The Post Office down the road has run out of all stamps except 3c and nc stamps, where n is any whole number. They have a large amount of each of these. The post master isn’t too worried though. He tells the person on the counter that he can make up every amount from (2n – 2)c onwards. Is that right? Can you justify it or can you find an amount of postage bigger than (2n – 2)c that the Post Office can’t make up?
Solution
It would be a good idea to test the post master’s conjecture out by trying some simple values of n, say n = 5 and 7. For n = 5 you should be able to show that everything from 8 = (2× 5 – 2)c on can be made up. For n = 7 you can show that everything from 12 = (2 × 7 – 2)c on can be made up. So those two cases make it seem as if the post master is right.
Now try n = 6. With some quick calculations you will find that you can only make up multiples of 3. If n = 6, 2n – 2 = 10. And you certainly can’t make up 10c postage with only 3c and 6c stamps.
Extension
Can you find a rule for the values of n for which the postmaster’s rule is true?