A ruby can belong to two people, but it cannot be split without losing its value. Justice, then, does not consist in cutting the stone, but in finding a compensation that both brothers can consider fair.
The trial of the cadi
Riddle statement
Two brothers inherit a ruby that cannot be broken. Both have the same right over the stone, but only one can keep it.
To avoid an arbitrary decision, they go to the cadi. The cadi makes them pass separately and asks each one how much money the entire ruby is worth to him.
One answers that he values it at 120 dinars. The other, in 60 dinars.
The cadi knows that both have answered sincerely. He decides that the ruby should stay with the person who values it most, but he must compensate the other for his part fairly.
How many dinars should he pay?
Show solution
Solution
Answer: you must pay 45 dinars.
Suppose that one heir values the entire ruby at \(A\) dinars and the other at \(B\), with \(A > B\).
As both have the same right over the stone, what really changes hands is not the entire ruby, but half of it. It corresponded to the heir who did not stay with him.
For those who value the ruby at \(A\), that half is worth \(\dfrac{A}{2}\). For those who value it at \(B\), it is worth \(\dfrac{B}{2}\).
If the compensation paid is \(C\), the advantage of the person who receives the ruby is:
and the advantage of the person who gives it up is:
A fair compensation equals both advantages:
Clearing:
With the values of the puzzle, \(A = 120\) and \(B = 60\):
The check confirms the balance. For someone who values the ruby at 120 dinars, half a stone is worth 60; pays 45, and gains an advantage of 15. For someone who values it at 60 dinars, half a stone is worth 30; receives 45, and also gains an advantage of 15.
Both benefit equally. The ruby remains in the hands of the person who values it most, and the advantage of that decision is shared equally between the two.