A logic puzzle? Now?

I’m glad I’m back writing, but need a break between remote teaching, the UFT, the politicians and the pandemic… That’s a lot of scary stuff and frustration and unknowns… Let’s squeeze in some math to lower the temperature.

Last week I ran discussions with no outside assignments. And some of the discussions were logic puzzles, run like problem solving sessions. And I dusted off this old favorite about leprechauns, and as the kids pushed to an answer, someone said “is this the same as the wolf and sheep problem?”

I did not know the wolf and sheep problem (which I told him). But when we were done, I looked it up. Brand new problem for me, but fits right in with some of my favorites: the pirates and leprechauns.  Here it is – try to reason it out for yourself:

I took the language from a page that seems to be selling a logic course – but I prefer the title I found on Braingle:

### Survival of the Sheep

On an island in a far away country there is a population of 100 wolves and 1 sheep. They are the only two living species on the island. The following facts are known to be true:

• There is grass covering the whole island (grass is not considered as a living species for the purposes of the problem).
• The sheep can survive just by eating grass throughout its lifespan.
• As the grass is being eaten, it instantaneously grows back. No matter how many times it gets eaten, it will always grow back. It is therefore suitable to state that the island has an infinite supply of grass.
• The wolves themselves, unlike the sheep, are part of a very rare and intelligent species. They are actually perfectly rational beings, and can be considered as being infinitely intelligent.
• Similarly to the sheep, the wolves can also survive by eating grass throughout their whole lifespan.
• As one might imagine, the wolves prefer eating sheep than eating grass.
• If the sheep were to be eaten, it could only be eaten by a single wolf (the wolves cannot share their prey). However, there is catch:
• In this faraway land it is known that after a wolf eats a sheep, the wolf itself will become a sheep and it will therefore be in danger of being eaten by other wolves.
• All wolves are perfectly aware of this.
• If a wolf knows for sure that eating the sheep will cause him to be eaten by another wolf, then it prefers to eat grass instead.
• In the same way, if the wolf knows that eating the sheep will not put him in danger, it will eat the sheep.

Given all these facts and given the scenario from the very beginning, the question which must be answered is the following:

Will the sheep be eaten?

April 23, 2020 am30 9:56 am 9:56 am

No, because every wolf will try to wait out every other wolf to be the *last* wolf such that when it eats the 2nd to last wolf and becomes a sheep, there will be no other wolves to eat it. Who would go first under such circumstances?

April 23, 2020 am30 9:59 am 9:59 am

I would add that the wolves could figure out what I just posted above, since they are rational etc. But it holds because if a wolf were willing to break ranks and go first, then that same impulse would likely be true of other wolves in the population, thereby ensuring that any wolf who goes first would be eaten by the next wolf with the same reasoning. So they will not eat sheep because if any wolf eats a sheep, they will ALL eat sheep, and that reinforces the reasoning to stay safe by not eating one.

• April 23, 2020 am30 11:11 am 11:11 am

But if you know everyone is afraid to break ranks – why not be the brave one? No one else, by that reasoning, will come after you.

We need to dig deeper.