Impossible Will

Logic Level 4

After passing away, a rich farm owner had left his 3 sons a dubious will, which stated that:

The eldest son will inherit $\frac{1}{2}$ of all horses in the farm.

The middle son will inherit $\frac{1}{3}$ of all horses in the farm.

The youngest son will inherit $\frac{1}{4}$ of all horses in the farm.

However, no matter how hard the 3 brothers tried to follow these divisions, they could not settle at the right numbers, and any horse couldn't be divided into pieces or fraction. Disputed, they consulted their family's lawyer about this impossible matter.

Then after giving some thoughts, the wise lawyer promised that each heir would earn the number of horses as the will instructed though each of them would need to donate one horse to him afterwards as the lawyer fee.

After signing such agreements, the lawyer instantly distributed the horses into the right amounts for the 3 brothers according to the will, without selling or buying any horses into the farm. As a matter of fact, he did not keep any horses for his fee and only said so to help out for his late master.

How many horses were there in the farm?

The answer is 36.

3 solutions

Worranat Pakornrat
Dec 17, 2016

The original will was impossible to follow because $\dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} = \dfrac{13}{12}$ , which exceed $1$ or $100 \%$ , so no matter what the number of horses is, such division will never be settled into such amounts.

When the lawyer took out $3$ horses from the $3$ brothers, the exceeded fraction had become whole number once more.

That means, for $x$ as the number of horses :

$\dfrac{x}{2} - 1 + \dfrac{x}{3} - 1 + \dfrac{x}{4} - 1 = \dfrac{13x}{12} - 3 = x$

Hence, $\dfrac{x}{12} = 3$ ; $x = \boxed{36}$ .

When dividing these $36$ horses, the eldest son would originally have $\dfrac{36}{2} = 18$ horses, but after "donating" one to the lawyer, he would have $17$ horses.

Then the middle son would originally have $\dfrac{36}{3} = 12$ horses, but after "donating" one to the lawyer, he would have $11$ horses.

Finally, the youngest son would originally have $\dfrac{36}{4} = 9$ horses, but after "donating" one to the lawyer, he would have $8$ horses.

Clearly, $17 + 11 + 8 = 36$ , and the lawyer did not take any horse and unselfishly helped ending the dispute.

I see why an attorney was needed for this one.

Michael Mendrin - 4 years, 5 months ago

Yes, he turned an impossible to a possible one. ;)

Worranat Pakornrat - 4 years, 5 months ago

Another attorney could have argued that those fractions needed to be normalized by multiplying by $\frac{12}{13}$ , so that the sum would be exactly $1$ . Then if the farm had $39$ horses, the boys would get $18$ , $12$ , and $9$ horses respectively, and the attorney then out of the kindness of his heart could have waivered his attorney fees, even though he wouldn't have to and could have lawfully collected $3$ horses for himself. If this had taken me an hour to figure this out, I would have collected the $3$ horses myself, but it didn't.

Michael Mendrin - 4 years, 5 months ago

@Michael Mendrin – Well, maybe the father might want to teach his children lessons about sharing with wits. If they all agreed to "donate" each horse independently at first, they wouldn't need the lawyer as well. (Just a thought...)

Worranat Pakornrat - 4 years, 5 months ago

@Worranat Pakornrat – That's another angle that would work, each son donates a horse to their father's estate, settles the matter, and each takes his horse back. The attorney would be there only to satisfy probate court.

Michael Mendrin - 4 years, 5 months ago

Great question :) I love puzzles like these :)

Yatin Khanna - 4 years, 6 months ago

Thank you. Me, too. 😉

Worranat Pakornrat - 4 years, 6 months ago

AMAZING (and confusing :p)

Manuel Kahayon - 4 years, 5 months ago

Well, not all maths are straight forward. ;)

Worranat Pakornrat - 4 years, 5 months ago

I have a question (doubt) for everybody, this proof is based in the equation $\dfrac{x}{2} - 1 + \dfrac{x}{3} - 1 + \dfrac{x}{4} - 1 = \dfrac{13x}{12} - 3 = x$ for x being the number of horses. My question is the following one: If the lawyer in the end is not going to receive any horses, why in the LHS of this equation we are substracting 3 (horses), althought there is a signed contract for making it?,i.e, everybody knows that $\frac{13}{12} \neq 1$ , then if $\frac{13}{12}x \neq x$ why this equation substracts 3 horses of $\frac{13}{12}x$ horses? shouldn't it be to substract 3 horses of x horses?...

Guillermo Templado - 4 years, 5 months ago

The question specifically stated that the brothers would earn horses they deserve first, so the lawyer couldn't just take away 3 horses. Divide first, donate second.

Worranat Pakornrat - 4 years, 5 months ago

Then if x = 36 and the first son must receive 18 horses and later he has to give one horse, in the end he would have 17 horses, wouldn't he? but in this case, in the end he doesn't give any horse, does it? So he should continue having 18 horses.... The same reasoning can be done for the rest of his brothers/sisters.... and then $x =18 + 12 + 9 = 39 \neq 36 = x$ ... doesn`t it?

Guillermo Templado - 4 years, 5 months ago

@Guillermo Templado – Each of them did donate one horse: 17 + 11 + 8 = 36 as illustrated in my solution.

Worranat Pakornrat - 4 years, 5 months ago

@Worranat Pakornrat – To whom did the horses donate?i.e, Who received all 3 horses?

Guillermo Templado - 4 years, 5 months ago

@Guillermo Templado – Supposedly to the lawyer, but he waived them to end the disputes between the brothers. Someone had to admit their loss, so the lawyer used the legal gap of donation to force each brother to admit that loss.

Worranat Pakornrat - 4 years, 5 months ago

@Worranat Pakornrat – Ok, it has almost conviced me, I have deleted my report... But, now suppose that the lawyer wants the 3 horses,furthemore there is a signed contract, i.e, they must legally give the 3 horses,,it's an obligation, what happens then?

Guillermo Templado - 4 years, 5 months ago

@Guillermo Templado – What we really need here for this discussion is an attorney!

Michael Mendrin - 4 years, 5 months ago

@Michael Mendrin – haha,ok, everything is ok now...

Guillermo Templado - 4 years, 5 months ago

@Guillermo Templado – Thank you for your understanding. If so, another dispute would ensue as the lawyer had become the fourth share and the brothers could claim that they already "donated" or lost 1 horse each.

In short, this will is like a game of greed: there are more demands over the supply. Everyone can't take all what they want: they must learn to admit some loss to make the deal work and compromise among the brothers. Maybe, this is what the father was trying to teach them. :)

Worranat Pakornrat - 4 years, 5 months ago

This is similar to Birbal Akbar stories But a good puzzle😋😋😋😎

Arun Krishna AMS - The Joker - 4 years, 4 months ago

Thanks. I'll read Birbal Akbar then. Lol...

Worranat Pakornrat - 4 years, 4 months ago

S Aditya
Jan 24, 2017

It is quite obvious that the answer will be a common multiple of (2,3,4,).A general pattern is followed which shows that (n/2)+(n/3)+(n/4) exceeds n by 'k'.Here n=12 k.As the lawyer asks for total three horses,therefore he manages each extra horse by not giving them to the sons and neither keeping them.Here k=3 (no. Of extra horses).Therefore total no. of horses =(12 k)=36.

Jingyang Tan
Dec 25, 2016

$\frac{1}{2}$ + $\frac{1}{3}$ + $\frac{1}{4}$ = $\frac{13}{12}$
IN order to make the share between the bros to 1, The lawyer has to take $\frac{1}{12}$ the horses.
The lawyer takes 3 horses = $\frac{1}{12}$
Whats left is $\frac{12}{12}$ = 12 X 3= 36

Impossible Will

The answer is 36.

3 solutions

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