 10 50 10 50
10 10 10 50 50
10 10 10 10 50
10 10 10 50 10
10 10 50 50 10
10 50 50 10 10
50 50 10 10 10
50 10 50 10 10
There are 20 possible sequences in £1.30, which proves my formula correct.
Hear is the updated table with the results for £1.30:
General Investigation.
I have found that my formulas relate to the coins used.
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
