Price of precious stone broken into pieces

Let price $P = k w^2$. Original price $= k (18^2) = 324k$.

Let the weights of the 4 pieces be $w_1, w_2, w_3, w_4$ such that $1 \le w_1 < w_2 < w_3 < w_4$ and $\sum w_i = 18$.

To maximize $\sum w_i^2$, make weights as spread out as possible: $\{1, 2, 3, 12\} \implies 1^2 + 2^2 + 3^2 + 12^2 = 1 + 4 + 9 + 144 = 158$

To minimize $\sum w_i^2$, make weights as close as possible: $\{3, 4, 5, 6\} \implies 3^2 + 4^2 + 5^2 + 6^2 = 9 + 16 + 25 + 36 = 86$

Difference in total price $= k(158 - 86) = 72k = 288000 \implies k = 4000$.

Original price $= 324k = 324 \times 4000 = 1,296,000$.