Question

The numbers $1, 2, 3, 4, 5$ are to be arranged in a circle. An arrangement is bad if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n.$ Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?

$1$

$2$

$3$

$4$

$5$

Solution:

Any single number covers sums $1$ through $5.$ If a consecutive block sums to $n,$ the remaining numbers form a consecutive block summing to $15 - n,$ so sums $10$ through $14$ are automatically covered as well. Thus an arrangement is bad only if it fails to produce $6$ or $7.$

If $6$ cannot be formed, then $1$ and $5$ are not adjacent, and working through the cases forces the arrangement $14352.$ If $7$ cannot be formed, then $2$ and $5$ are not adjacent, forcing $23154.$

These are the only two bad arrangements up to rotation and reflection.

Thus, the correct answer is B.

2014 AMC 12B Problem 18

Solution:

Problem 18 in Other Years