Caliban's Will
Richard Harter
think of it as the ultimate logic puzzle. I have posted a version
of it before in rec.puzzles and have run it in a fanzine. At first
sight it seems outrageously incomplete. However with a bit of
inference... One of the reasons that I like it is that I have never
seen any other puzzle quite like it; another is that it is quite
subtle in the levels of reasoning required. Here then, for your
delectation, is Caliban's Will.
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| Caliban's Will by M.H. Newman |
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When Caliban's will was opened it was found to contain the following
clause:
"I leave ten of my books to each of Low, Y.Y., and 'Critic,' who are
to choose in a certain order.
No person who has seen me in a green tie is to choose before Low.
If Y.Y. was not in Oxford in 1920 the first chooser never lent me
an umbrella.
If Y.Y. or 'Critic' has second choice, 'Critic' comes before the one
who first fell in love."
Unfortunately Low, Y.Y., and 'Critic' could not remember any of the
relevant facts; but the family solicitor pointed out that, assuming the
problem to be properly constructed (i.e. assuming it to contain no
statement superfluous to its solution) the relevant data and order
could be inferred.
What was the prescribed order of choosing; and who lent Caliban an
umbrella?
----------------------------------------------
If you tackle this, remember that every statement is necessary.
Also remember that the requirement (that the puzzle represented by
the three statements is properly constructed) is not part of the
original puzzle -- it is a meta-statement about the base puzzle.
[People regularly get confused about this point.] One further
point you have to consider -- was the family solicitor correct in
claiming (a) that the relevant data could be inferred and (b)
that the order be inferred.
----------------------------------------------
Solution in one week. Arguments about the solution immediately
afterwards, if experience is any guide.
--
Richard Harter, Software Maintenance and Development Systems, Inc.
Net address: jjmhome!smds!rh Phone: 508-369-7398
US Mail: SMDS Inc., PO Box 555, Concord MA 01742
This sentence no verb. This sentence short. This signature done.
cate...@pirates.armstrong.edu
Low, Critic, Y.Y.
Ronnie Kon
>
>When Caliban's will was opened it was found to contain the following
>clause:
>
>"I leave ten of my books to each of Low, Y.Y., and 'Critic,' who are
>to choose in a certain order.
>
>1. No person who has seen me in a green tie is to choose before Low.
>
>2. If Y.Y. was not in Oxford in 1920 the first chooser never lent me
>an umbrella.
>
>3. If Y.Y. or 'Critic' has second choice, 'Critic' comes before the one
>who first fell in love."
>
>Unfortunately Low, Y.Y., and 'Critic' could not remember any of the
>relevant facts; but the family solicitor pointed out that, assuming the
>problem to be properly constructed (i.e. assuming it to contain no
>statement superfluous to its solution) the relevant data and order
>could be inferred.
>
>What was the prescribed order of choosing; and who lent Caliban an
>umbrella?
Assume that all statements are necessary. Why is statement #1 present?
Clearly if no-one has seen Caliban in a green tie the statement would be
meaningless. Therefore, at least one of YY and Critic have seen Caliban in
a green tie (although I admit that it is difficult to picture Caliban in a
green tie. Must be a different one...). Thus Low must be either first or
second.
Why is statement #3 present? Again, the if clause must be true, or the
statement says nothing and could have been left out. This means that Low
is not second, so s/he is first. Thus YY must have been the first to fall
in love, and the order is
Low 'Critic' Y.Y
Why is the second statement present? We have determined the order without
it. It only serves to tell us that Low lent Caliban an umbrella, and that
Y.Y. was in Oxford in 1920. (We assume the if clause is true, as above).
Since this answers only tangenital questions, and not the point of the
will, clearly our original assumption--that all statements are
necessary--is false.
Since we cannot now assume the if clauses to be always true, any
solution must hold even when we assume that they are all false. As this
leaves us with no information whatsoever, no solution is possible.
Ronnie
-------------------------------------------------------------------------------
Ronnie B. Kon | "Was it a millionaire who said
k...@groundfog.stanford.edu | 'Imagine no possessions'?"
...!{decwrl,ames}!mindcrf!ronnie | -- Elvis Costello
-------------------------------------------------------------------------------
Dave Seaman
>>1. No person who has seen me in a green tie is to choose before Low.
>>
>>2. If Y.Y. was not in Oxford in 1920 the first chooser never lent me
>>an umbrella.
>>
>>3. If Y.Y. or 'Critic' has second choice, 'Critic' comes before the one
>>who first fell in love."
>
>Clearly if no-one has seen Caliban in a green tie the statement would be
>meaningless. Therefore, at least one of YY and Critic have seen Caliban in
>a green tie (although I admit that it is difficult to picture Caliban in a
>green tie. Must be a different one...). Thus Low must be either first or
>second.
Ok, so far...
>Why is statement #3 present? Again, the if clause must be true, or the
>statement says nothing and could have been left out. This means that Low
>is not second, so s/he is first. Thus YY must have been the first to fall
>in love, and the order is
>
> Low 'Critic' Y.Y
Statement #3 implies either that Low is second, or 'Critic' is not last. You
can't conclude from the statement that Low is not second. However, you can
rule out the ordering Low, Y.Y., 'Critic', because it violates both clauses.
The three remaining possibilities are:
(1) Low, 'Critic', Y.Y.
(2) Y.Y., Low, 'Critic'
(3) 'Critic', Low, Y.Y.
From the second clue it is tempting to conclude that Y.Y. lent Caliban an
umbrella at Oxford in 1920, and therefore Y.Y. must be first choice. However,
it can't be as simple as that, because then the third clue would be rendered
unnecessary. There are only two orderings in which Y.Y. is first, and one of
those is ruled out by the first clue.
Likewise, if 'Critic' were the first chooser, then clues 1 and 2 alone would be
sufficient to deduce the order. The only way the third clue can contribute to
the solution is if Low is the first chooser. Therefore, the order must be
Low, 'Critic', Y.Y.
--
Dave Seaman
a...@seaman.cc.purdue.edu
Timothy Yi-chung Chow
Several attempted solutions have been posted, but all have contained
flaws. Some have overlooked the word 'never' in the second clue;
others, in spite of Mr. Harter's warning, have confused the meta-clue
with a clue. Following is, I believe, a correct solution, except that
my answer to the question about the umbrella makes me uneasy.
>When Caliban's will was opened it was found to contain the following
>clause:
>
>"I leave ten of my books to each of Low, Y.Y., and 'Critic,' who are
>to choose in a certain order.
>
>No person who has seen me in a green tie is to choose before Low.
>
>If Y.Y. was not in Oxford in 1920 the first chooser never lent me
>an umbrella.
>
>If Y.Y. or 'Critic' has second choice, 'Critic' comes before the one
>who first fell in love."
>
>Unfortunately Low, Y.Y., and 'Critic' could not remember any of the
>relevant facts; but the family solicitor pointed out that, assuming the
>problem to be properly constructed (i.e. assuming it to contain no
>statement superfluous to its solution) the relevant data and order
>could be inferred.
>
>What was the prescribed order of choosing; and who lent Caliban an
>umbrella?
Let us assume that, had the three friends had perfect memories, they
would have been able to deduce the correct order using only the three
given clues (i.e., without assuming the meta-clue that the puzzle is
properly constructed) and that they would have needed all three clues.
If neither Y.Y. nor 'Critic' had seen Caliban in a green tie, the first
clue would have contributed no information. Hence either Y.Y. or
'Critic' or both have seen Caliban in a green tie.
If both Y.Y. and 'Critic' had seen Caliban in a green tie, the first and
third clues would have been enough for the three friends to deduce the
order. Hence exactly one of Y.Y. and 'Critic' have seen Caliban in a
green tie.
Thus Low is either first or second, and moreover the friends could have
deduced this fact from the first clue alone.
If Y.Y. had been in Oxford in 1920, then the second clue would have
contributed no information. Hence Y.Y. was not in Oxford in 1920.
If Low had lent Caliban an umbrella, then the three friends would have
been able to deduce the correct order from the first two clues alone
(remember that the friends know that exactly one of Y.Y. and 'Critic'
had seen Caliban in a green tie, and furthermore they knew WHICH ONE had
seen Caliban in a green tie; of course they also know that Y.Y. was not
in Oxford in 1920). Hence Low has never lent Caliban an umbrella.
If Low were the ONLY one never to have lent Caliban an umbrella, then
the second clue would have been enough to establish that Low was to
choose first, and hence the first clue would have been superfluous.
Hence either Y.Y. or 'Critic' or both never lent Caliban an umbrella.
But it could not have been both, for then the second clue would have
contributed no information. So exactly one of Y.Y. and 'Critic' lent
Caliban an umbrella.
If 'Critic' had both lent Caliban an umbrella and seen Caliban in a
green tie, then the friends would not have been able to deduce the
order from the three clues, since they would not have been able to
decide between the order Low, 'Critic,' Y.Y., and the order Y.Y., Low,
'Critic' (in this latter case, the friends would have gained no
information from the third clue, but WE cannot eliminate the
possibility on this account).
Similarly if Y.Y. had both lent Caliban an umbrella and seen Caliban in
a green tie, then there would have been indecision between the orders
Low, 'Critic,' Y.Y. and 'Critic,' Low, Y.Y.
So either Y.Y. is the umbrella-lender and 'Critic' is the
green-tie-seer, or vice versa. The order is thus Low, Y.Y., 'Critic,' or
Low, 'Critic,' Y.Y.
From the third clue it follows that the order is Low, 'Critic,' Y.Y.,
and that Y.Y. was the first to fall in love.
We still do not know which of 'Critic' and Y.Y. is the umbrella-lender
and which is the green-tie-seer. A quick check shows that either of
the two possible assignments yields a situation where the three friends
could have deduced the order using all three clues, but could not have
deduced the order using any two out of the three clues (if they had
perfect memories).
To summarize: given our initial assumption, the order is Low, 'Critic,',
Y.Y.; either 'Critic' or Y.Y. lent Caliban an umbrella, but not both,
and we cannot determine which; the solicitor was correct in saying that
the order could be inferred from the assumption, but incorrect in saying
that all the data could be inferred from the assumption.
--
Tim Chow tyc...@phoenix.princeton.edu
Keith Ramsay
More spoiler, of course.
In article <10...@idunno.Princeton.EDU>, Timothy Yi-chung Chow writes:
> If 'Critic' had both lent Caliban an umbrella and seen Caliban in a
> green tie, then the friends would not have been able to deduce the
> order from the three clues, since they would not have been able to
> decide between the order Low, 'Critic,' Y.Y., and the order Y.Y., Low,
> 'Critic' [...]
You didn't mention the case that Low was the first to fall in love,
which would eliminate the order Low, 'Critic', Y.Y. More simply, if
'Critic' had both seen Caliban in a green tie and lent him an
umbrella, the second clue would be redundant, as 'Critic' already
cannot choose first, because he must choose after Low.
> Similarly if Y.Y. had both lent Caliban an umbrella and seen Caliban in
> a green tie, then there would have been indecision between the orders
> Low, 'Critic,' Y.Y. and 'Critic,' Low, Y.Y.
Likewise for this reasoning.
I got the same conclusion as you did, though. It seems like there must
be something else in the puzzle we're missing, because it asks us who
lent Caliban an umbrella, and it looks as though either 'Critic' or
Y.Y. could have.
--
If you want to contact me after this week,
Keith Ramsay write c/o Harvard math dept., 1 Oxford St.,
ram...@zariski.harvard.edu Cambridge MA, 02138.
Keith Ramsay
In article <5...@smds.UUCP>, Richard Harter quotes the author's
solution:
> (3) Either Low is second or Critic is not last.
This is not equivalent to the original (3) which specified that either
Low is second or Critic is *before the first to fall in love*.
Now to rule out the case where 'Critic' saw the green tie and Y.Y.
lent the umbrella, (and also presumedly was first to fall in love, to
use the originally specified version of (3)) we have the mistaken
reasoning:
> For if Critic is a T, then by (1) Low precedes Critic
Yes. YLC, LYC, LCY are the orders allowed by (1).
> and hence (3) allows only "Low, Critic, Y.Y"; (2) is superfluous.
The order YLC (Y.Y., Low, 'Critic') is also allowed by (3), since
neither Y.Y. nor 'Critic' is second.
Perhaps the author fell to the fallacy that "if A then B" doesn't hold
(or is "redundant") when A and B are false, although this seems
implausible since he rephrased (3) in the form "either not A or B".
I checked (under the conditions that 'Critic' saw the green tie, Y.Y.
was not in Oxford in 1920, lent the umbrella, and was first to fall in
love) that no two of the three conditions suffice to determine the
order but all three do. (1&2 allow LYC; 1&3 allow YLC; 2&3 allow CLY,
all in addition to LCY which is allowed by all 3.)
It would be better to alter the puzzle so that the question is "Who
first fell in love?" rather than "Who lent Caliban the umbrella?" In
the original form it is somewhat disappointing.
Richard Harter
[... confirms that the author was wrong ...]
> It would be better to alter the puzzle so that the question is "Who
> first fell in love?" rather than "Who lent Caliban the umbrella?" In
> the original form it is somewhat disappointing.
Certainly the author should have done so. Actually I find the notion
that the puzzle is sufficiently subtle that even the author erred to add
to the interest. If you feel really challenged perhaps you can modify
the puzzle so that the solution is unique.
Terence Donahue
In article <5...@smds.UUCP> r...@smds.UUCP (Richard Harter) writes:
>
> Finally, Y.Y. is a T, and Critic is a U. For if Critic is a T, then
> by (1) Low precedes Critic and hence (3) allows only "Low, Critic, Y.Y";
> (2) is superfluous. I.e. Critic (only) lent Caliban an umbrella.
>
> ??? Is that final paragraph correct ???
>
I don't think so. If Critic is a T, then by (1) Low precedes Critic, leaving
the possibilities: LCY, LYC, YLC. But (3) allows LCY and YLC, since YLC gives
Low the second choice, which is allowed by (3). So you need (2) to eliminate
YLC as the solution indicated above. You cannot say that the predicate to (3)
must be true (as was done with (2)) since the predicate adds information to the
puzzle if it is false (whereas in (2) if the predicate is false, it adds no
information). By making a similar argument with C and Y reversed, I claim that
one cannot deduce whether Critic or Y.Y. lent Caliban an umbrella.
Terry
tdon...@next.com
Peter Jansen
<
< [... confirms that the author was wrong ...]
<
<< It would be better to alter the puzzle so that the question is "Who
<< first fell in love?" rather than "Who lent Caliban the umbrella?" In
<< the original form it is somewhat disappointing.
<
<Certainly the author should have done so. Actually I find the notion
<that the puzzle is sufficiently subtle that even the author erred to add
<to the interest. If you feel really challenged perhaps you can modify
<the puzzle so that the solution is unique.
<--
Well, I don't know about the "really challenged" and all that, but
how about changing condition 1 to:
"No person who has seen me in a green tie is to choose AFTER CRITIC",
leaving the rest of the original problem statement the same.
I think there is now just one solution (which I am sure I can safely
leave as an exercise to the reader :-) ).
<Richard Harter, Software Maintenance and Development Systems, Inc.
<Net address: jjmhome!smds!rh Phone: 508-369-7398
<US Mail: SMDS Inc., PO Box 555, Concord MA 01742
<This sentence no verb. This sentence short. This signature done.
-- Peter Jansen
--