 # 362 logic/zoo.p

## Description

This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

# 362 logic/zoo.p

I took some nephews and nieces to the Zoo, and we halted at a cage marked

Tovus Slithius, male and female.
Beregovus Mimsius, male and female.
Rathus Momus, male and female.
Jabberwockius Vulgaris, male and female.

The eight animals were asleep in a row, and the children began to guess
which was which. "That one at the end is Mr Tove." "No, no! It's Mrs
Jabberwock," and so on. I suggested that they should each write down
the names in order from left to right, and offered a prize to the one
who got most names right.

As the four species were easily distinguished, no mistake would arise in
pairing the animals; naturally a child who identified one animal as Mr
Tove identified the other animal of the same species as Mrs Tove.

The keeper, who consented to judge the lists, scrutinised them carefully.
"Here's a queer thing. I take two of the lists, say, John's and Mary's.
The animal which John supposes to be the animal which Mary supposes to be
Mr Tove is the animal which Mary supposes to be the animal which John
supposes to be Mrs Tove. It is just the same for every pair of lists,
and for all four species.

"Curiouser and curiouser! Each boy supposes Mr Tove to be the animal
which he supposes to be Mr Tove; but each girl supposes Mr Tove to be
the animal which she supposes to be Mrs Tove. And similarly for the oth-
er animals. I mean, for instance, that the animal Mary calls Mr Tove
is really Mrs Rathe, but the animal she calls Mrs Rathe is really Mrs
Tove."

"It seems a little involved," I said, "but I suppose it is a remarkable
coincidence."

"Very remarkable," replied Mr Dodgson (whom I had supposed to be the
keeper) "and it could not have happened if you had brought any more
children."

How many nephews and nieces were there? Was the winner a boy or a girl?
And how many names did the winner get right? [by Sir Arthur Eddington]

logic/zoo.s

Given that there is at least one boy and one girl (John and Mary are
mentioned) then the answer is that there were 3 nephews and 2 nieces,
the winner was a boy who got 4 right.

Number the animals 1 through 8, such that the females are even and the
males are odd, with members of the same species consecutive; i.e. 1 is
Mr. Tove, 2 Mrs. Tove, etc.

Then each childs guesses can be represented by a permutation. I use
the standard notation of a permutation as a set of orbits. For
example: (1 3 5)(6 8) means 1 -> 3, 3 -> 5, 5 -> 1, 6 -> 8, 8 -> 6 and
2,4,7 are unchanged.

 Let P be any childs guesses. Then P(mate(i)) = mate(P(i)).

 If Q is another childs guesses, then [P,Q] = T, where [P,Q] is the
commutator of P and Q (P composed with Q composed with P inverse
composed with Q inverse) and T is the special permutation (1 2) (3 4)
(5 6) (7 8) that just swaps each animal with its spouse.

 If P represents a boy, then P*P = I (I use * for composition, and
I for the identity permutation: (1)(2)(3)(4)(5)(6)(7)(8)

 If P represents a girl, then P*P = T.

 and  together mean that all girl's guesses must be of the form:
(A B C D) (E F G H) where A and C are mates, as are B & D,
E & F G & H.

So without loss of generality let Mary = (1 3 2 4) (5 7 6 8)
Without to much effort we see that the only possibilities for other
girls "compatible" with Mary (I use compatible to mean the relation
expressed in ) are:

```    g1:  (1 5 2 6) (3 8 4 7)
g2:  (1 6 2 5) (3 7 4 8)
g3:  (1 7 2 8) (3 5 4 6)
g4:  (1 8 2 7) (3 6 4 5)
```

Note that g1 is incompatible with g2 and g3 is incompatible with g4.
Thus no 4 of Mary and g1-4 are mutually compatible. Thus there are at
most three girls: Mary, g1 and g3 (without loss of generality)

By  and , each boy must be represented as a product of
transpostions and/or singletons: e.g. (1 3) (2 4) (5) (6) (7) (8) or
(1) (2) (3 4) (5 8) (6 7).

Let J represent John's guesses and consider J(1).
If J(1) = 1, then J(2) = 2 (by ) using  and Mary J(3) = 4, J(4) =
3, and g1 & J => J(5) = 6, J(6) = 5, & g3 & J => J(8) = 7 J(7) = 8
i.e. J = (1)(2)(3 4)(5 6)(7 8). But the [J,Mary] <> T. In fact, we
can see that J must have no fixed points, J(i) <> i for all i, since
there is nothing special about i = 1.

If J(1) = 2, then we get from Mary that J(3) = 3. contradiction.

If J(1) = 3, then J(2) = 4, J(3) = 1, J(4) = 2 (from Mary) =>
J(5) = 7, J(6) = 8, J(7) = 5, J(8) = 6 => J = (1 3)(2 4)(5 7)(6 8)
(from g1)
But then J is incompatible with g3.

A similar analysis shows that J(1) cannot be 4,5,6,7 or 8; i.e. no J
can be compatible with all three girls. So without loss of generality,
throw away g3.

We have Mary = (1 3 2 4) (5 7 6 8)
g1 = (1 5 2 6) (3 8 4 7)

The following are the only possible boy guesses which are compatible
with
both of these:

```  B1: (1)(2)(3 4)(5 6)(7)(8)
B2: (1 2)(3)(4)(5)(6)(7 8)
B3: (1 3)(2 4)(5 7)(6 8)
B4: (1 4)(2 3)(5 8)(6 7)
B5: (1 5)(2 6)(3 8)(4 7)
B6: (1 6)(2 5)(3 7)(4 8)
```

Note that B1 & B2 are incombatible, as are B3 & B4, B5 & B6, so at most
three of them are mutually compatible. In fact, Mary, g1, B1, B3 and
B5 are all mutually compatible (as are all the other possibilities you
can get by choosing either B1 or B2, B3 or B4, B5 or B6. So if there
are 2 girls there can be 3 boys, but no more, and we have already
eliminated the case of 3 girls and 1 boy.

The only other possibility to consider is whether there can be 4 or
more boys and 1 girl. Suppose there are Mary and 4 boys. Each boy
must map 1 to a different digit or they would not be mutually
compatible. For example if b1 and b2 both map 1 to 3, then they both
map 3 to 1 (since a boy's map consists of transpositions), so both
b1*b2 and b2*b1 map 1 to 1. Furthermore, b1 and b2 cannot map 1 onto
spouses. For example, if b1(1) = a and b is the spouse of a, then
b1(2) = b. If b2(1) = b, then b2(2) = a. Then b1*b2(1) = b1(b) = 2
and b2*b1(1) = b2(a) = 2 (again using the fact that boys are all
transpostions). Thus the four boys must be:

```B1: (1)(2)...    or (1 2)....
B2: (1 3)...     or (1 4) ...
B3: (1 5) ...    or (1 6) ...
B4: (1 7) ...    or (1 8) ...
```

Consider B4. The only permutation of the form (1 7)... which is
compatible
with Mary ( (1 3 2 4) (5 7 6 8) ) is:

(1 7)(2 8)(3 5)(4 6)

The only (1 8)... possibility is:

(1 8)(2 7)(3 6)(4 5)

Suppose B4 = (1 7)(2 8)(3 5)(4 6)

If B3 starts (1 5), it must be (1 5)(2 6)(3 8)(4 7) to be compatible
with B4. This is compatible with Mary also.

Assuming this and B2 starts with (1 3) we get B2 = (1 3)(2 4)(5 8)(6 7)
in order to be compatible with B4. But then B2*B3 and B3*B2 moth map 1
to 8. I.e. no B2 is mutually compatible with B3 & B4.

Similarly if B2 starts with (1 4) it must be (1 4)(2 3)(5 7)(6 8) to
work with B4, but this doesn't work with B3.

Likewise B3 starting with (1 6) leads to no possible B2 and the
identical reasoning eliminates B4 = (1 8)...

So no B4 is possible!

I.e at most 3 boys are mutually compatiblw with Mary, so 2 girls & 3
boys is optimal.

Thus:

```Mary = (1 3 2 4) (5 7 6 8)
Sue  = (1 5 2 6) (3 8 4 7)
John = (1)(2)(3 4)(5 6)(7)(8)
Bob  = (1 3)(2 4)(5 7)(6 8)
Jim  = (1 5)(2 6)(3 8)(4 7)
```

is one optimal solution, with the winner being John (4 right: 1 2 7 & 8)

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