Problem 587

We begin by naming the boxes as shown.

[[IMAGE0]]

Of the five answers given, the integer which cannot appear in box $M$ is 20. Why?

Since boxes $F$ and $G$ contain different integers, the maximum value that can appear in box $K$ is $8\times9=72$.

Since boxes $H$ and $J$ contain different integers, the minimum value that can appear in box $L$ is $1+2=3$.

Next, we consider the possibilities if 20 is to appear in box $M$.

If 3 appears in box $L$ (the minimum possible value for this box), then box $K$ must contain 60, since $60\div3=20$.

However, there are no two integers from 1 to 9 whose product is 60 and so there are no possible integers which could be placed in boxes $F$ and $G$ so that the product in box $K$ is 60.

If any integer greater than or equal to 4 appears in box $L$, then box $K$ must contain at least $4\times20=80$.

However, the maximum value that can appear in box $K$ is 72.

Therefore, there are no possible integers from 1 to 9 which can be placed in boxes $F,G,H,$ and $J$ so that 20 appears in box $M$.

The diagrams below demonstrate how each of the other four answers can appear in box $M$.

[[IMAGE1]]

Hide/Reveal Description of Diagrams

From left to right, the boxes in the top row contain the numbers 6, 8, 1, and 2, the boxes in the middle row contain 48 and 3, and the box in the bottom row contains 16.From left to right, the boxes in the top row contain the numbers 8, 9, 1, and 2, the boxes in the middle row contain 73 and 2, and the box in the bottom row contains 24.
From left to right, the boxes in the top row contain the numbers 6, 7, 2, and 4, the boxes in the middle row contain 42 and 6, and the box in the bottom row contains 7.
From left to right, the boxes in the top row contain the numbers 4, 9, 1, and 3, the boxes in the middle row contain 36 and 4, and the box in the bottom row contains 9.