h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Glad to have helped you out there, Icy.


Meerkats:

Either the problem is very simple, or (more likely) I have misunderstood the question. Would you mind clarifying a couple of things for us:

Can two meerkats watch each other?

Please explain what you mean by 'the meerkats are a [distinct] distance apart'?

Does there have to be an odd number of meerkats?

If a meerkat is watching the nearest to it, then that nearest one must be watching it. So they must be watching each other in pairs.
There must therefore be the odd one who is watching but is not being watched.

To answer your points, RE:

1. Yes

2. 'Distinct distances apart' means that they're not all the same distance apart.

3. Yes, in this situation.


CD, but you're thinking on the right lines, but you can't say that 'they must be watching each other in pairs'.

1...................2.....3

1 watches 2, 2 watches 3, 3 watches 2, and no kat watches 1. No?

OK, in the situation you describe, no meerkat is watching 1. Now, how about a field of any odd number of meerkats?

If only 1 Meerkat is, say, 40ft. away from the nearest one, and all others are within 40 ft of each other. Then that one would not be being watched but would only be watching the nearest one to it.

That is correct, of course, CD.

This is one of those logical arguments which is slightly tricky because we can all see it's common sense.

If you consider what you've just said along with your earlier argument (the nearest two have to watch each other) can you find the logical argument which proves this for the whole field?

(must dash - back later)

I've been playing with my small change... ...and it seems to me that as long as one beast is remote from the rest of the field, and he watches his nearest relation, then the remainder of the family - whether an odd or even number - can watch and be watched (at least under the conditions you have given us so far).

Now I must go.

If they are all equi-distant (i.e. a large circle) then it is possible for all meerkats to be viewed by one other meerkat.
If they are a varying distance apart, then I agree that at least one meerkat would not be viewed by any other meerkat, if there are an odd number. This would most likely be the one furthest apart from any other meerkat if they followed the strict logic of looking at the meerkat nearest to them.

'the one furthest apart from any other meerkat'. That's quite a difficult concept to describe.

1->................2->............3->..........4->........5->......6->....<-7

And 8 etc - provided that the distance between each animal reduces as we go down the line. (Permitting the final animal to look back at his immediate antecedent ensures that the number doesn't matter).

Who is looking at number 1?

Exactly. That is why there will always be one left who is not being looked at, if a strict policy of looking at the nearest is adhered to.

I wonder why Icy has left us here to stew. He *is* a great poser though, so there may well be more to this than meets the eye.

I await 'the word'.

I haven't seen a cast-iron logical argument which shows that at least one meerkat isn't being watched. Your example in which you line them up is fine, but they're not all neatly lined up in their field. In fact they could all be floating in three-dimensional space, come to think of it.

Czar Dene was probably the nearest with his initial post, but it needs expanding.

O.K. Some pairs may be watching each other. In order to watch another Meerkat it involves 2 Meerkats each time, the one watching and the one being watched. Because of the odd number of Meerkats, there must be at least 1 Meerkat not being watched.

"Some pairs may be watching each other."

There can be only one 'pair' - which would end the series (by definition).


"Your example in which you line them up is fine, but they're not all neatly lined up in their field. In fact they could all be floating in three-dimensional space"

But it's the distance that rules, isn't it?


'A' looks at 'B' because 'B' is the closest.
Since 'B' is the closest to 'A', then no other cat could be closer -otherwise 'A' would be looking at *that* kat instead of 'B'.
Ergo, the first Kat can be watched only by 'B' (in a field of a mere two meerkats).

However, this works for both odd and even numbers...

Let's call a halt. I think Czar Dene is just about close enough, but the classic explanation is this:

The nearest meerkats must be watching each other.

We ignore these meerkats, then find the next closest pair of meerkats in the field. If either of them was watching one of the meerkats we have ignored, then at least one meerkat is being watched twice, so therefore at least one meerkat out of the whole set isn't being watched.

Alternatively, they are watching each other, so we ignore them, too, and repeat the previous step.

As there is an odd number of meerkats, we will eventually run out of pairs, and the last meerkat is not being watched.

Does that make sense?

Icy

Hmmm. "you can't say that 'they must be watching each other in pairs'."

Seems I let that lead me astray. Silly boy! Nevertheless, I have shown that an even number of meerkats can also leave one of their number unwatched [even amongst a field of decidedly odd meerkats ("In fact they could all be floating in three-dimensional space") ].

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