Example research essay topic: Acute First Three - 2,193 words

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The Pay Phone Problem Introduction This coursework is about finding all the possible combinations for putting in to payphones various diffrent 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 cours work, 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 10 p and 20 p 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 70 p. These are all the combinations for 10 p. 10 p There is only 1 combination for 10 p. These are all the combinations for 20 p. 10 10 20 There are 2 combinations for 20 p These are all the combinations for 30 p 10 10 10 20 10 10 20 There are 3 combinations for 30 p These are all the combinations for 40 p 10 10 10 10 20 20 10 10 20 10 20 10 20 10 10 There are 5 combinations for 40 p These are all the combinations for 50 p 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 50 p These are all the combinations for 60 p 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 60 p These are all the combinations for 70 p 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 70 p Now I have collected all my results I shall make a results table. Results Tabl 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 formula 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 ad together. Therefore 13 + 21 = 34. 34 = T 8 To test out this they I have done a list for 80 p to check that my formula works, according to the pattern 80 p should have 34 sequences in it. 80 p 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 formula. Investigation 2 This is an investigation to show the different combinations of putting in a 10 p and 50 p 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 40 p as I assume that if I started with a 10 p, the first three results would all be the same, as the 50 p would be redundant in any call less than 50 p; 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 40 p 10 10 10 10 There is 1 combinations for 40 p These are all the combinations for 50 p 50 10 10 10 10 10 There are 2 combinations for 50 p These are the combinations for 60 p 50 10 10 10 10 10 10 10 10 50 There are 3 combinations for 60 p These are the combinations for 70 p 10 10 10 10 10 10 10 50 10 10 10 50 10 10 10 50 There are 4 combinations for 70 p These are the combinations for 80 p 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 80 p These are the combinations for 90 p 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 90 p 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 sequence 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 formular's 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 formular's work. As I found out, the last two parts of the equation 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 formular's relate to the coins used. Tn = Tn - 1 + T 10 p coin - 5 = 50 p 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 10 p and 20 p coins: - 1 = 10 p coin - 2 = 20 p 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 10 p 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. I have noticed that for both the 2 investigations there is a shortcut for one of the answers. In the first formula using 10 p and 20 p you can predict the total of the coins.

In this example it would be 30 p. This is also true of the experiment using the 10 p and 50 p. If you found out the combination for the 10 p and 50 p then you could give the answer for 60 p (the sum total of 10 p and 50 p). 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 20 p coin and a 50 p coin or even tried using a pound coin.

Research essay sample on Acute First Three