MathDB

Let

(a_1, a_2,\ldots , a_n)

be a permutation of

1, 2, \ldots , n,

where n \geq 2. For each

k = 1, \ldots , n

, we know that

a_k

apples are placed at the point

k

on the real axis. Children named

A,B,C

are assigned respective points

x_A, x_B, x_C \in \{1, \ldots , n\}.

For each

k,

the children whose points are closest to

k

divide

a_k

apples equally among themselves. We call

(x_A, x_B, x_C)

a stable configuration if no child’s total share can be increased by assigning a new point to this child and not changing the points of the other two. Determine the values of

n

for which a stable configuration exists for some distribution

(a_1, \ldots, a_n)

of the apples.

Problem Statement