Question 83
A set of consecutive integers is balanced if it can be divided into two subsets of equal size such that:
(i) the subsets have no common elements;
(ii) the sum of the elements in one subset equals that in the other;
(iii) the sum of their squared elements is also equal.
For example, $\{7,8,9,10,11,12,13,14\}$ is balanced because it can be divided into $\{7,10,12,13\}$ and $\{8,9,11,14\}$, and
$$\begin{aligned}
7+10+12+13&=8+9+11+14,\\
7^2+10^2+12^2+13^2&=8^2+9^2+11^2+14^2.
\end{aligned}$$
a) Verify that $\{1,2,3,4,5,6,7,8\}$ is balanced.
b) Show that every set of eight consecutive integers is balanced.
c) Show that no set of four consecutive integers is balanced.
(i) the subsets have no common elements;
(ii) the sum of the elements in one subset equals that in the other;
(iii) the sum of their squared elements is also equal.
For example, $\{7,8,9,10,11,12,13,14\}$ is balanced because it can be divided into $\{7,10,12,13\}$ and $\{8,9,11,14\}$, and
$$\begin{aligned}
7+10+12+13&=8+9+11+14,\\
7^2+10^2+12^2+13^2&=8^2+9^2+11^2+14^2.
\end{aligned}$$
a) Verify that $\{1,2,3,4,5,6,7,8\}$ is balanced.
b) Show that every set of eight consecutive integers is balanced.
c) Show that no set of four consecutive integers is balanced.