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More ways to get Braingle...

Survival of the Sheep

Logic puzzles require you to think. You will have to be logical in your reasoning.


Puzzle ID:#9026
Fun:*** (3.03)
Difficulty:*** (2.57)
Submitted By:sakirski***
Corrected By:boodler




There is an island filled with grass and trees and plants. The only inhabitants are 100 lions and 1 sheep.
The lions are special:
1) They are infinitely logical, smart, and completely aware of their surroundings.
2) They can survive by just eating grass (and there is an infinite amount of grass on the island).
3) They prefer of course to eat sheep.
4) Their only food options are grass or sheep.

Now, here's the kicker:

5) If a lion eats a sheep he TURNS into a sheep (and could then be eaten by other lions).
6) A lion would rather eat grass all his life than be eaten by another lion (after he turned into a sheep).

1) Assume that one lion is closest to the sheep and will get to it before all others. Assume that there is never an issue with who gets to the sheep first. The issue is whether the first lion will get eaten by other lions afterwards or not.
2) The sheep cannot get away from the lion if the lion decides to eat it.
3) Do not assume anything that hasn't been stated above.

So now the question:
Will that one sheep get eaten or not and why?


The sheep would remain untouched.

In fact, the sheep would remain untouched if there is an even number of lions on the island, and would be eaten immediately if there is an odd number of lions on the island.

Here's the reasoning:
Consider a scenario with just one lion and one sheep: The lion will eat the sheep. Why? Because after he eats it and turns into a sheep himself, there aren't any lions on the island to eat him, so he is happy.

Now look at a scenario with 2 lions and 1 sheep. Here the sheep would remain unharmed. Why? Because if any one of them eats it, and turns into a sheep himself, he knows that he awaits certain death because he will then be a sheep and the other lion will be the only lion on the island and nothing will stop him from eating the sheep.

So now we know for a fact 1 lion and 1 sheep - sheep gets eaten. 2 lions and 1 sheep - sheep doesn't get eaten.
We can now make a conclusion about 3 lions and 1 sheep: the sheep will definitely be eaten, because the lion that eats it will know that by eating he leaves behind 2 lions and 1 sheep (himself). And as we already know 2 lions and 1 sheep is a situation where the sheep survives.

You can use the same logic to go on to 4 lions and 1 sheep, and then all the way to 100 or 1000, but it will always be true that with an odd number of lions the sheep gets eaten and with an even number the sheep doesn't.


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Dec 18, 2002

thats quite good well done
Dec 21, 2002

That was a good one.
Keep it up
Jan 07, 2003

Although it is the most interesting teaser I've ever read (good job), I don't know if this makes sense. You have 3 lions and one sheep. You suggest that one would eat the sheep. This makes no sense. If a lion eats a sheep, he will get eaten, whether there is an even number or an odd number. So unless there is only one lion and one sheep, the lion would eat the sheep because he would be inevitably eaten. You're using false logic to suggest that because another lion won't get eaten, the first lion (and so on.....until there is only one lion left) will be eaten. Once again...If I've offended anyone, or if I'm incorrect in my logic, email me at
Jan 07, 2003

In response to "pemalova". I am suggesting that if there are 3 lions, 1 of them would eat the sheep. Why I am suggesting this you may have missed. Let's go back for a second. You have to agree that 1 lion and 1 sheep is a no-brainer, the sheep is eaten as there is no threat to the remaining lion. Now for 2 lions, the reason that the sheep will not be eaten, isn't because there is more than one, but specifically because there is 2. HEAR ME OUT, what I am saying is lets look closer at the 2 lion scenario. I am one of the lions, and if I eat the sheep I will leave behind one sheep (myself) and one remaining lion, who then will have a guaranteed freebie in eating you. Therefore when there is 2 lions, they cannot eat the sheep as they are guaranteeing the remaining lion a worry-free snack. If this is a fact, then a 3-lion scenario will go as follows. Any lion (of the 3) can happily eat the sheep because he knows that he leaves behind 2 lions, who cannot touch him; they cannot touch him because if they do they will guarantee the remaining one lion that freebie snack. I hope this makes it more clear.
This situation is actually a pretty standard game theory puzzle, and can probably be found in one form or another in most math books that cover game theory.
Jan 11, 2003

You said with an odd number of lions the sheep is eaten and survives with an odd number.

I see where the sheep is eaten with one lion and lives with two. I understand about getting eaten with htree lions. But at four and above wouldn't the sheep survive no matter what?
Jan 12, 2003

resp. dgtw.

You are quite right. for example if there were 27 lions to use a random odd number then a lion would not eat the sheep because he would become the next lions dinner.

and for 56 lions the first lion would not eat the sheep as it would become supper.

The theory does work for 1 2 and 3 sheep but after this i cannot see how it works.

(let me know if i'm missing something)
Jan 12, 2003

To dgtw & cnic: I think I can understand why you are thinking this way. This is probably because when you work this thing out you think you get to 2 conclusions that contradict eachother, depending on the way you look at it. when there are 27 lions, you think the L27 wouldn't eat since he would be L26's supper, just because "well, why not!". This makes more sense than the real answer: L27 can safely gobble up sheepy since he knows that L26 won't eat him, because L26 knows that L25 WOULD eat him... and so on until L5. Why would L5 eat? because L4 won't eat, because L3 would, because L2 wouldn't, because ONE LION WOULD EAT!!! Don't just look at the numbering as "lion #1", but also as "a situation where there is 1 lion". I hope this was more clear than it seems, and if this doesn't work, the best way is to start from one lion and one sheep and work your way up.
Jan 12, 2003

One would think that the greater the number of lions, the less likely a lion is to eat the sheep, simply because there *may* be more lions willing to take the chance that they will be "last to take a chance." (whew!) This logic, of course defies mathematical justification and relies entirerly on emotion. Make sense?
Jan 18, 2003

Neato, though i got lost the first time i read it so i had to reread it.
Jan 18, 2003

so if there are 30 lions.
and 1 sheep

naturally a lion would NOT eat the sheep. as another lionwould eat him.

But the same surely applies to 31 lions.

the sheep would not get eaten.

tell me....what would happen if there were 2 sheep and all the same rules applied
Jan 19, 2003

you said, "so if there are 30 lions. and 1 sheep naturally a lion would NOT eat the sheep" that right there is exactly why the 31st lion would eat! if at 30 the lion wouldn't eat, it doesn't matter for the 30th lion if the animal up for eating is the sheep itself or an ex-lion, he wouldn't eat! however, having 2 sheep would probably make it more interesting...
Jan 22, 2003

The sheep will always be eaten. One lion can kill the sheep and all the lions can eat it. They will then all turn into sheep and they can frolick peacefully in the land of infinite grass.
Jan 28, 2003

what a wonderful logic.
u have used the statement like lions r infinately logical.otherwise ur logic would be wrong
Jan 28, 2003

what a wonderful logic.
u have used the statement like lions r infinately logical.otherwise ur logic would be wrong
Jan 28, 2003

what a wonderful logic.
u have used the statement like lions r infinately logical.otherwise ur logic would be wrong
Jan 28, 2003

what a wonderful logic.
one thing i would like to state is that if there was no statement like infinately logical ur logic would be wrong
Jan 29, 2003

For some reason I read it as 100 lions and 100 sheep, which made the puzzel interesting, though rather easier. If there are equal or more sheep than lions, the lions can eat peacefully. Lets say that there are 100 and 100. One lion eats one sheep. Now there are 100 sheep and 99 lions. Now the lion/sheep, being smarter than the normal sheep (assuming he retains his logic), simply has to stay as far as possible from the lions. The lions, being logical and therefore efficient, will always go for the closest sheep, and will thus never eat anyu of the lions who are always huddleing around at the back!
(user deleted)
Feb 06, 2003

I understand the problem and solution, and like it very much, but I think the ending is mis-stated. Reading carefully, it says "it will always be true that with an even number of lions the sheep gets eaten and with an odd number the sheep doesn't." This is directly contradictory to the original correctly stated answer, that, "(I)n fact, the sheep would remain untouched if there is an even number of lions on the island, and would be eaten immediately if there is an odd number of lions on the island."

Great teaser!
Feb 07, 2003

Wow! It's amazing that after all this time someone finally noticed a mistake in the answer. Other than the fact that the last line of the answer is backwards, this is a great logic teaser.
Feb 07, 2003

Ooopps, that was my dyslexia acting up! Sorry about that, obviously the last line in the answer is meant to read the other way (Odd=eaten sheep, Even=not eaten).
Feb 11, 2003

The even and odd logic doesn't add up. Like others said if there are three lions or more it is an entirely different decision to be made. I personally think however, the first sheep would be eaten even in this situation due to the fact that the lions are very logical. The first lion would get the satifaction of the sheep and take the risk of being eaten himself. By doing this he shows that a lion was willing to take this risk, which in turn would deter other lions from eating him because it would give them the impression that if they take the same risk future lions will follow in turn and eat them.
Feb 13, 2003

To save people the trouble of having to post questions or doubts here, I provide my e-mail address: Feel free to direct all your questions and doubts to me if you like.

But for the record let me tell you that before you do, I guarantee that this problem is 100% valid, and that it has been done in game theory text books over and over again. SO before you disagree or decide that it's faulty, spend some more time thinking about it.
Feb 28, 2003

All I can say is, I LOVED your riddle 'survival of the people' and i think i luv this one even more
... GREAT job, I didnt spot a flaw (excepting the last line of the answer, but thats already been mentioned)! Keep 'em comin', you write GREAT teasers! n.n;
Mar 20, 2003

nice riddle!
Apr 03, 2003

Surely the answer is No the sheep won't get eaten purely because of point six. A lion would rather eat grass than risk being eaten by another lion? Why is the rest of the logic neccesary?
Apr 03, 2003

That's the point, the answer isn't always NO because a lion isn't always risking getting eaten! Sometimes the lion can, for a fact, pre-determine that he will not be eaten - therefore there would be no risk.
(user deleted)
May 25, 2003

well, if I was one of the lions, i wouldnt eat the sheep unless i was the only lion. because if there were more than me, i would know that i would end up being eaten. I got confused way back in the beginning! lol
May 28, 2003

This is the best teaser I have ever seen! Really good!
Oct 25, 2003

It was funny when it said the lion would turn into a sheep when it ate it. I liked it.
Feb 01, 2004

another great teaser!!
Mar 04, 2004

Good teaser, like most of the others I initially thought, what a no-brainer, of course the sheep would be untouched. but I was wrong.

The only thing I would say is that the teaser should be re-stated to say that there were 99 lions on the island. That way, someone (like me!!!) would not feel smug that he got the answer right, without getting anywhere near the logic!!!!

(though that was just ego initially, I sorted myself out afterwards!!)

Good riddle.
May 19, 2004

in response to something said a while ago. the lions are infinatly logical. so i think that whoever said that one lion would kill and then share the sheep is right. then they'll all be sheep and be deliriously happy about eating grass for the rest of their lives, because eating grass is what sheep like best. (nowhere in the problem does it say that only one lion can eat the sheep.)
Jun 02, 2004

This is definately one of my favorites. Awesome!!!
Jul 01, 2004

that's... awesome. I do wonder, though... let's compare the 5 and 6 lions. If the 5 eats the sheep, 4 would not eat him because 4 would then be eaten by 3, and so on. If 6 ate the sheep, then we return to the 5 lion 1 sheep scenario. Being entirely logical, a lion would eat the sheep, knowing that any other lion eating him would be eaten. If each lion followed this line of reasoning, then each lion would eat until there is only 1 sheep left. However, being supremely logical, each lion would predict this outcome, thus refraining from devouring the sheep. however, knowing that the others would each arrive at the same conclusion, he would eat the sheep. thus arriving back at the beggining. would this problem, then, be a paradox of self-refutement, much like the sentence "this sentence is false"? I'm going to keep thinking about this one.
Jul 01, 2004

hmmm... in retrospect, it was probably a mistake to submit a comment without reading the other comments first. Now that I have thoroughly shown my ignorance to the world, I plan to sulk for a few days. Anyway, I now understand how foolish my question was, and how you should never quetion the great sakirski. I'm sorry.
P.S. thanks einat16, great explination.
Jul 27, 2004

I enjoyed this a lot! I can understand it using my brilliant logic Hehe. To some of the people who didn't understand it: It doesn't matter how many lions are left that COULD eat him (the lion who turned into a sheep). It's whether they would WANT to eat them based on what the lion after them would do. So you have to open your mind a little wider, and think in the Lion's perspective. You wouldn't want to get eaten, would you now?!?
Dec 10, 2004

A verry interesting riddle, indeed. But... if you think about it if there are 101 lions, and the sheep gets eaten, you would be left with 100 lions and a sheep.

That answer makes no sense to me. But if the lions are smart and agree to each eat a part of the original sheep, they will all become sheeps and they will all be happy eating grass.

I don't know. Thats my own opinion!
Feb 11, 2005

great teaser with a good solution. perfectly logical.
Feb 13, 2005

To the favorites!
May 17, 2005

I follow the reasoning but I don't see the answer as necessarily true. All the lions are infinitely smart and logical, and know that all the others are too. So they each go through the described reasoning, where you don't eat the sheep when the number of lions is even. So the sheep appears safe with 100 lions. But suppose a lion does eat the sheep. The 99 remaining lions have now seen a precedent: an infinitely smart and logical lion has eaten a sheep while the number of lions is even. The 99th lion can now no longer rely on the 98-lion scenario being a safe one, and so may not eat the sheep. Maybe he will eat the sheep, thinking that would reestablish the fact that a sheep gets eaten with an odd number of lions, and so scare off lions frome eating him. But now we have strayed from the purely logical analysis, and have factors such as private decisions, risk aversions, psychology, etc., which cannot be governed by mere logic.
May 22, 2005

that was hard for me

Good one
May 23, 2005

1. Can more than one lion share the sheep? If they were infinitely smart, could they all eat some and all turn in to sheep?

Also, I still think that Rule #6 would mean none of the lions would eat the sheep while there are other lions around...

Anyway, this was a creative logical puzzle. Seems to have gotten a lot of people thinking.
Jun 23, 2005

keep up the good teasers that was great even though it was hard for me thx
Jul 15, 2005

Wow. That's a teaser!! I understand the logic in the answer, but I did not get it right. Because... I did not assume that the first lion knew that he would be turned into a sheep. If he doesn't know that (how could he? and I was told NOT to assume anything I wasn't told), then he will eat the sheep no matter what. Then the question is if the next lion will eat him or not (of course he now knows that he will turn into a sheep, since he has seen it done). Then you are back to the problem as solved in the answer (but with 99 lions).
Sep 11, 2005

This is the best teaser I have seen in a long time.
It really got the old grey matter working overtime
Definitely going on my Faves list
(user deleted)
Apr 20, 2006

In response to netgoof, the same logic you use dictates that the sheep will never be eaten in my opinion. If the lions are infinitely smart and logical then they will all come to the same conclusion given the same dilemma. Therefore if one lion eats the sheep it's a guarantee that the following lion will come to the same conclusion, if the lion doesn't eat the sheep then likewise the following lions will come to the same conclusion. It seems a bit of a paradox therefore, the only way the lion can eat the sheep and be safe is if he comes to the conclusion that he doesn't eat the sheep.
May 12, 2006

first of all this teaser was garbage, the logic works for the first three scenarios, 1 lion 1 sheep, thats obvious, 2 lions 1 sheep, the lion will not eat the sheep because he will in turn be eaten, 3 lions 1 sheep, the first lion will eat the sheep because he know the second lion would not eat him with the third and last lion waiting in the wing, now lets follow your logic to 4, ok even number no lion eats, fine, now 5 lions, NO LION EATS AGAIN since there will be 4 lions waiting to eat him, you can just apply the logic of 1 lion and 3 lions to all odd numbers, this was an awful teaser, and there is no logical answer to this because assumptions have to be made to justify any answer
May 12, 2006

i agree with MrIxolite "Surely the answer is No the sheep won't get eaten purely because of point six. A lion would rather eat grass than risk being eaten by another lion? Why is the rest of the logic neccesary?" this makes sense, and sakirski how can we assume that "there would be no risk" for some lions, personally i like farmerben's reasoning, that regardless the number, the first lion would eat, the second lion would eat that one, and the rest would see that and not even mess with the sheep anymore
May 12, 2006

by the way i meant you CANT just apply the logic for 1 and 3 lions to all odd numbers
May 12, 2006

albuquer and all others who didn't get the logic...Where do I begin. Well let's adress point 6. A lion would rather eat grass then risk...

The key there is "RISK". If there is no risk then he can eat sheep. Now if the lions were not logical and acted based on past performance, precidence, established common law, then there would risk associated with eating a sheep every single time.

BUT that is not how they work.The are governed by logic, and logic is like math, it's an exact science.

So lets start from scratch. It seems that it's easy for everyone to grasp the concept of 1 lion and 1 sheep. The lion has no one to be scared of, he is alone, he can safely eat the sheep. It's also easy to understand that 2 lions and one sheep, will never even try to eat the sheep cause then the remaining lion is guaranteed safe food (cause he will be left alone). Thus we've established 2 lions and 1 sheep is a standstill where everyone eats grass.

NOW, 3 lions. Why is this an unstable position? Why is it safe for one of the lions to eat the sheep? Easy because if any of them eats then only 2 lions remain, and we already know that 2 lions will not eat sheep. SOOO, now we know that 3 lions will certainly result in one of them grabing up that sheep.

NOW, if we know for a fact that 3 lions will result in eaten sheep, we can now make a conclusion about 4 lions. 4 LIONS will not eat sheep cause the result of that action is 3 lions, where we already know the sheep will get eaten. So we know that 4 lions results in no sheep eating at all. Based on that certain knowledge we can speak to 5 lions. In the case with 5 lions, all the lions know that once there are 4 of them left, noone will dare touch the sheep (as we've deducted above), therefore whoever of the 5 can get to the sheep first will eat it, and by that action leave behind only 4 lions, who will not touch the sheep, since that would leave 3 who would touch the sheep, cause that would leave 2 who would not touch the sheep, cause that would leave just 1 who will definitly eat the sheep.

I really hope you don't need me to go all the way to 100 to illustrate how the logic works.
Jul 08, 2006

When I first came to this site, this was one of the first teasers I saw. Now, after a long(ish) time, it is still my favorite teaser on the entire site. However, I agree with some of the comments saying you should change it to 99 instead of 100 lions, just to make it so people's first guesses are wrong, instead of right for the wrong reason.

Great teaser!
Mar 02, 2007

If there are more then 2 lions the sheep won't be eaten.
Mar 02, 2007

deepsea your answer is wrong. Check out some of the comments/explanations above if the solution explanation isn't very forthcoming.
(user deleted)
Oct 02, 2007

I say the lions split the sheep into 100 equal pieces and then all turn into sheep. Everyone gets sheep and no one gets eaten!
Jan 29, 2008

The puzzle asks if one of the 100 lions would eat the sheep and why or why not.

The answer is No by the sole nature of # 6) "A lion would rather eat grass all his life than be eaten by another lion (after he turned into a sheep)".

There's no condition to force the lion to eat the sheep if his survival is guaranteed by the status quo

Based on #6, the 100 lions would rather eat grass than be eaten, and since there's an infinite amount of grass on the island, sheep and lions live happily everafter.
Jan 30, 2008

moocho, there is condition #3, which could get a lion to eat a sheep if a lion is certain that he will not be eaten by another lion afterwards.

Think about it some more. Look at the explanation, and maybe read some of the comments posted here especially in the beginning, they may help clear things up for you.
May 14, 2008

One more piece that screws everything up. One way or another, an assumption must be made. Either you have to ASSUME that the lions will live forever (or die in pairs) such that there will always be an even number. Or, you must assume that lions are mortal and thus, when one dies, that would change from an odd to an even (or vice versa) condition. Logically, this will mean that no lion would eat a sheep ever except in the one lion, one sheep scenario. Maybe you can infer from clue 2 that they can "survive" indefinitely since there is infinite grass. Anyway, I understand the point that the teaser creator is assuming the lions do not die.
May 24, 2008

I guess "myhalcyondaze" is right about that. Though I would argue that if I we have to make the assumption about the lions living there indefinitly, we have to also mention that they will not kill each other in a petty fight over a she-lion, and that people won't come and take one away in a cage. But I don't think too many people are considering these possibilities when dealing with the problem at hand. Though to be fair I should say something like "the lions are all male and all the same age and will all die at the same time many years from now". I should also add that "there are no other external factors and the island is a universe within itself".
Jan 06, 2009

I understand the basis for the reasoning here, but the extension of the simple case cannot be extended to 100 lions. The lions would not eat the sheep with 99 lions either.

This situation is similar to a classic puzzle that demonstrates the fallacy of over-extending conditional logic. It goes like this:

A prisonor is sentenced to be executed in the next seven days. However, to avoid being cruel, the day the prisonor is executed must be a surprise. The prisonor reasons that he can't be executed on the 7th day because he would know which day he will be executed. He reasons that he can't be executed on the 6th day since he would know they would have to execute him on that day since they can't execute him on the 7th day. If he can't be executed on the 6th day then he surely can't be executed on the 5th day either since he would know they would have to execute him on that day because they can't wait until the 6th day. Following this chain of reasoning, the prisonor concludes that he can't be executed without violating the judge's orders. On the 4th day the prisonor is surprised when the executioner arrives at his cell.

The logic that applies to the end cases cannot be extended indefinitely, thus the completely logical lions will not eat the sheep if there are 99 or 100 lions.
Oct 05, 2010

Yep. I completely agree with you

I am so happy now. Because all the way, i was seen that "when even they are safe, when odd they can eat the sheep".
Yes it "is" true, but I don't like that answer.
Nov 29, 2010

-Mind blown-
I just thought that because no lion wanted to get eaten afterwards, they wouldn't go after that sheep
Jun 17, 2011

This is probably my favorite teaser ever. It's also the cause for much controversy as evident in the comments. Brilliant.
Nov 20, 2011

I thought that the sheep wouldn't get eaten because sheep eat grass, and if the lion ate the sheep and turned into one, he would have to eat grass anyway, so why even eat the sheep? lol. I'm dumb.
Jul 10, 2012

The question is: ''Will that one sheep get eaten or not and why?''.

The answer: That one sheep will get eaten.

Why?: If one lion eats the first sheep he turns into a sheep. However, since all lions are infinitely logical, smart, and completely aware of their surroundings + ''A lion would rather eat grass all his life than be eaten by another lion'' that one lion (who is now a sheep) won't be eaten. If he would get eaten by another lion it is it contradiction with the assumption that lions act logical (since a lion does not act logical if he prefers to eat grass all his live to be eaten by another lion and then still eats a sheep knowing that he will be eaten by another lion).

Therefore, odd or even numbers of lions does not matter. It is only the first sheep that will get eaten since the lion who eats the first sheep knows that (via his infinite logic) 1) he prefers sheep to grass and 2) he will not be eaten by other lions since that would be illogical.
Feb 22, 2013

^^ Some people just don't get it. Ok, I'll give this a shot.

See, after eating a sheep, it is possible that the lion (now turned into a sheep) will NOT be eaten. How?

Assume 3 lions and 1 sheep. First lion now thinks- should I eat this sheep?

Case 1: He eats the sheep and turns into a sheep himself.
2nd lion now thinks- should I eat this sheep now (the one who was initially a lion.) The answer now is obviously NO because if he does, he is sure to be eaten by the third lion who faces no risk as there will be no more lions.
Net result- the 1st lion ate the sheep. 2nd lion did not eat the lion-converted sheep. Hence, the smartest move is that the initial Sheep IS eaten.

If you're a little open-minded instead of just assuming you got the right answer in a "shorter" way, you can easily extend this to 4 then 5 lions.

(I'm not even sure anyone's going to come back and read this but it was fun explaining. Great teaser, by the way. Going into my favorites.)
Mar 23, 2014

This game can be solved exactly by backward induction. Let us designate the sheep 0 and the lions with the positive integers 1, 2, 3, ..., 99, 100. Let us imagine that the lions' proximities to the sheep are the same as their order on the number line, with lions with lower numbers closer to the sheep than lions with higher numbers. Should 1 try to eat 0? That depends on the decisions that 2-100 make, so we shall start our analysis with 100, assuming that he would be the last remaining lion should all the lions decide to go for the sheep. If 100 is the last remaining lion, he will eat 99, since there are no more lions left to eat him. If 99 and 100 were the last remaining lions, 99 would not eat 98 because he knows that 100 has no incentive to not eat 99 should 99 become a sheep. Therefore 99 will not eat. What about 98? 98 knows that 99 will not try to eat 98, so 98 can safely eat 97 should it get to that point. Therefore 97 should avoid eating 96. Therefore 96 can eat 95. Therefore 95 should avoid eating. 94-1 will progress similarly, eat, not eat, eat, not eat, eat, ... , eat, not eat.

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