The Farmer’s Will

My amigo CyberDave sent me this puzzler:

This is what they call HORSE SENSE:
A farmer died leaving his 17 horses to his three sons.
When his sons opened up the Will it read:
My eldest son should get 1/2 (half) of total horses;
My middle son should be given 1/3rd (one-third) of the total horses;
My youngest son should be given 1/9th (one-ninth) of the total horses.
As it's impossible to divide 17 into half or 17 by 3 or 17 by 9,
the three sons started to fight with each other.
So, they decided to go to a farmer friend who they considered quite smart,
to see if he could work it out for them.
The farmer friend read the Will patiently, after giving due thought,
he brought one of his own horses over and added it to the 17.
That increased the total to 18 horses.
Now, he divided the horses according to their fathers Will.
Half of 18 = 9. So he gave the eldest son 9 horses.
1/3rd of 18 = 6. So he gave the middle son 6 horses.
1/9th of 18 = 2. So he gave the youngest son 2 horses.
Now add up how many horses they have:
Eldest son  9
Middle son  6
Youngest son  2
TOTAL IS 17
Now this leaves one horse over, so the farmer friend takes his horse back to his farm.
Problem Solved!

I’m sure the intended reaction is “Huh?” But it’s pretty easy to explain this puzzler.

There are two problems with the Will. First, this distribution leads to fractional horses. And second, the fractions (1/2, 1/3, 1/9) do not add up to One. Even if you could do the division, there would be left-over horse.

Let’s restate the fractions using their lowest common denominator (54). According to the Will, the oldest son receives 27/54 (half) of the horses; the middle son receives 18/54 (a third) of the horses; the youngest son receives 6/54 (a ninth) of the horses. These add up to 51/54, leaving 3/54 of the horses not going to anyone. How much is 3/54 of 17 horses? It is 0.944444444 - in other words, just about one horse.

Let’s compare what the sons “should” have received from the Will versus what they actually did receive.

	Should have received	Actually received
Oldest	8.5	9
Middle	5.66666666666 …	6
Youngest	1.88888888888 …	2
Total	16.0555555555 …	17

The part about the other farmer bringing his horse over and then taking it back is just a smokescreen. It covers up the fact that all the numbers have to be rounded up to get to 17.

If you still don’t see the trickery, let’s suppose we aren’t distributing 17 horses but rather 17 dollars.

Then, the oldest son gets $8.50, the middle son gets $5.67, and the youngest gets $1.89, and everyone is happy. Of course, these amounts add up to $16.06 and not $17, but the Will doesn’t say the sons are to receive 17 of anything. The Will says only that each son will receive a certain fraction of 17. The additional (and unstated in the Will) constraint that we impose on the distribution is that the amounts have to be whole numbers because we don’t want to chop up a horse. By choosing to round the numbers up, the sons altogether receive almost one more horse than they were left by the Will. As we’re talking about horses, that rounding up seems logical (and humane).