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"Find the smallest value of a such that for every integer ≥ a, it is possible to produce n cents of postage using 4 cent and 7 cent stamps."

My only guess of how to start this is to set a = 11 cents because it is the smallest value obtained with both coins.

Am I on the right track? Where do I go from here? Frankly, I'm not sure I even entirely understand what the problem is asking.

hellojuicebox
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    Regarding understanding of the question: you noticed that it is possible to make 11 cents using those stamps. However, there are many other values (5 cents, 13 cents, etc.) that are impossible to make using those stamps. It turns out that the number of these "impossible" values is finite, and it is actually possible to make any large value with these stamps. The question is asking when this point starts (or equivalently, when the impossible values stop). – angryavian Feb 07 '20 at 22:15
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    This problem has a number of fun names, including the Chicken McNugget theorem. – angryavian Feb 07 '20 at 22:16
  • As a hint: note that if you can achieve $n$, you can achieve $n+4$ always, as you just add another 4 cent stamp. This means the first time we have four consecutive values that can be made, we are done. – WaveX Feb 07 '20 at 22:17

1 Answers1

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One way to solve this problem is to write the positive integers into seven columns.

$$\begin{array}{cccc} 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 8 & 9 & 10 & 11 & 12 & 13 & 14 \\ 15 & 16 & 17 & 18 & 19 & 20 & 21 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \end{array}$$

We will mark values that can be made using 4-cent and 7-cent stamps by circling the number.

First, we can circle the multiples of $7$ (you can obtain such values using only 7-cent stamps). This eliminates the last column.

Next, we can circle each multiple of $4$ since these values can be obtained using only 4-cent stamps. However, once you circle a multiple of $4$, you can actually circle all the numbers below it in the same column, since those numbers are obtained by adding a multiple of 7 to the current value (add 7-cent stamps).

Thus, you just need to circle multiples of $4$ until you hit every column, from which you can then find $a$ by adding one to the largest uncircled number.

This is a nice way to visualize things (esp. when working with students), but it takes a bit more effort to prove the general result $a= mn-m-n+1$.

angryavian
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