Upon Maharaja's death all his sons (n in number) were summoned by his lawyers to get acquainted with his will. The relevant part directed them to an ornamented jewelry box filled with extremely expensive diamonds, similar in size, color and cut along with detailed, albeit peculiar instructions:
My firstborn son will pick up 1 diamond and take 1/(n+1) of the rest,
My second son will pick up 2 diamonds and 1/(n+1) of the rest,
My third son will pick up 3 diamonds and 1/(n+1) of the rest, etc.
Although there were some unresolved issues (what if some quantity will not be divisible by n+1, why not divide equally, and what happens with the rest after the youngest brother?) - there was no one to answer and the brothers acted exactly "to the selfsame tune and words" of their deceased father.
Strange as may this seem, no problems of cutting occurred and none of the brothers complained about the fairness of the beloved Maharaja.
The original problem, as compiled by B.A. Kordemski, relates the above story concerning 6 brothers and asks how many diamonds were initially in the box.
Out of respect to my audience, I generalized and modified the will to n sons, and now I ask how the quantity of diamonds was augmented each time another son was born?
The narrative appears in Математические завлекалки издание 1998
What a clevel maharajah! There are n^2 diamonds and each son winds up with n of them as follows:
When it is the k'th son's turn, there are n^2 - (k-1)n diamonds left.
After he takes his initial k diamonds, there are n^2 - (k-1)n - k left.
But this = (n+1)(n-k), so 1/(n+1) of this amount is just (n-k).
The kth son gets k + (n-k) = n diamonds.
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And when the kth son is born, the diamond supply needs to be increased to k^2.
It was (k-1)^2, so the number of diamonds to be added is k^2 - (k-1)^2 = 2k-1.
Another way: When the kth son is born, he immediately stands to inherit k diamonds, and each of his k-1 older brothers stands to inherit an additional 1 diamond. k + (k-1)*1 = 2k -1 additional diamonds that need to be added to the inheritance.
Edited on July 17, 2023, 8:26 pm
Clever maharajah! (Spoiler)
