A farmer has 90 sheep. He wants to take them to the market across the river and sell them. There is no way to cross except via a ferry. The price the farmer pays the ferryman for his service is half the number of sheep that the farmer take across.
What is the minimum number of sheep that the farmer can get away with paying the ferryman?
Edited 21 February 2006 at 10:11 PM for consistency of wording. Changed references to ‘you’ to ‘the farmer’.
I think this problem needs a bit more description. I’m assuming the farmer must eventually take all 90 sheep, and he has unlimited trips to do it in (because if he can take any amount of sheep, the answer is 0; if he has to take all sheep but has only one trip, the answer is 45).
If the ferryman is not willing to cut sheep up, then the farmer can simply do 90 trips with 1 sheep each and avoid paying at all. So the answer in this case is 0.
If the ferryman is not willing to cut sheep up but requires a payment of at least one sheep per crossing, and he’s nice enough to round down, the farmer can make 30 crossings of 3 sheep each, giving one sheep to the ferryman each trip. So the answer in this case is 30.
If the ferryman is willing to cut up sheep (and the farmer lets him do it), the answer is 45.
Hi Rohan. Yes, the farmer wants to take all the sheep across — there’s no advantage to him leaving any sheep behind.
The ferryman will round up odd numbers so if the farmer takes 3 sheep, he will have to pay two.
No cutting of sheep is required 🙂
If the ferryman always rounds up, then that means the farmer always loses at least half his sheep on every trip. So the best possible case means he loses half of his sheep in total, so he may as well take all 90 sheep at once, paying the ferryman 45.
I feel like I’ve missed something… some kind of trick 🙂
I also suspect that it’s a think outside the box type of question.
Taken as is, I agree with rohan, and say that you have to pay 45. But there are other options:
1) Build your own ferry.
2) Beat the piss out of the ferryman, throw him overboard, and ferry yourself across.
🙂
The farmer has a unique medical condition, which makes him violently ill when he thinks about hurting anyone. Sorry, BHB 🙂
The answer is less than 45.
I will give you a clue. It’s an order of operations question. I don’t think it’s a trick question. In fact, once we figure out the answer, I would be fascinated as to why people (including myself) didn’t think of the solution. That’s why I posted this question in the first place.
I’m reading it over and over again. The only other thing I can think of has to do with the wording.
“A farmer has 90 sheep.”
“The price you pay the ferryman for his service is half the number of sheep that you take across.”
The question implies that the farmer is someone who is not me. Therefore, the price listed is just for me, but not the farmer. But I doubt this is a riddle based on semantics.
You said it was an order of operations question? If the ferryman gets half of what is taken across I don’t see many operations happening, just the splitting of the sheep.
I bet I’m going to feel real dumb when you tell us the answer, eh?
Apologies, my wording was inconsistent. I should have written ‘the farmer’ instead of ‘you’.
Here’s the last clue: the ‘operations’ I’m talking about aren’t confined to mathematical operations. Think about the order of events. At what stage do you (the farmer) pay the ferryman?
I claim that this is a trick question, precisely because it is ambiguous. It is ambiguous in exactly the right location, so that it shows how we all make the same default assumption when faced with a lack of extra information. (Or, equivalently, how we tend to interpret the situation in the same way without any further description.)
Thus, there are multiple correct solutions, each corresponding to different interpretations. For example, 45 is a correct solution, although I believe most people wouldn’t realise all the assumptions they are making when deriving it (which is related to your order-of-operations clue).
However, it is clear that what you are after is the solution to the ‘non-standard’ interpretation. An interpretation that is true to the spirit of the puzzle (e.g. you can’t build your own ferry) whilst still being different enough to be surprising.
Isn’t it fascinating how we become so accustomed to ‘reading between the lines’ that we don’t notice it when we do it?
It doesn’t really say when the farmer has to pay the ferryman. It does imply that he pays at the end of the trip, but I don’t see how that changes things. The only other solution I can think of is that he pays the farmer zero sheep and just splits the profits from selling the sheep at the market.
I would hope that the problem doesn’t imply that the farmer pays at the end of the trip. I mean “what do the farmer pay the ferryman for his service” in the same way as we say “how much do you pay for a movie ticket?”
So here’s a partial solution. Say you (the farmer) pre-pay or (or at least pre-arrange) your paymement to the ferryman (as we would do to buy a ticket for the bus).
“Okay, mate, how many sheep are you taking across?”
“I am taking X sheep. So here’s X/2 for your fee, my good man.”
X is not 90 because you’ve paid some of them away. I mean, why would you bother taking some sheep across that you’ve just paid and can’t sell?
So, once you’ve paid your toll, how many sheep would you take across? Isn’t that less than 90?
Can someone offer me a number for the number of sheep paid?
Then it would be 60 because half of that is the 30 he left behind.
Oi, I can’t believe I didn’t think of that. Thank god I don’t work in finance.
BHB said that this could be a ‘think outside the box’ question and considering how long it took all of us to ge the answer, maybe some lateral thinking was required.
My puzzlement is when did the box get put in? The problem is straightforward. Beldar said it was ambiguous. I don’t think it was deceptively ambiguous. Okay, so maybe if I had said ‘half the number of sheep the farmer wants to take across’, then that would have led us to the answer more quickly. But my statement (present continuing tense) was legitimate too (a la ‘How much do you pay for a bus ticket?’)
Why do we assume that we have 90 sheep and therefore we pay 45? It’s clear when you say, “Let x be the number of sheep we take across. Therefore, x + 0.5x = 90.”
Dad says that the solution I’ve offered for this problem is the businessman’s solution — careful interpretation, minimal effort, big payoff.
Is that our problem? We’re too literal and not entrepreneurial to cut it with the best of them?
Or (as Joee suggested to me via email) that we’re not a bartering society anymore and aren’t used to thinking this way. The money track is so disassociated with the goods track that we can’t think straight when they cross over.
So many questions.
Well, I just assumed it was a riddle. I never take things at face value, which is probably just as much of a flaw as always taking things at face value.
And I’m an American. Of course I would have said 45…we Americans are pretty wasteful like that. 😛
Not everything in the world is pre-paid, like bus tickets. In some cases you even get a choice whether to pre-pay or post-pay. Thus, I don’t expect people to assume one or the other by default in this context. Despite the bus ticket analogy, this situation feels sufficiently unfamiliar to me that I wouldn’t make that connection between the two.
> Why do we assume that we have 90 sheep and
> therefore we pay 45? It’s clear when you
> say, “Let x be the number of sheep we take
> across. Therefore, x + 0.5x = 90.”
Careful. You are making an unstated assumption here, just like most people are when they say the answer is 45. The ferryman might have a post-pay policy going. (A keenly entrepreneurial ferryman might consider this.)
I believe that ‘half the number of sheep the farmer wants to take across’ is the way most people interpret this problem – they do not think about goods vs money, bartering or “order of operations”.
As to the “businessman’s solution”, I dispute this being the one of “minimal effort”, since it required more exertion of my brain cells. Although, maybe businessman’s brains are wired differently. (Or maybe you meant “minimal effort” to mean minimal cost?)
Too literal? If we were, then we would realise there were multiple interpretations. Our problem is that we are more than literal – we read between the lines (but in a non-optimal manner in this case).
Joee’s suggestion sounds right to me. I wonder what it would be like to live in a bartering society? Instead of always worrying about how much money I have (not that I do), instead I would constantly be thinking about the value of any particular item I happen to have in my possession. Maybe that society will value their surroundings more, instead of being obsessed with value being primarily measured with money?
“Despite the bus ticket analogy, this situation feels sufficiently unfamiliar to me that I wouldn’t make the connection between the two.”
Why? I want to know why people by default assume it’s a post-pay system when most things in Western society are pre-pay. In fact, I don’t think it’s even sensible to run the ferry service as a post-pay system. Shouldn’t you define how many sheep you want to take across first, then determine the payment, regardless of the time you actually hand the sheep over?
When I say ‘minimal effort’, I do mean ‘minimal cost’. Can you imagine the cost in resources, time and planning it would take to build our own ferry?
Anyway, that wasn’t the point of the question. The point was to work through the scenario you’re given in the smartest way possible (don’t worm your way out of the riddle by inventing an uncle who is really rich so you can borrow money from him to build a bridge than charge people a toll and become a transport tycoon).
I believe that the ’30 sheep’ answer is the smartest solution within the constraints of the riddle (otherwise, a riddle becomes an exercise in making up stories).
What I want to know is:
a) Why we assume a post-pay arrangement. It seems a curious and baseless assumption. Is there a clear bias in the words of the riddle? The ambiguity is deliberate — but why the consistent assumption that leads to 45?
b) Why, even when told that 45 is not the answer, we have such difficulty conceiving the sensible and logical paying arrangment that leads to 30. Such a big pay off for what should be a straightforward mental exercise.
I’m not sure about the bartering culture element. I still think that living as we do, we still should have come up with the answer. I can’t see a significant barrier to the solution, unless the words were misleading.
Witness the common reaction (which was also my reaction) when told the solution — “Oh my god, why didn’t I think of that?”
Other riddles, you say, “Oh! That’s cool/clever/stupid. I’d never have gotten that.”
I think I’ve worked it out. I think I’ve pinpointed what the barrier was. It’s the opening two sentences:
“A farmer has 90 sheep. He wants to take them to the market across the river and sell them.”
People therefore assume he wants to sell ‘them’ (i.e. the 90 sheep). We set up this box (‘Oh, so he wants to sell 90 sheep’) and get stuck inside the box.
Imagine if the problem said. “A farmer has 90 sheep. He wants to sell as many as he can at the the market across the river.” Or even, “He wants to sell the sheep at the market across the river and make as much money as possible.”
No box. No barrier.
> most things in Western society are pre-pay
I’ve always though most things were post-pay. For example, you go into a store, browse around, fill up your shopping basket, and then pay. You get your house painted, then you pay. You fill up your gas tank, then you pay. You get your electricity bill, for electricy you have already consumed, and then you pay. You work for a large company and place an order for 100,000 hammers, the goods arrive and then you pay (minus, perhaps, a deposit).
But lots of things are pre-pay too. Transport tickets, phone calls (sometimes), amusement park rides, private school fees.
It’s hard to weigh up the two, but I still have the impression most things are post-pay. If you pay by credit card, then certainly it will be post-pay!
> What I want to know is:
>
> a) Why …
> b) Why …
I still suspect Joee’s suggestion has an element to play here. We think of sheep being goods and money at the same time, and forget that one aspect interacts with the other. First we think of them as goods, so we know that we have 90 sheep we are taking across. Then we think of them as money, so we cost the trip at 45 sheep.
The fact that the common reaction is “Oh my God” suggests that this isn’t a hard thing to think about at all, and so it is just our usual thinking habits that have led us down the wrong track. This, then, suggests to me that in a different society, e.g. a bartering society, people might not get so attached to the 45 solution. For this reason, I think the opening two sentences aren’t really the main barrier (although they help enhancing it). I would predict that people will still be as stumped as before even with the two sentences changed.
Here are some thoughts I had earlier, which are possibly off on a tangent now, but may be interesting to some:
We have sheep, and they function as goods as well as money. Let’s call the money version “sheep dollars”. You can think about the ferryman payment problem by thinking of converting some of your sheep to sheep dollars so that you can pay the ferryman.
How many sheep do you need to convert? Okay, so we’ve discussed that question enough already. But now we introduce a twist…
The ferryman runs a pre-paid service in one direction across the river, and a post-paid service in the other. Why? The ferryman has a farm on one side of the river where he keeps his sheep. Thus, he would like to be paid on that side of the river – he doesn’t want to take his sheep across if they are already on the right side, and he doesn’t want to pay transport fees to get his goods to the right side if they aren’t already. (You might claim that he doesn’t need to pay transport fees because he owns his own ferry, but in transporting his goods he is possibly losing out on time serving customers, so he is paying in one way or another.)
Now, us western society people like things simple, so we’d prefer if the ferryman just has a pre-pay or post-pay system. Here is one way to interpret the situation in pre-pay manner. Each sheep functions as either a sheep or a sheep dollar. To get a single sheep across the river, the ferryman charges 1 sheep dollar in one direction but only 0.50 sheep dollars in the other. You first convert your goods to ‘money’, and then you take the ferry across, paying the ferryman upfront like you would a bus ticket. (The ferryman acts as both a currency exchange agent as well as transport provider.) This gives us the solutions we expect: 45 sheep at $1/sheep in one direction (total cost = $45), 60 sheep at $0.50/sheep in the other (total cost = $30).
By introducing different prices, we can conveniently hide the pre-pay/post-pay mechanism. The situation hasn’t changed, just the way we think about it. But we can also do this another way: instead of thinking of having different prices in the different directions, let’s suppose that the cost is always the same – it costs $1 to ship a sheep. Now what’s changed is that the ferryman has different exchange rates on the different sides of the river: $1 for a sheep on one side, $2 for a sheep on the other.
The situations are equivalent. In the first case, the different prices reflect different costs of transport, and in the second the different exchange rates reflect the different value (to the ferryman) sheep have by virtue of being on a particular side of the river.
If you want to get even more twisted, you can think of “left bank sheep dollars” and “right bank sheep dollars”, with the ferryman demanding payment in right bank sheep dollars (and running an exchange service with the appropriate exchange rate). But we won’t go there…
I wonder if expert barterers think in a way that is similar to one of the above? Probably not – I bet it’s just me putting my non-bartering society spin on things.
Hahaha! Sheep dollars! Baaaaa… I can imagine the green notes with a sheepish looking sheep on it.
I believe that while barterers might not think like that, a lot of other people do. Instead of thinking of the location-value of sheep, consider our extremely complex (to me) system that puts a time-value on money. That is, money now is worth more than money in the future. This premise (which is based on the premise that people want the certainty and flexibility of money now over potentially greater amounts in the future) has given rise to the concepts of interest, future discounting, net present value, inflation, and (I believe) the driving pressure to consume materials (and destroy the environment) now because there is reduced value in conserving them for the future.
So, Beldar, keep extrapolating with your sheep dollars and you’ll eventually end up with a system that destroys the environment 🙂 Or results in the efficient use of the ferry system.
Baaa… all those sheep will destroy the environment. 🙂
I read something interesting last week. It was written by an economist who was complaining about the “religion of environmentalism”. His claim was that our obsession with teaching children to recycle, use mass transport, avoid pollution, etc, was a form of indoctrination. He was alarmed that his (very young) kids were coming back home from school being brainwashed about the importance of sorting their rubbish, taking the bus and not littering. He subsequently spent some effort in “de-programming” his kids, and complained to the teachers.
I think he has made a good observation that this form of environmentalism is indoctrination. However, I think he has missed an important point about teaching young children (and, to an extent, teaching in general) – it is difficult to extract moral content from factual content and still keep it engaging and meaningful (or, as some like to describe it, to keep it human).
Why would you want to de-program kids from behaving more environmentally?
We’re indoctrinated in lots of things like buying big thirsty cars (in the case of the USA), wanting to look tanned, wanting to own houses. I think sustainable behaviour is a very worthwhile brainwashing syllabus.
If the argument is about how we teach kids about sustainability — indoctrination or understanding — then of course, it’s better to teach understanding. But indoctrination is much better than nothing. Hence my confusion — why deprogram people? Who is being harmed by sorting rubbish, taking public transport and not littering?
The economist (let’s call him Mr E) didn’t agree with what his kids were taught, so that’s why he wanted to de-program them. I don’t think he shares your enthusiasm for sustainability brainwashing.
Mr E was complaining more generally about environmentalism as dogma, and the lack of respect for other people’s beliefs. He contrasted the way environmental beliefs are treated as compared to religious beliefs (in the usual sense that we think of them) – if you hold a certain religious belief, say Catholicism, then your school teacher would usually respect this and not try impose a different set of religious beliefs on you. On the other hand, they will certainly indoctrinate you in environmentalism, and will express confusion (or, in the extreme, indignation) if asked to justify this.
I think it’s hard to teach kids in a completely non-indoctrinating way. For example, books for children often use the concepts of Good and Evil, because kids understand and respond to these, but their use entails indoctrination.
Who is harmed? Mr E might argue that sorting rubbish takes up your time, taking public transport harms car producers and associated industries, while littering provides employment for street cleaners and makes your life more convenient.
While interesting to think about, I can’t say I agree with Mr E’s ‘religious’ beliefs. In a letter he sends to his child’s school teacher, he mentions that he doesn’t believe in the common adage “with privilege comes responsibility”. That, to me, sounds like a dangerously anti-social attitude to life.
Is Mr E arguing against indoctrination or environmentalism?
That is, if we agree that environmentalism is good, is it therefore acceptable to indoctrinate our children in it?
Or is he saying that there will always be people will disagree with the prevailing social thought, therefore indoctrination shouldn’t be used?
Does he consider the teaching of children the times tables or manners by rote ‘indoctrination’?
Both, it seems.
It doesn’t look like indoctrination is compatiable with his beliefs. Although, I don’t know what his response would be if we proposed to indoctrinate kids into his favourite brand of economics.
Times tables are not indoctrination, they are as close to fundamental truth as you can get! 🙂
I told this problem to Nick. He sat quietly for about three minutes and came up with the right solution!
So the reason we have trouble getting the answer is simple — we’re just not smart enough! 🙂
Which problem? The one about the sheep, or the one about whether indoctrination is okay or not? 🙂
(Either way, Nick IS very clever. Good boy. *pat*)