Solution to Riddle of the Week

Solution to Riddle of the Week: Getting the Goat

Difficulty level: Moderate

Michael Stillwell
 
By Jay Bennett

What's interesting about this puzzle is that there is an infinite number of possibilities for the number of cows that the brothers start with, and yet the amount of money owed the brother who gets the goat will always be the same. You can revisit the original question here.

SOLUTION

The brother who gets all lambs must pay $2 to the brother who gets the goat to equalize the share of the profits from the cow sales.

Let's take a closer look at this. If we assume that the brothers started with 10 cows, then they sold those cows for 10 dollars each (because each cow is sold for as many dollars as the brothers have cows), netting the two brothers $100. Then they would have enough money to buy 10 lambs for $10 apiece. But in this case, the brothers have no money left over to buy a kid goat, which the riddle tells us they do. So the brothers could not have started with 10 cows. This examples tells us something us: They could not have started with a number of cows that would sell for a dollar amount that is cleanly divisible by 10, or else they would have no money left over for the goat.

Here's another piece of information that proves crucial to solving the riddle: The amount of money the brothers receive for selling the cows must be a perfect square. Because each cow is sold for as many dollars as the brothers have cows, the amount of money they receive will always be a number multiplied by itself.

Let's try another number for starting cows, say, 15. In this scenario, the brothers sell each cow for $15 dollars, netting them a profit of $225. They can then buy 22 lambs for $10 apiece, leaving $5 left over to buy the lamb. However, this example would give the brothers 22 lambs and one goat. That's a total of 23 animals, and the question tells you the brothers end up with an even number of animals, so 15 cannot be the right number of starting cows.

This failed example gives us the final piece of information we need to solve the riddle. Because the final number of animals must be even, we know the brothers must have an amount of money that allows them to buy an odd number of lambs. That way, once you add in the goat, you've got an even number. In other words, the digit in the tens place of the amount of money they receive from selling cows must be odd. We already know that amount of money will be a square number, so let's take a look at the first 20 square numbers.



We have discovered two key pieces of information: The amount of money received from cow sales cannot have a factor of 10, and it must have an odd number in the tens place. Assuming the brothers have between 1 and 20 cows to start, this conditions are met for 4 cows, 6 cows, 14 cows, and 16 cows.

Let's assume the brothers have 14 cows, so they sell them for $196. This would give them 19 lambs and a goat that cost $6. One brother gets 10 lambs worth $100, the other gets 9 lambs and a goat collectively worth $96. The first brother must pay the second brother $2 so the total transaction has an equal value of $98 per brother.

But here's the thing: If you plug any of the numbers in that work for the riddle, you will find that the goat must have cost $6 (the number in the ones place). If the goat cost $6, then the brother who receives it is owed half the difference of the cost of a lamb and the cost of the goat, or: 1/2 ($10 - $6) = $2. This is the answer to the riddle, the brother who gets the goat will always be owed $2.

If you look at a full list of square numbers, it will quickly become apparent that all numbers that work for the starting number of cows are the ones with 4 or 6 in the ones place. The brothers could have started with 16 cows, or 2,116 cows, and the one will still owe the other $2 when all is said and done.

It helps to practice a little math if you are going to get into dairy farming.