Thursday, June 23, 2011

A Roman Puzzle

One noble Roman dictated a will on his deathbed requesting to split his possessions between his pregnant wife and the unborn child in the following way. If the child is a boy, he should get 2/3 of the total amount, leaving 1/3 to the wife. If the child is a girl, she should get 1/3, leaving 2/3 to the wife. The nobleman died and a few months later his wife gave birth to twins: a boy and a girl. The smartest men and women of the empire were called to resolve the inheritance dispute. One of them - Stevenus Goodmanus - came up with a solution and a proof that it follows the will as closely as possible. What is this solution?

Image by Steven O'Donald, distributed under CCL.

This puzzle celebrates our 20+ puzzle winner, known nowadays as SteveGoodman18.

Answers accepted all day long on Friday June 24th and Saturday June 25th, on our Family Puzzle Marathon. They will be hidden until Sunday morning (EST) and everyone who solved it will get a puzzle point.

thelittlebird said...

I think that the inheritance should be split first, 1/3 playing into the GIRL calculation and 2/3 into the BOY calculation. With the GIRL calc, the daughter would receive 1/3 of 1/3 = 1/9 of the inheritance and the mother receives 2/3 of 1/3 = 2/9. With the BOY calc, the son would receive 2/3 of 2/3 = 4/9 of the inheritance and the mother would receive 1/3 of 2/3 = 2/9 of this part of the inheritance. The final split would be 1/9 for the daughter, 4/9 for the son, and 4/9 for the mother. She deserves the extra for having to raise twins by herself!!

How about if we divide the inheritance in half, then share each half according to his instructions. The boy would get 2/3 of his half, and the girls 1/3 of her half. In summary, the mother would get 1/2 the total, the boy would get 1/3, and the girl 1/6.

Anonymous said...

I would suggest the mythical Stevenus Goodmanus is a cad but that said I'd propose the boy get 4/7, the wife get 2/7, and the girl get 1/7 because this preserves the relative ratios from the original will (the boy gets twice as much as the wife, and the wife gets twice as much as the girl).

--D.

Laura Lynn Walsh said...

W=wife's inheritance; G=girl's inheritance; B=boy's inheritance; T=total.
W= 2G; W=1/2 B; W + G + B = T
W + G + B = T
W + 1/2 W + 2W = T
W = 2/7 T
G = 1/7 T
B = 4/7 T
Wife's amount is half of girl's; boy's is double wife's.

Pamela Foster said...

The mother should get 100%, because she will care for the children as needed.

Tom said...

That does sound Roman, indeed. The guy wanted his infant son to get twice what his wife gets, and wanted his daughter to get half what his wife gets. Nice guy. But anyway, onward to the math of it.

Daughter gets x … about 14.3% (one seventh)
Wife gets 2x --- about 28.6%
Son gets 4x … about 57.1% plus a taste

That would make the ratios about right, if that were his intention. Daughter gets ripped off the worst by far, which was probably his intention anyway. I think women would not have been called to resolve the dispute.

We can also design some other adjustments that would be more equitable. How about if BOTH the son and the wife chip in maybe 20% of their anticipated take toward the unwanted daughter. Like:
Son gets 66% times 0.8 = 52.8%
Wife gets 33% times 0.8 = 26.4%
Daughter gets remainder, 20.8% but she‘d never get it in a Roman court.

Or find some middle ground between those two solutions?

Bean said...

My first instinct is to divide it 1/3, 1/3, 1/3, ending the inherent sexism of the system.

More mathematically, half should be split between mother and son according to the "son" system, with the other half split between mother and daughter according to the "daughter" system. So the son would get 2/6 = 1/3, with the mother getting the other 1/6 from that half. The daughter would get 1/6, while the mother would get the other 2/6 from the half designated for the mother/daughter split. In the end, the mother would get 1/6 + 2/6 = 1/2.

Mother: 1/2, son 1/3, daughter 1/6.

Dennis (of Dennis and Katrina) said...

From the will, it seems the father wants a boy to inherit twice what the mother does, and a girl to inherit half what the mother does. The total of all three inheritances must be 1 (or 100%). We know Mom getting 1/3, or 33% won't work. 30% doesn't work either (30%+60%+15% =105%). 25% is too low (total of 87.5%). Iterations and increasing the number of decimal places leads us to the following approximation:

• Mom gets 28.5714% of the estate
• Boy gets 57.1428% of the estate
• Girl gets 14.2857% of the estate

This does leave 0.0001% of the estate unaccounted for. I submit that this is either too small a difference in local currency to matter - much like a fraction of a penny to us - or that Stevenus Goodmanus keep it as a reward for figuring it out!

Teacher Christine said...

1/3 for the boy, 1/2 for the mom, and 1/6 for the girl

anne-marie said...

Hello,
I think that the husband wanted to give to the mom half as much as their son or twice as much as their daughter.
It should be 4/7 for the boy, 2/7 for the mom and 1/7 for the girl.

SteveGoodman18 said...

The will wanted a son to get twice as much as the wife, or the wife to get twice as much as a daughter. Both of these can be accomplished if the son gets 4/7, the wife 2/7, and the daughter 1/7 of the inheritance.

This seems to be a better solution that splitting the pot 50/50 from the start and then dividing each half - the wife gets too much in that scenario.

kj said...

Solution 1:
The Roman nobleman said to give his wife 1/3 if they had a son, and 2/3 if they had a daughter, so she should receive all.

This leaves nothing for the children, which was not the full intent of his will.

Solution 2:
The Roman felt if they had a son, his son should receive twice what his wife receives, and if they had a daughter, his wife should receive twice what the daughter receives.

Let d = the fraction the daughter receives.

Then, 2d = w = the wife's inheritance,
and 2*2d = 4d = s = the son's inheritance

d + 2d + 4d = 1
7d = 1
d = 1/7
w = 2/7
s = 4/7

This maintains the ranking the father envisions in his family heirarchy. The son is worth twice the mother, who is worth twice the daughter.

But with this solution the mother receives less than if she had given birth to a son, since 2/7 < 1/3. Also, the father felt the son was worth twice the daughter, and with this solution he is worth four times his sister.

Solution 3:
One might expect the wife to receive an average of the two possible inheritances her husband envisioned, and the children should split the rest with the son receiving twice what the daughter receives.

In this scenario,
w = 1/2
s = 2d
1/2 = s + d
= 2d + d
= 3d
=> d = 1/6, s = 1/3

Annie said...

If a boy, boy gets x Mom gets 1/2x
If a girl, girl gets 1/2x Mom gets x

x + 1/2x +1/2x+x = 1 (total inheritance)
3x=1 therefore, x=1/3

Boy gets 1/3, girl gets 1/6, mother gets 1/6 + 1/3 = 1/2

Ilya said...

the fair way to divide would be such that when mom's share taken together with the son's share satisfies the will, and when mom's share taken together with the daughter's share also satisfies the will. so let's make a system of equations that represents this. let's say son's share is X, daughter's is Y and mom's is Z. from that:
X+Y+Z = 1
X = 2*Z
2*Y = Z

solving the above, we get X = 4/7, Y = 1/7 and Z = 2/7. That way, the son gets twice as much as the mom, and the daughter gets half as much as the mom, just like the will specified.

Anonymous said...

As we see there are may be many interpretations to "following the will as closely as possible."
According to some historical evidence, (e.g. C. Koval's book of Mathematical Mix) dated 1975, Roman wise man (presumably Stevenus Goodmanus) told that it is all about proportions. Proportions of the wife's, boy's and girl's inheritance declared in the Will will be preserved if boys will get 4/7th, wife 2/7th and the daughter 1/7th.

It is fair? Definitely not! Mom may sue demanding doubling her portion after the twin delivery. And knowing Roman history, the daughter is likely to poison her brother and kill her father, getting it all eventually.

Since this puzzle may be interpreted in a few different ways, I decided to award puzzle point to everyone who dared to answer: thelittlebird, Andree, D(who are you?), Laura Lynn Walsh, Pamela Foster, Tom, Bean, Dennis, Teacher Christine, anne-marie, SteveGoodman18, kj, Annie, Ilya.

Annie said...

With the Roman "solution" the girl loses again! Instead of 1/2 of what a boy would get, she gets 1/4!!

David said...

Hi, I'm D (although I used an initial because my son and I share it and I figured we might want to try some puzzles together). I try to follow the puzzles when I can and once in a while I've tried to submit answers but I guess this is the first time it's actually posted (and been useful?).

It's funny because my instinct, like Dennis', was to start working the relative ratios as percentages but I realized it wasn't going to work and then I realized it worked quite nicely as a fraction. I'm often asking my kid to convert fractions to percentages but this was a good reminder that plain old fractions not only work as well but sometimes work better.

Maria said...

Annie - I see your point. While the Roman solution preserves proportions of mom's to son's parts and mom's to daughter's parts, it violates the proportions of son's to daughter's parts. Your solution does the opposite. I like it that mom gets 1/2!

Unknown said...

The wife should get .33 of the total inheritance, and the boy gets 2/3 of the remaining inheritance, so he gets .44 and the girl gets .22