School-Safe Puzzle Games

## The Alpaca Riddle

Can you think through and tease out logically what happened in the scenario below?-thanks to Roel for submitting!

There is a story of a man who left seventeen alpacas to his three sons. He left half the alpacas to his eldest son, a third to his middle son and a ninth to the youngest. The three set up to divide their inheritance but couldn’t negotiate a solution – because seventeen could not be divided between two or three or nine. The sons finally consulted an old wise woman. After pondering the problem, the old woman said, “See what happens if you take my alpaca.” So then the sons had eighteen alpacas. The eldest son took his half – that was nine. The middle son took his third – that was six. And the youngest son took his ninth – that was two. Nine and six and two made seventeen. They had one alpaca left over. They gave it back to the old wise woman.

You can submit your explanations below; will wait until Thursday or Friday to unmask!

### 30 Comments to “The Alpaca Riddle”

1. seelesscacti | Profile

1/2, 1/3, and 1/9 are 17/18 of a whole.

2. Shawn | PUZZLE GRANDMASTER | Profile

1/2 + 1/3 + 1/9 does not = 1.0, but rather 17/18. The father set his kids up for failure from the beginning! The old woman realized that by adding an alpaca to the mix she could match the common denominator of 18 and allow for an even distribution, and with a remainder of 1.0 to boot – she gets her alpaca backa!

They also could have solved the problem, and spared themselves a trip to see “Crazy Betty,” by each calculating their share of alpacas and “rounding up.”

oldest = 1/2 * 17 = 8.5, so 9 alpacas
middle = 1/3 * 17 = 5.67, so 6 alpacas
youngest = 1/9 * 17 = 1.89, so 2 alpacas

Technically, each son got more than their allotted share.

The boys took away from this experience:

1. intact alpacas instead of lopped-off pieces (the likely alternate scenario)
2. validation of the feeling that the oldest brother always gets the better deal
4. lifelong indebtedness to some old lady who happened to own an alpaca

3. billybob | Profile

A half, a thrid, and a ninth only equal 95%.

4. Hendy | Profile

1/2 or 9/18
1/3 or 6/18
1/9 or 2/18

(9+6+2)/18 = 17/18
That does not account for the whole lot of alpacas. One cannot divide 17 alpacas in half, thirds, or ninths without slaughter involved. Half of an alpaca would be rather messy too. Pull out all the organs and divide them in half and then cut the remaining parts in half. What kind of sick father would setup an inheritance this way anyhow?

5. Someone | Profile

1/2 + 1/3 + 1/9 = 17/18, not 1, thats why they had one left over.
They didn’t actually divide the alpacas up correctly, if they had they would have had 17/18 of an alpaca left over.

6. DoodleFish | Profile

This works because 1/2 + 1/3 + 1/9 = 9/18 + 6/18 + 2/18 = 17/18
As the fractions of alpacas left to the three sons add up to less than one, it is possible to borrow an extra alpaca and then give it back at the end.

7. Diego | Profile

1/2 + 1/3 + 1/9 = .9444, by dividing the 18 alpacas that way you will be left with .0556 that was not given to anyone which is equivalent to 1 alpaca every 18. The old woman just made it easy to do the math, they could just have rounded up the numbers

8. marsyk | Profile

With the instructions the man left there would be 1/18 of the alpacas left
Because 1/2, 1/3 and 1/9 = 4,5/9 + 3/9 + 1/9 = 8.5/9 (appr. 94,44%)

What effect this has is related to how many alpacas there are to share.
The three sons were to share 17 alpacas. Unless they’d settle with pieces of alpaca meat two of them would be disappointed – because 4.5/9, 3/9, 1/9 of 17 is correspondingly 8.5, 5.66 and 1.82.

The wise women figured out that if she tried to find the number of alpacas that times the loss from each alpaca (100-94.44=5,66) would equal 100 she’d found a solution. 5,66*X=100

With the help of some algebra she came up with 18.

PS. The man demonstrated impressive foresight.

9. money226156 | Profile

the boys just needed an even number to divide them between themselves.

10. michaelc | Profile

The old man gave away a total of 17/18 of his 17 alpacas to his 3 sons. Once the woman put in her 1 alpaca the 1/18 share went back to what was hers to begin with. He goofed!

Hey, come to think of it, I think the father had something going on with the old wise woman! That’s why she had the other alpaca!

11. suineg | PUZZLE MASTER | Profile

cool puzzle indeed, I think one explanation is this:
you have to divide 17 that its prime which is impossible but for itself so:
the old woman(very gifted one)knew that 1/2+1/3 +1/9=17/18; not 100% of the 17 alpacas; but if you add 1 to the 17 alpacas you got 18 alpacas; 1 alpaca is 1/18 of the proportion that is exactly what you have left when the parts are divided between then brothers, that equals 1 alpaca, cool.

12. jasc | Profile

1/2 + 1/3 + 1/9 + 1/18 = 1

13. Obiwan | Profile

1. 1/2 + 1/3 + 1/9 = 9/18 + 6/18 + 2/18 = 17/18.
2. The father did not give 1/18 of the total to any of the sons.
3. Temporary addition of one alpaca to the herd makes each of the
fractions (e.g., 1/2) equivalent to whole numbers. That is, 1/2 of 18 = 9,
whereas 1/2 of 17 = 8.5 animals. This son gets a half animal extra.
Some sons get extra, some less. When these deltas are added to the 1/18
that was not assigned, the sum is zero.
4. Essentially, the temporary addition of one animal allow for 17/18s of the
temporary herd to be divided into whole animals, with 1/18 of the heard
being “extra” so that it can be returned to the wise woman.

14. Obiwan | Profile

PS: this is a rather old riddle.

15. the_god_dellusion | Profile

yes.
Well they didnt take the fractions that were suggested by their father of the 17 apacas but rather of the 18 apacas. Also note that a half plus a third plus a ninth add up to 17/18th’s.

So they took the fractions of the 18 Alpaca’s so since the fractions ‘coincidently’ add up to 17/18th’s there will only be 17 out of the 18 Alpaca’s included in the dictribution and there will be one left over.

Please note that the fractions of the original 17 Alpaca’s were not inherited by each son.

16. aaronlau | Profile

1/2 + 1/3 + 1/9 = 17/18
Strictly speaking, the father did not leave all of his assets, 1/18 was supposed to be left behind! Which the sons “took” to split among themselves.

17. alexc | Profile

1/2 + 1/3 + 1/9 equals 17/18 which means that if divided how the father wanted there will always be a remainder

18. MFox | Profile

The wise woman knows about least common denominators. She figured out that the LCD of 1/2, 1/3, and 1/9 is x/18. So when she added up these fractions, 9/18 + 6/18 + 2/18, she got 17/18, which is less than 1. She realized that whatever the size of the herd, by the dead father’s math there would be 1/18th of the herd left over, and if the number wasn’t a multiple of 18, some poor alpaca would be amputated. So by adding an alpaca, she preserved their health, created a herd divisible by 2, 3 and 9, and a remainder of one alpaca. Pretty wise indeed.

Personally, I would have summed the weights of the alpacas, and tried to subdivide the herd so that each son got the appropriate proportion of the total alpaca mass, irrespective of the number of entire alpacas. If the numbers didn’t quite work out, I bet it would have been possible to shave some alpacas to make it work, and I’d have kept the wool as payment.

19. MFox | Profile

It occurs to me that my solution would run up against the same problem, so that I’d inevitably be receiving 1/18th of the total herd-weight in wool.

20. fuzzy | Profile

) This is quite easy. The problem was that the father did not will his entire herd to begin with. His will said to give:
1/2 + 1/3 + 1/9 = 9/18 + 6/18 + 2/18 = 17/18
18 is the common denominator.
That’s why it’s so easy to do with 18 and have 1 left over, and near-impossible with 17.

I bet what happened was he had 18 alpacas to begin with, and he just gave one away to the woman, who was his lover. And then he made sure the numbers worked out for his sons.

21. bobg | Profile

it isn’t possible! but it is the closest approximation of full alpacas.

9/17 != 1/2
6/17 != 1/3
2/17 != 1/9

they did NOT split the animals according the the father’s wishes, but they did the best without killing any animals

22. Mashplum | PUZZLE MASTER | Profile

The father bequeathed only 1/2 + 1/3 + 1/9 or 17/18 of the herd to his sons. If the remaining 1/18 herd or 17/18 of an animal was divided among the sons using the same ratios, the first would get 17/36 of an alpaca to go with the 8 1/2 alpacas he already had. This is just shy of 9, by 1/36 to be exact. But after the second divvy there would still be 1/18 of the 17/18 unclaimed. This could be chopped up too, but there would always be a fraction left. As that fraction approaches zero, the number of alpacas claimed by each son will approach their final tally.

If we focus just on the eldest son we get (1/2)[17 + 17/18 + 17/(18*18) + 17/(18^3) + 17/(18^4) …]. Consider the infinite series (1/2)*[a*r^(n-1)] where a=17, r=(1/18), and n is the set of positive integers. Since |r| < 1 the series is convergent and its sum is (1/2)*[a/(1 – r)] = (1/2)*(17/[1 – (1/18)]) = 9. Therefore, 9 is the correct number of alpacas for the eldest son. Similarly, 6 and 2 are appropriate for the other two.

So what does the wise woman do? By adding her animal, she becomes a 1/18 part owner of the herd. When the sons claim their 1/2, 1/3, and 1/9 respectively, all of these are now whole numbers. There is still 1/18 left, but it no longer makes sense to divide that remainder amongst them since there is a 4th owner who deserves it.

23. cameo | Profile

The man did not divide his alcapas to his sons to a total of 100%. There is an unappointed 1/18 share. So the old woman let them take her alcapa knowing that at 18 alcapas, the sons would be working their shares at a number that is a multiple of 2,3 and 9 and there would be 1 left unaccounted for.

24. joe | Profile

Well for starters splitting any number by a half , then a third, then a ninth doesnt get you all of them. By adding one to 17 you do get a number divisible by 2, 3 and 9 leaves one left over whch is given back of course.
Something like that?

25. joe | Profile

Sorry my first line should read splitting any number into a half, a third and a ninth does not give you the whole…

26. RK | Founder | Profile

If you had a hard time understanding this problem, check out one of the more detailed explanations provided by:
-Shawn,
-Mashplum
-Marsyk
– Obiwan
– MFox
– Fuzzy (wow, you and Michaelc sure think alike!)
-the_god_delusion (makes some good points which may also really help you understand the problem better)
-Diego
-Cameo
-BobG and Hendy (the animal activists!)
-DoodleFish

Hmmm, it’s lunch time here, and I’m going to grab a burger. Alpaca, anyone?

27. maybethisislove | Profile

if they gave one back to the women then there is 17…. not 18
plus think about it! the father gave half of 17? thts not a whole number! so he realy only got 8 or 9

28. Tor.Q | Profile

1/2 + 1/3 + 1/9 not = to 1… does make sense…