The price of a precious stone is directly proportional to the square of its weight. Sita has a precious stone weighing $18$ units. If she breaks it into four pieces with each piece having distinct integer weight, then the difference between the highest and lowest possible values of the total price of the four pieces will be $288000$. Then, the price of the original precious stone is

When we say "price is directly proportional to the square of weight," it means:

Price = k × (Weight)²

where k is a constant that depends on the type of precious stone.

Think of it this way: if we double the weight, the price becomes 4 times (2² = 4). If we triple the weight, the price becomes 9 times (3² = 9).

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Given:

Original stone weight = 18 units

Stone broken into 4 pieces with distinct integer weights

Sum of all 4 weights = 18 (since total weight is conserved)

Difference between highest and lowest possible total prices = 288000

We need to find: Price of original stone

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Since Price = k × (Weight)², the total price of 4 pieces will be:

Total Price = k(w₁² + w₂² + w₃² + w₄²)

where w₁, w₂, w₃, w₄ are the weights of the 4 pieces.

Key Insight: To maximize the sum of squares, we want weights to be as far apart as possible. To minimize the sum of squares, we want weights to be as close together as possible.

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To maximize $w₁² + w₂² + w₃² + w₄²$, we need one very large weight and the rest as small as possible.

Since all weights must be distinct positive integers and sum to 18:

Smallest possible weights: 1, 2, 3

Remaining weight: 18 - (1 + 2 + 3) = 12

Maximum weights: 1, 2, 3, 12

Maximum sum of squares: $1² + 2² + 3² + 12² = 1 + 4 + 9 + 144 = 158$

Maximum total price = 158k

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To minimize $w₁² + w₂² + w₃² + w₄²$, we need weights as close to each other as possible.

18 ÷ 4 = 4.5, so we want weights around 4 or 5.

Trying weights close to 4.5:

Weights: 3, 4, 5, 6

Check: 3 + 4 + 5 + 6 = 18

All distinct

Minimum sum of squares: $3² + 4² + 5² + 6² = 9 + 16 + 25 + 36 = 86$

Minimum total price = 86k

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Given: Difference between maximum and minimum prices = 288000

Maximum price - Minimum price = 158k - 86k

= 72k = 288000

72k = 288000

k = 288000 ÷ 72 = 4000

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Price of original stone = k × (original weight)²

= 4000 × 18²

= 4000 × 324

= ₹1296000

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This problem beautifully demonstrates how mathematical optimization works in real life. When breaking something valuable, the way we break it significantly affects the total value!