Prove that any postage greater than 17 can be made using 4 and 7 cent stamps

Question

Please use strong induction for the problem. I know that regular induction doesn't work. I assume there is a proof by logic by simply saying that 18, 19 and 20 cents can be made using these stamps and that you just need to subtract a 7 cent stamp and then add two four cent stamps to get the next higher cent value. However, that is not formaly strong induction.

For the general case (where the values may be different from 4 and 7), see for example Largest integer that can't be represented as a non-negative combination. Or google for the Frobenius problem. — hmakholm left over Monica, May 01 '16 at 03:38

score 2 · Answer 1 · answered May 01 '16 at 03:07

2

Hint 1 $$18=7+7+4 \\ 19=7+4+4+4\\ 20=4+4+4+4+4 \\ 21=7+7+7$$

Hint 2: If $n \geq 21$ and you know that $P(1), P(2),.., P(n)$ is true, then $P(n-4)$ is true.

answered May 01 '16 at 03:07

N. S.

132,525

score 2 · Answer 2 · answered May 01 '16 at 03:09

2

Hint: If you can make any four consecutive denominations you can make any afterward by adding four-cent stamps. So all you have to do is show combinations adding up to 18 to 21 cents.

answered May 01 '16 at 03:09

Syd Henderson

2,061

Prove that any postage greater than 17 can be made using 4 and 7 cent stamps

2 Answers2

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