Le Monde puzzle [#1016]

An even more straightforward Le Monde mathematical puzzle that took a few minutes to code in the train to Cambridge:

  1. Breaking {1,…,8} into two sets of four integrals, what is (or are) the division into two groups of equal size such that the sums of the squared terms from each are equal? Same question for the set {21,…,28}.
  2.  Considering the integers from 1 to 12, how many divisions into two groups of size six satisfy the above property? Same question when the two groups are of different sizes.

The first code is

nop=TRUE
while (nop){
 s=sample(1:8)
 nop=(sum(s[1:4]^2)!=sum(s[5:8]^2))}

with result

1 6 4 7

while the second set leads to the unique [drifted] solution (up to symmetries)

21 24 26 27

and the divisions for the larger set {1,…,12} is unique in the equal case, and are four in the unequal case.

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