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Julia Li Stamp Investigation This piece of worksheet will be presenting the process of investigating a pattern for stamps.1.The table below shows which postage values of 1 cent up to 20 cents can/cannot be representedfrom 3 cent stamps and 5 cent stamps. Postage Stamps Postage Stamps 1 No 11 5+3+3 2 No 12 3+3+3+3 3 3 13 5+5+3 4 No 14 5+3+3+3 5 No 15 5+5+5 6 3+3 16 5+5+3+3 7 No 17 5+3+3+3+3 8 5+3 18 5+5+5+3 9 3+3+3 19 5+5+3+3+3 10 5+5 20 5+5+5+5(Table 1)2.After postage value 7, every postage value can be represented. Therefore 7 is the largest postagenumber that cannot be represented according to Table 1.3.The two values of stamps has been changed to 3 cent stamps and 7 cent stamps. Which leads to adiﬀerent table: Postage Stamps Postage Stamps 1 No 11 No 2 No 12 3+3+3+3 3 3 13 7+3+3 4 No 14 7+7 5 No 15 3+3+3+3+3 6 3+3 16 7+3+3+3 7 7 17 7+7+3 8 No 18 3+3+3+3+3+3 9 3+3+3 19 7+3+3+3+3 10 7+3 20 7+7+3+3(Table 2)
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Therefore the postage value 11 is the largest number that cannot be represented out of 3 cent stampsand 7 cent stamps. (Source: Table 2)4.The country issues two stamps - 3 cent stamp and n cent stamp.Working with diﬀerent values of n to find the largest postage value that cannot be represented.3 cent stamps & 4 cent stamps (Table 3) 3 cent stamps & 6 cent stamps (Table 4)Largest number: 5 Largest number: Infinite3 cent stamps & 8 cent stamps (Table 5) 3 stamps & 9 cent stamps (Table 6)Largest number: 13 Largest number: Infinite
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3 cent stamps & 8 cent stamps (Table 7) I have chosen to replace number 4, 6, 8, 9 & 10 for n, because all of the numbers are larger than 3. I also think 5 others are enough for many.Largest number: 17Final table showing all of the largest post numbers that cannot be represented to compare: 3 cent Diﬀerence n cent Diﬀerence Largest postage value stamp stamp that cannot be represented 3 +1 4 +1 5 3 +2 5 +2 7 3 +3 6 N/A Infinite 3 +4 7 +4 11 3 +5 8 +5 13 3 +6 9 N/A Infinite 3 +7 10 +7 17(Table 8)5.Finding pattern Let x=Largest postage value that cannot be represented According to Table 8, n - 3 = x - n, because their diﬀerences are equivalent. Too simplify thisequation: n-3=x-n= n-3+n=x= 2n - 3 = x
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Therefore the rule is: 2n - 3 = The largest postage value that cannot be represented through 3cent stamps and n cent stamps. Although if n is a multiple of 3, the rule won’t work, because Table 4 & 6 has proven that thelargest postage value that cannot be represented is infinitely large. If n is a multiple of 3, n is justadding on more 3s. The amount you add won’t change a fact, the amount stays as a multiple of 3forever. Not every number can be represented out of 3, only 1/3 of all positive numbers can berepresented from 3 and its multiples. So there’s no end of postage number that can be represented,the pattern will go on like this:no, no, 3, no, no, 3+3, no no, 3+3+3... My rule: 2n - 3 only applies when n isn’t a multiple of 3. Table 8 has shown the diﬀerences.Every time working on a pattern, you usually try to find the diﬀerence between the numbers, and Ifound the diﬀerence.6.The text below shows my prediction answering for numbers 21-30 using my rule.The numbers 21, 24, 27 & 30 will not be done for the reason of being a multiple of 3.22. 2n - 3 23. 2n - 3= 2 * 22 - 3 = 2 * 23 - 3= 44 - 3 = 46 - 3= 41 = 4325. 2n - 3 26. 2n - 3= 2 * 25 - 3 = 2 * 26 - 3= 50 - 3 = 52 - 3= 47 = 4928. 2n - 3 29. 2n - 3= 2 * 28 - 3 = 2 * 29 - 3= 56 - 3 = 58 - 3= 53 = 55
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7.Justifying my rule.I am choosing to pick a random number to replace n: 13The table below proves that 23 is the largest postage number that can’t be represented. Post Stamps Post Stamps Post Stamps age age age 1 No 11 No 21 3+3+3+3+3+3+3 2 No 12 3+3+3+3 22 13+3+3+3 3 3 13 13 23 No 4 No 14 No 24 3+3+3+3+3+3+3+3 5 No 15 3+3+3+3+3 25 13+3+3+3+3 6 3+3 16 13+3 26 13+13 7 No 17 No 27 3+3+3+3+3+3+3+3+3 8 No 18 3+3+3+3+3+3 28 13+3+3+3+3+3 9 3+3+3 19 13+3+3 29 13+13+3 10 No 20 No 30 3+3+3+3+3+3+3+3+3+3Using my rule: 2n - 3n = 13= 2 * 13 - 3= 26 -3= 23This justifies that my rule works.8.Further math: Extending the problem This problem can be extended by including 4 cent stamps and n cent stamps to compare with3 cent stamps and n cent stamps. Then we can try to find another pattern, if there’s anotherdiﬀerence and a formula to convert.
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