1. 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 represented
from 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 postage
number 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 a
different 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)

2. Therefore the postage value 11 is the largest number that cannot be represented out of 3 cent stamps
and 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: Infinite
3 cent stamps & 8 cent stamps (Table 5) 3 stamps & 9 cent stamps (Table 6)
Largest number: 13 Largest number: Infinite

3. 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: 17
Final 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 this
equation:
n-3=x-n
= n-3+n=x
= 2n - 3 = x

4. Therefore the rule is: 2n - 3 = The largest postage value that cannot be represented through 3
cent 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 the
largest postage value that cannot be represented is infinitely large. If n is a multiple of 3, n is just
adding on more 3s. The amount you add won’t change a fact, the amount stays as a multiple of 3
forever. Not every number can be represented out of 3, only 1/3 of all positive numbers can be
represented 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 I
found 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 = 43
25. 2n - 3 26. 2n - 3
= 2 * 25 - 3 = 2 * 26 - 3
= 50 - 3 = 52 - 3
= 47 = 49
28. 2n - 3 29. 2n - 3
= 2 * 28 - 3 = 2 * 29 - 3
= 56 - 3 = 58 - 3
= 53 = 55

5. 7.Justifying my rule.
I am choosing to pick a random number to replace n: 13
The 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+3
Using my rule: 2n - 3
n = 13
= 2 * 13 - 3
= 26 -3
= 23
This 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 with
3 cent stamps and n cent stamps. Then we can try to find another pattern, if there’s another
difference and a formula to convert.