1
Problem You have an inexhaustible supply of two kinds of stamps, a cent and be cent.
Which amounts can be made using just these stamps? Which amounts cannot be made?
In this lesson we will be exploring a modified version of this problem where we set a=5 and try to answer the question for b=2, b=3, b=4 and more general values for b.
The goal of this investigation is to get students engaged in a problem that is easy to understand, is easy enough to start mathematical exploration without much prompting, and where the solution is not obvious. The process in formulated to suggest the kind of approach a mathematician would use to attack the same problem if he or she only had secondary school mathematics at his or her disposal. For more information on this and for other examples please refer to the resource section.
You have an inexhaustible supply of 5 cent and 8 cent stamps
(i) Which of these amount can be made using 5c and 8c stamps only?
23c, 42c, 76c, 951c, 13632c, 27c, 17c, 11c, 14c, 26c, 28c
(ii) Make a complete list of the amounts between 1c and 99c which cannot be made.
Total Amount 0 1 2 3 4 5 6 7
Can be made
Cannot be made
2 You have an inexhaustible supply of 5c and 13c stamps.
(i) Which of these amounts can be made using 5c and 13c stamps only?
23c, 24c, 25c, 26c, 73c, 87c, 351642c
17c, 84c, 36c, 37c, 47c, 48c, 49c
8 9

(ii) Make a complete list of the amounts between 1c and 99c which cannot be made.
Total Amount 0 1 2 3 4 5 6 7 8
Can be made
Cannot be made
3 Complete and extend the below table until you think you can say exactly which amounts can be made using 5c and 2c stamps only, and which amounts cannot be made.
9
Total Amount
Good?
0 1 2 3 4 5 6 7 8
Bad?
4. Complete and extend the table below until you think you can say exactly which amounts can be made using 5c and 3c stamps only, and which amounts cannot be made.
5.
Suppose you have only 5c and 4c stamps. Use a table like the one below and extend it until you think you can say exactly which numbers are good (can be made) and which are bad (cannot be made)?
6.
Suppose you have only 5c and 6c stamps. Draw up a table like the one in problem 5 and extend it until you think you can say exactly which numbers are good and which are bad.
9

7.
Suppose you have only 5c and 7c stamps. Draw up a table like the one in problem 5 and extend it until you think you can say exactly which numbers are good and which are bad.
a 5 5 5 . 5 5 b
2 3 4 . 6 7
Apparent cut off point
3/4 7/8
8.
i)Use your solutions to problems 3-7 to complete the table below. ii) For each of the pairs a, b below draw up a table like the one in problem 5
(a) a=5, b=8;
(d) a=5, b=11;
(b) a=5, b=9;
(e) a=5, b=12;
(c) a=5, b=10
(f) a=5, b=13 iii) Use your solution to (ii) to extend the above table as far as b=13.
9.
(i) Look at the list of cut off points you complied in problem 8. Try to find a pattern which would you guess the likely cut-off point when a=5, b=14.
(ii) Check your guess by drawing up a table like the one in Exercise 5 for a=5, b=14.
(iii) Guess the likely cut-off point where a=5, b=15. Then check whether your guess was correct.

(iv) Guess the likely cut-off point when a=5, b=16. Then check whether your guess was correct.
10.
(i) Suppose a=5 and that b is some number which is not a multiple of 5. Guess a formula which gives the cut-off point in terms of b. Then check whether your formula really works.
(ii) Suppose a is any old number (not necessarily a=5). Can you guess a formula which gives the cut-off point in terms of a and b? If you can guess such a formula, check whether it works for these pairs.
( a ) a=6, b=7;
Hints to the Problems
( b ) a=6, b=8 ; (c) a=6, b=9.
9.
(i) 3/4, 7/8, 11/12, ...., 19/20, 23/24, 27/28, 31/32,...., 39/40, 43/44
Solutions
1.
(i) 23, 42, 76, 951, 13632, 26, 28 (27c, 17c, 11c, 14c cannot be made.)
(ii) 1, 2, 3, 4, 6, 7, 9, 11, 12, 14, 17, 19, 22, 27. a 5 5 5 5 5 5 5 5 5 5 b
2 3 4 6 7 8 9 11 12 13
Apparent cut off point
3/4 7/8 11/12 19/20 23/24 27/28 31/32 39/40 43/44 47/48
2.
(i) 23, 25, 26, 73, 872, 351642, 84, 36, 48, 49. (24c, 17c, 37c, 47c cannot be made)
(ii) 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 27, 29, 32, 34, 37, 42,
47.
3.
It looks as though 1c and 3c are the only amounts which cannot be made.
4.
It looks as though 1c, 2c, 4c, and 7c are the only amounts which cannot be made.

5.
It looks as though 1, 2, 3, 6, 7, and 11 are the only bad numbers.
6.
It looks as though 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, and 23 are the only bad numbers
8.
10.
(i) Guess: Cut off point between 4b-5/ 4 ( b -1 ) .
(ii) Guess: Cut off point between ab( a+b )/( a-1 )( b-1 ). Testing seems to confirm this guess: for example, when a=4 and b=7 the apparent cut off point occurs between
17/18=4x7-(4+7)/(4-1)(7-1)
Resource
A. Gardiner. Discovering Mathematics, The Art of Investigation, Dover Publications, Inc.
Mineola, New York