*A mad king, nearing death, wanted to pass his entire fortune to his two sons. The king wanted history to record him as the richest man who ever lived; therefore, he wanted neither son to receive too much of his fortune. The king also wanted to sow unrest in the kingdom; therefore, he wanted one son to receive substantially more than the other.*

*As the king's counsellor, what fraction of his fortune would you recommend giving to each heir in order to fulfill his wishes?*

*Can you give a rationale according to which yours is the best possible recommendation?*

My wife's initial recommendation was to pass two-thirds of the fortune to one son and pass one-third of the fortune to the other son. That way, you sow unrest because one son gets twice as much as the other son; and at the same time, you protect the king's legacy because the king's fortune is one and a half times what the richer son's fortune will be. Likewise, most readers' recommendations landed in the 60%–70% range. (I also received some Machiavellian political calculation and clever recommendations for trust arrangements.)

Looking in particular at my wife's proposal of two-thirds/one-third, a person might observe that if the king cares equally about the two considerations in the problem (the legacy aspect and the unrest aspect), then he might be unsatisfied with the two-thirds/one-third solution—because the legacy ratio (1.5) isn't as great as the unrest ratio (2). If we care equally about the legacy ratio and the unrest ratio, shouldn't we seek a solution in which the two ratios are the same?

So if 1 represents the entirety of the king's fortune, and

*x*is the fraction given to the son who gets more, and 1 −*x*is the fraction given to the son who gets less, then let's arrange matters so that the legacy ratio 1/*x*and the unrest ratio*x*/(1 −*x*) are the same:
1/

*x*=*x*/(1 −*x*).
There is one positive solution to this equation, and it's the reciprocal of the golden ratio φ:

*x*= ½(Sqrt[5] − 1) = 0.61803… = 1/φ.

(Making the comparison in the opposite sense, the ratio of the king's fortune to the larger share is the golden ratio: 1/

So by this analysis, we give one son about 62% of the fortune and give the other son about 38% of it.

*x*= 1/(1/φ) = φ.)So by this analysis, we give one son about 62% of the fortune and give the other son about 38% of it.

The golden ratio φ = ½(Sqrt[5] + 1) = 1.61803… is famous from architecture and the sciences. You can read about it on Wikipedia, or buy Mario Livio's book

*The Golden Ratio: The Story of Phi, the World's Most Astonishing Number*. (I haven't read it.)
This diagram from Wikipedia shows why φ enters into our problem:

In the context of our problem, the two sons' shares are

Reader jeff came up with this same answer, using a spreadsheet rather than an equation:

*a*and*b*, the king's fortune is*a*+*b*, and the total amount will be divided in the golden ratio when (*a*+*b*)/*a*=*a*/*b*; that is, when the total fortune is to the larger share as the larger share is to the smaller.Reader jeff came up with this same answer, using a spreadsheet rather than an equation:

A 50/50 split would maximize Wealthiest but it would minimize Discord. A 100/0 split would maximize Discord but it would minimize Wealthiest. The midpoint of 75/25, therefore, seems like the right choice BUT it doesn't feel right. We're trying to find out the best possible difference between the king's wealth and the younger son's wealth while also finding the best possible difference between the younger's wealth and the elder's wealth.

I created a spreadsheet (table below) to do the calculations for me and came up with 61.8% for the younger son and 38.2% for the elder. That way the mad king's wealth is 1.618 times greater than his younger son's inherited wealth and the younger son's wealth is 1.618 times greater than his older brother's.

Large Share Small Share WealthDiscord(King/ Large) (Large/ Small) 0.75 0.25 1.333 3 0.7 0.3 1.429 2.333 0.667 0.333 1.499 2.003 0.64 0.36 1.563 1.778 0.63 0.37 1.587 1.703 0.623 0.377 1.605 1.653 0.622 0.378 1.608 1.646 0.621 0.379 1.61 1.639 0.62 0.38 1.613 1.632 0.619 0.381 1.616 1.625 0.6180.3821.6181.6180.617 0.383 1.621 1.611 0.616 0.384 1.623 1.604 0.615 0.385 1.626 1.597 0.614 0.386 1.629 1.591 0.613 0.387 1.631 1.584

This leads to the golden ratio because column 1 of the spreadsheet contains values of

*x*, column 3 contains corresponding values of 1/

*x*, column 4 contains corresponding values of

*x*/(1 −

*x*), and the method of the spreadsheet is to zoom in until column 3 equals column 4, that is, until 1/

*x*=

*x*/(1 −

*x*).

***

But maybe we don't care about the two ratios equally? When I gave this riddle to my daughter, she said that one son should get

^{501}/_{1000}of the fortune while the other son should get^{499}/_{1000}of the fortune. To this suggestion I replied, "That's helpful for making the king the richest person ever, but it doesn't sound like very much unrest to me." Her answer: "If my sister got even a little more than me, I'd be furious."
Well OK then!

***

Another way to approach the problem is to define an objective function and maximize it. For example, take the objective function to be

*f*(

*x*) = Min{1/

*x*,

*x*/(1 −

*x*)} (½ <

*x*< 1)

Maximizing

*f*(*x*) means setting things up so that the lesser of the two ratios is as large as it can possibly be. This is a kind of "minimize the worst case" approach, typical of min-max problems.
If you graph

*f*(*x*) on the interval ½ <*x*< 1 (see the heavy blue curve in the image below), you can see that the maximum value occurs where 1/*x*=*x*/(1 −*x*), that is, where both ratios equal the golden ratio φ. Now we can tell the king not only that φ yields equal legacy and unrest ratios, but also that any other approach would have led to one of the ratios being smaller than it is.
Will the king go for it? Maybe, maybe not. Perhaps, like my daughter, the king has a different take on sibling rivalry.