MathDB

Source: 2017 Canadian Open Math Challenge, Problem C4 —-- Let n be a positive integer and

S_n = \{1, 2, . . . , 2n - 1, 2n\}

. A perfect pairing of

S_n

is defined to be a partitioning of the

2n

numbers into

n

pairs, such that the sum of the two numbers in each pair is a perfect square. For example, if

n = 4

, then a perfect pairing of

S_4

(1, 8),(2, 7),(3, 6),(4, 5)

. It is not necessary for each pair to sum to the same perfect square.
(a) Show that

S_8

has at least one perfect pairing. (b) Show that

S_5

does not have any perfect pairings. (c) Prove or disprove: there exists a positive integer

n

for which

S_n

has at least

2017

different perfect pairings. (Two pairings that are comprised of the same pairs written in a different order are considered the same pairing.)

Problem Statement