Postage stamp with $6$ and $7$ cents question

Question

What is the largest postage in cents that cannot be paid exactly with an unlimited supply of $6$-cent and $7$-cent stamps?

Any hint so that I can proceed?

You might be interested in watching Numberphile: How to order 43 Chicken McNuggets on youtube, which touches on a similar problem, where the packs are 6, 9, and 20 (as opposed to your example which uses 6 and 7). — JMoravitz, Apr 18 '15 at 22:34
related : http://math.stackexchange.com/questions/1181222/proof-that-every-number-%e2%89%a5-8-can-be-represented-by-a-sum-of-fives-and-threes/1181253#1181253 — Fermat, Apr 19 '15 at 19:17
See this answer for the general case of this well-known Frobenius problem. — Bill Dubuque, Apr 19 '15 at 22:08

score 1 · Accepted Answer · answered Apr 19 '15 at 19:14

You are interested in solving the coin (Frobenius) problem for $n =2$ and coins $6, 7$. Although there is no general answer, in case $n = 2$ there is one: because $6$ and $7$ are coprime, the result will be $6 \cdot 7 - 6 - 7 = 29$.

score 0 · Answer 2 · answered Apr 18 '15 at 21:57

0

An idea :

If $n= 7a+6b$, then:

if $b\neq 0$, $n+1 = 7(a+1)+6(b-1) $
if $b=0$, $n+1 = 7(a-5)+6(b+6)$

This gives you 34 as an upper bound of the number you're looking for

Now, you can examine the number lower than 35

answered Apr 18 '15 at 21:57

Tryss

14,310

I have absolutely no idea what you did there. – Aspiring Mathlete Apr 19 '15 at 07:25

score 0 · Answer 3 · answered Apr 19 '15 at 19:11

Let's say you have $n$ cents in six and seven cent stamps. You can obtain $n + 1$ cents if you have at least one six cent stamp, because you can replace it by a seven cent stamp. On the other hand, if your pile of stamps contains only seven cent stamps, then you can only obtain $n + 1$ cents by replacing $5$ of the seven cent stamps by $6$ six cent stamps. You can use strong induction with the procedures mentioned above to show that you can obtain any total of at least $35$ cents using only six and seven stamps. Thus, you must check whether you can construct totals that are less than $35$ cents using only six and seven cent stamps.

Postage stamp with $6$ and $7$ cents question

3 Answers3