On Friday I posted this puzzle….

There exists a grassy field that contains no other food.  11 goats could last there for 7 days before they run out of food.  10 goats could last 8 days there.  How long could 3 goats survive there?

If you have not tried to solve it, have a go now.  For everyone else the answer is after the break.

In the first scenario there are 77 goat days of food.  In the second case, there is 80 goat days.  Therefore the field produces enough grass each day to sustain 3 goats, and so the 3 goats could live there forever.

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UKhere and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

1. Kristian says:

Oooh, and I thought I was super clever, for figuring that out.

Oh well. I love this solution much more than any of the 18-26 days answers gained from extrapolating.

Happy new year!

2. The Masked Twit says:

Elegant solution.
Better than mine – I had to use equations.
I’m looking forward to seeing the posts from those who had more than one answer on Friday
Toodlepip

3. Niva says:

Yep the 3 goats can eat forever! That delighted me when I got it because it was so unexpected based on my tackling similar puzzles in the past.

1. Eddie says:

How did you do it? I couldn’t do this puzzle? I got totally bogged down with rates of growth of grass and couldn’t make the equations work… 😦

2. Niva says:

@Eddie: The two pieces of info (11 goats can eat for 7 days, 10 goats can eat for 8 days) can only be reconciled if the food (grass?) is growing at a rate that will feed 3 goats per day. Here’s how I did it … In the second scenario (10 goats for 8 days), after 7 days 70 goat-days of food would be gone. They need 80 goat-days of food to last 8 days, We know that over 7 days there would be 77 goat-days of food. So 80 – 77 = 3 goat-days of food must have grown between day 7 and day 8. Hope this helps.

3. Eddie says:

Thanks. Yes, the equations do work. The grass grows at the same rate as 3 goats in the field can eat it.

4. Chris Martin says:

Give me some of that grass! 🙂

1. UDontnome says:

😀

5. Richard has (as he sometimes does) given the correct answer to a question which is different from the one he asked. The original question specified the time that 10 or 11 goats would last *before they ran out of food*, but then asked how long 3 goats would *survive*. The unknown variables needed to answer the question concern the time it would take the goats to starve to death *after* the grass has run out, and how much nutrition they would get from cannibalising each other. Thus I was correct in my original answer, when I stated that we didn’t have enough information.

1. But if the field produces enough food for 3 goats a day it doesn’t matter how long they will survive after they run out of food, as they won’t run out of food.

2. wheeliebinchimaera says:

These goats die the moment there is no more food left !

3. Dave's Neighbour's Mum says:

You must be a hoot at parties, John.

4. Dave's Neighbour's Mum's (American) Neighbor says:

Heeehheeee *snerk* Ahhhahahahahahaha!

5. Tony says:

John Loony (@JohnLoony) I do hope your joking. Most people realise that this was posted as a puzzle and as a puzzle that implies certain rules about how puzzles are interpreted. You take the facts provided are create a puzzle type question which produces a puzzle type answer. You don’t put your most pedantic head on, be as smug as possible and be an ass about it in front of everyone else who understand how these thing work. You were joking though yeh?

6. Goatse says:

@johnloony even if we agree that you’re correct (which I absolutely do not, as you’re just making a childish justification for being wrong), then you still lose a whole lot of points for not even understanding the question (one that everyone else seems to). Not sure why Richard draws so much hate in his comments. Don’t like his puzzles? Go back to FarmVille.

6. 8 goats can eat for 10 days implies at least 80 goat days of food. If there are in fact only 80 goat days of food, and the grass grows too slowly to replenish, then 11 days could supply 80/11 = 7.27 goats. So far this fits the original question as it is put – the question gives no information about grass growth. Since you can’t have .27 goats then 7 goats could survive 11 days, with a bit of grass left over. There is no indication in the question that the grass would grow sufficiently fast to replenish itself to support and specific number of goats.

You have to supply your own additional information to the problem, that the difference between the 7 and 8 goat cases is accounted for by the growth rate of the grass. The provided solution is a mistake based on the questioner assuming information that is not provided, but which might be guessed at or inferred.

The suggested solution is 80/3 = 26 days. Grass grows at something like 2 – 6 inches a month. 26 days is close to a month, so will the field replenish itself enough? Do goats require 2 – 6 inches a month? We now need to know the volume of grass requried per goat, and the area, in order to figure out if 2 – 6 inches provides enough volume.

So, ‘puzzles’ like this are not problems that have a definite solution, but problems for which all the questioner’s information has not been provided. You get the puzzle ‘right’ (i.e. the answer that agrees with that of the questuioner) if you happen to guess the missing details. You can come to a quite different and valid answer by assuimg different unspecified conditions, or by taking the puzzle at face value and working with what was given.

1. Pete says:

Plus the grass will stop growing over a cold winter so rate of replenishment will drop.

2. Anonymous says:

No. You can’t have partial goats, but you can certainly have partial days. If there were a remainder in the 11-goat 7-day scenario, it could be accounted for in hours (and minutes, seconds, etc.) the only logical way to resolve the discrepancy is to deduce that the grass is growing. It wouldn’t exactly be a riddle if it gave you the answer, now would it?

7. The Moron says:

I agree with Ron Murphy and The Loonster. There is so much information missing from the puzzle. How old are the goats? Their gender mix? Their sexual orientation? Do they like eating sour grapes?
Come on Richard – let’s have a proper puzzle this Friday.

1. The Moron says:

Having read all the subsequent whingeing and nit-picking I now think that the puzzle is even better.
Well done Ricardo!!

8. Stevie says:

I do enjoy all of the clinical evaluations from those that over think these puzzles.
I gave it no thought from a maths point of view, I’m not even sure that I could, as on seeing the puzzle reasoned that the three goats would not overtake the grass growth rate.
Surely, if all of the information isn’t given, the solution needs to be theorized.
Or to put it another way; get over it!

1. smoodaboo says:

Well said Stevie!!

9. “…and so the 3 goats could live there forever.”

Wait, what? Goats are immortals?!?

10. Anonymous says:

The three goats could live there for five days, until a freak meteor strike kills them all.

Not based on information provided in the question, but random extra information I have created to devise my answer.

That, or 26 days.

11. Rubyreddevon says:

I’d like to say that I worked it out mathematically, perhaps I did but basic animal husbandry provided the answer instantly.

12. Ron is correct that the proper answer >=26 since one doesn’t know for sure what the rate of grass growing is.

13. John Rowland says:

Nice one – (since I got it!)

14. ctj says:

11 goats lasting for 7 days sets a minimum of 77 gdf (goat days of food) and a max of 88 gdf, because we know they run out of food before day 8. 10 goats lasting for 8 days sets a min of 80 gdf and a max of 90 gdf. so the field contains somewhere between 80 and 88 gdf. thus, 3 goats will last between 26 and 29 days.

the solution provided is wrong because it relies on information that is not provided in the question, and which is not necessary to solve the puzzle.

1. Kristian says:

My take on it: The two scenarios don’t provide min/max values. Ideally I think the riddle would’ve stated “7 days 10 hours, 32 minutes, 5 seconds”, or whatever exact time, the food ran out: The values stated are exactly 7 days and 8 days, so the scenarios must have each 77 gdf and 80 gdf.

Assuming the values are exact, it produces another puzzle: how the gdf value change?

One solution is that the goats ate more on the 8th day – maybe it was cold, maybe one was giving birth – any guess goes.

Another answer is that the grass replenished at a rate of 3gdf per day.

Admittedly, this way there are countless viable explanations. But personally I think Richard’s solution was within reason and I liked the puzzle.

This question assumes not only that the grass has a growth rate, but also that the starting point is non-zero (i.e. the field is ‘overgrown’ on the first day). If you accept that the grass grows 3 goat days of food per day and you assume a starting point of zero (a well groomed field) eleven goats wouldn’t last even one day.

1. Smylers says:

Not really: what you’re calling assumptions could be derrived from the question — in that there obviously needs to be something to explain the non-linear difference in the numbers given.

However, what the answer given does presume is that there is that the food exactly runs out at the points stated. Otherwise ctj’s answer above is also valid. The question relies on there being exactly 77 goat days of food in the first scenario, not somewhere in the range 77–87.

2. Kristian says:

@Smylers the non-linearity could be explained in many ways, grass growth being one of them. Maybe the goats ate extra on the 8th day – maybe it was cold.

We introduced grass growth into the puzzle, because we know it from our own experience.

But we might aswell have introduced a locust swarm or cows from the neighboring field.

Or we could’ve made up a combination of them all. Whichever we pick, we can’t reliably predict the gdf values.

I admit I’m being nitpicky – I like Richard’s solution – but introducing new concepts into a riddle is problematic, because it also allows wizards and miracles. That said, I think grass growth is within reason, and I liked the puzzle.

16. Julia says:

I managed to work out that the grass keeps growing, but I failed to do the maths. 😦

17. Mike Torr says:

I don’t care about whether the puzzle has an unambiguous answer. I only care that ‘for ever’ seems to have become ‘forever’ and there is nothing I can do to prevent it. 😦

18. -M- says:

Not enough accuracy in these variables…
in the first case there are 7×11 = 77 food units
in the second case there are 8 x 10 = 80 food units
so three more, which means there is 3 more every day.

NOT SO FAST!

If there would only be 1 more food unit every day:
the 10 goats would have 80 food units and would run out of food after 80/10 = 8 days.
the 11 goats would have 79 food units and would run out of food after 79/11 = 7.182 days which is on day 7.

In that case there would be 72 food units available on the first day.
After 35 days, there are 107 food units. 3 goats eat that in 107/3 = 35.6 days. That means they will run out of food on day 36 in that case.

19. david dredge says:

A very good puzzle even though I didn’t get it. Thanks Richard!

20. wheeliebinchimaera says:

Its fairly obvious that there are two different sets of answers depending on how one interprets the question. My first approach was that the difference between the two given examples was because the grass didn’t run out exactly at the end of the 8th (or 7th) day (obviously this rare breed of goat dies the moment there is no food left) – this gave me between 26 and 29 days. As this type of puzzle generally has one answer I tried the second approach that the difference in goat food days was because of grass growth (and I think we have to assume that would be constant). This gives two simultaneous equations:
I + 8g = 80 & I + 7g = 77 (where I is the initial amount of grass and g is the growth rate per day). Solving tells us that there were 56 goat food days of grass to start and it grows at the rate of 3 per day.

21. Lazy T says:

yes, butt…..

1. Eddie says:

You’re really getting my goat.

22. NickC says:

If f = food per goat per day, x = available food already in the field, y = food growth per day then
11*7*f = x + 7*y or 77*f = x + 7*y ——– equation 1
and
10*8*f = x + 8*y or 80*f = x + 8*y ——– equation 2
Subtracting equation 1 from equation 2:
3*f = y or simply the rate of growth of grass = 3 times the consumption of one goat per day
thus the field can sustain 3 goats for the rest of their lives.
(Assumptions are steady rate of growth and steady consumption of food per day per goat)

1. Eddie says:

Totally agree. Nice one, NickC.

23. Surely though food growth rate is a function of the remaining food?

24. Anonymous says:

I agree with the above comments, no mention in the original question about time lapsing between two hypotheical scenarios.

25. Z says:

In the first scenario there are 77 goat days of food after 7 days. In the second scenario there are 80 goat days of food after 10 days. These are consistent with 70 goat days of food at the start and a groth rate of 1 goat-day of food each day. So 3 goats will consume all food after 35 days.

26. John says:

I still go with my original answer “Until they run out of food.”

27. anon says:

can I suggest a useful puzzle? its answers might help readers acquire \$\$, and isn’t that the HK raison d’etre? So here is the puzzle: please explain HIGH PROBABILITY TRADING in layman language, in 1-2 sentences. TNX!

28. Tony says:

Most people realise that this was posted as a puzzle and as a puzzle that implies certain rules about how puzzles are interpreted. You take the facts provided are create a puzzle type question which produces a puzzle type answer. You don’t put your most pedantic head on, be as smug as possible and be an ass about it in front of everyone else who understand how these thing work. If you dont understand get puzzles dont visit puzzle pages. Go to a maths challenge site.

1. One Eyed Jack says:

First time you’ve visited this blog?

29. Tony says:

I love the way people who can’t work it out blame the puzzle and get all pissy. Love em !

30. One Eyed Jack says:

The answer is 3-4 days in all cases as the goats are given no source of water. 😉

31. They could not live there forever.

I would say the goats could live there for 15 to 20 years (approx. life expectancy of a goat). 😉

32. Excelent info thanks i wanna more site like this 😉

33. The 3 goats will all die come winter, when the grass stops growing. Since we don’t know what time of year it is, the is insufficient information to answer the question of “how long 3 goats would survive?”

1. Billy the Kid aka The Partial Goat says:

Where in the question does it say that the field is in a climate where the grass stops growing in winter?

34. Henry says:

My first instinct was that this puzzler was a variation on the frog in the well, but I got stuck trying to work out how that trick would apply, so I put it aside. It wasn’t until I looked at it fresh the next day that it just hit me that the stupid grass is constantly growing (except during the dormant period in the winter, when presumably the shepherds will bring the goats back to the manger so the baby Jesus can play the drum for them, or whatever.)

1. Bill36 says:

One can easily over-think complicating factors: Goats are being born. As soon as a kid is born and able to eat grass, the four goats will eat grass faster than it is being replenished. No reason all goats die at the same instant, therefore as soon as the first one dies, we are back to a sustainable scenario, provided the three remaining emaciated goats are able to recover. Another complicating factor is that surely a pregnant/nursing goat will eat a bit more than the non-pregnant goats. I’m assuming, of course, that there is a mix of genders: two nannies and one billy. One nanny and two billies will result in one billy being butted out of the field.
Goats need water, so there must be a stream running through the field. Location: grass stops growing in cold weather. But if the field is located in the Miami area, it stops growing in the winter dry season. Well, the stream will keep it watered. Etc.
I go for a simple answer to a simple question. Keeping with just the information supplied, then Richard’s answer is quite satisfactory. And no, I didn’t get it, either.

35. If 3 goats could survive indefinitely, then wouldn’t larger groups of 7 or 8 groups eventually dwindle down to 3 surviving goats that could then survive indefinitely?

36. Anonymous says:

Pretty Clever Riddle