Cooked poisons
Timothy Y. Chow
University. I believe his name was Michael Rabin; perhaps someone at
CMU can verify or correct this. I have modified the original statement
of the puzzle slightly to eliminate ambiguities.
There is a world in which the inhabitants have a strange physiology.
If a healthy person ingests a poison, he will die within an hour unless
he ingests a stronger poison, in which case he becomes healthy again.
The poisons in this world are strictly linearly ordered in strength.
Moreover there are two kinds of poisons: magical and medical, which are
dispensed by the Royal Magician and the Royal Physician respectively.
No magical poison has the same strength as a medical poison. All of
these facts are common knowledge.
The King decides that he wants to find the strongest poison in the
land, because it will not only be very useful for eliminating enemies
but will also act as an antidote against any other poison. So he calls
in the Royal Magician and the Royal Physician and says, "I want each of
you to return here to my royal chambers at noon one week from now.
Bring two tablets of your strongest poison. To give you incentive to
bring your strongest poison, you must do the following: each of you
must eat one of the other's tablets and then eat one of your own
tablets. I will have trained observers present to make sure that you
cannot cheat. Then you will be watched for one hour, during which you
may not ingest any substances. The person who has the stronger poison
will of course survive and the other will die. This is unfortunate but
I have decided that it is worth it, in the interests of national
security. If I detect any attempt to circumvent these rules you will
both be executed. You may go now, but you must return at the specified
time."
The Royal Physician and the Royal Magician go off, both very disturbed,
because neither wants to die. Each has had some experience with the
other's poisons and knows that some of them are quite potent. Neither
of them is fully confident of having the strongest poison. Nor does
either have any way of getting access to the other's poisons. They
rack their brains all week trying to think how they can best ensure
their own survival.
The appointed time rolls around and the two Royal Servants return.
They follow the specified protocol exactly, and are watched carefully
for one hour. To everyone's astonishment, both Royal Servants keel
over and die within the hour. The Royal Coroner confirms that both
died of poisoning.
1. Explain what happened in the way the person who originally devised the
puzzle intended.
2. Cook the puzzle by finding three alternative explanations.
--
Tim Chow tyc...@math.mit.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons. ---Popular Mechanics, March 1949
RING, DAVID WAYNE
>There is a world in which the inhabitants have a strange physiology.
>for one hour. To everyone's astonishment, both Royal Servants keel
>over and die within the hour. The Royal Coroner confirms that both
>died of poisoning.
>1. Explain what happened in the way the person who originally devised the
> puzzle intended.
>2. Cook the puzzle by finding three alternative explanations.
It seems they both thought they would lose, so they each prepared a placebo
and took some weak poison before the test.
There are more sophisticated strategies which lead to the same result. Is
that what you mean by 'cooking' the puzzle?
Dave Ring
dwr...@zeus.tamu.edu
Matthew T. Russotto
[the poison puzzle]
>
>The appointed time rolls around and the two Royal Servants return.
>They follow the specified protocol exactly, and are watched carefully
>for one hour. To everyone's astonishment, both Royal Servants keel
>over and die within the hour. The Royal Coroner confirms that both
>died of poisoning.
>
>1. Explain what happened in the way the person who originally devised the
> puzzle intended.
>2. Cook the puzzle by finding three alternative explanations.
Here's one explanation:
Each brought completely inert tablets. But neither expected the other
to do this, so they each ate a tablet of their weakest poison
beforehand, so the other persons tablet would cure them rather than
kill them.
There are variations of this-- one brought inert tablets and expected
the other to do the same, while the other brought poison as ordered.
--
Matthew T. Russotto russ...@eng.umd.edu russ...@wam.umd.edu
Some news readers expect "Disclaimer:" here.
Just say NO to police searches and seizures. Make them use force.
(not responsible for bodily harm resulting from following above advice)
Jay Sipelstein
>Following is a puzzle invented by someone from Carnegie Mellon
>University. I believe his name was Michael Rabin; perhaps someone at
>CMU can verify or correct this. I have modified the original statement
>of the puzzle slightly to eliminate ambiguities.
[puzzle deleted]
This puzzle appeared on one of the CMU CS department bboards last summer.
It was developed by Michael Rabin (CS prof at Harvard). It also generated
lots of traffic on the bboard about what assumptions you're allowed to
make....
Jay Sipelstein
sipel...@cs.cmu.edu
Tom Carmichael
The only thing that I can think of is that both of the Royal Servants gave
the other a tablet which was not poison, therby killing themselves with
their own poison. I suppose that it is also possible that only one of them,
the one with the weaker poison, did this. I still haven't figured out _why_
they did this, only that they did.
Tom
--
"Sleep is a poor substitute for caffine."| Tom Carmichael
| tac...@hertz.njit.edu
- The Wizardry Compiled. | to...@shock.njit.edu
|
Hong Liang Xie
I think the original puzzle assumes that the two tablets that
each brought were the same poison. So if a non-poisonous tablet
was given to the other person, one himself should also took
the non-poisonous tablet, the consequence of this would be
no one got killed.
(SPOILER)
Both of them thought this way:
His poison is stronger. Hence if I took a very weak posion
beforehand, it could be cured by his poison. If the two tablets
I bring to the king were not poison at all, I would not be killed
by taking it after the weak poison I takes beforehand being cured
by his poison, but he would be killed by his won poison because
my tablet is not poison and hence can not be cured by his poison.
It ended up that:
Both people brought two non-poisonous tablets but took a weak poison
beforehand. They both got killed by the weak poions taken beforehand.
(I have not yet started thinking the alternative explanations.)
-Hong, CIS Dept, U of Penn
Timothy Y. Chow
inadvertently introduced new unintended solutions.
A. Answer the two original questions, with the protocol changed as follows:
instead of bringing two tablets, each brings a small bottle of liquid
poison, and each must drink from the other's bottle first and then from his
own bottle. This is closer to the original problem statement.
B. Analyze the differences between the two protocols.
To clarify some questions that have been raised. By "cooking" I mean
finding alternative sequences of events that lead to the same result.
The behavior of the two Royal Servants must be consistent with their
desire to save themselves. Thus, the suggestion that one of the servants
brought two placebos and the other behaved normally is not clearly correct;
why on earth would one of the servants bring two placebos? Carelessness is
technically a possibility, but there is a more satisfactory explanation.
Ajit Sanzgiri
>1. Explain what happened in the way the person who originally devised the
> puzzle intended.
>2. Cook the puzzle by finding three alternative explanations.
Can't guess what the original devisor (divisor ?) had in mind but
here are three possible explanations:
a) One of the two (but not both) brought along a placebo poison
(Don't ask me why.)
b) One of the two cheated and brought along a sample of the
other's poison. This happened to be exactly as potent as the
sample the other one brought along.
c) One of the two cheated and swallowed the tablets in the wrong
order. Unfortunately he was wrong.
Solution c) is capable of modifications (swallow one but give the
other one some other sample etc.) but we are beginning to strain
credibility.
Ajit Sanzgiri
Ken Arromdee
>By trying to eliminate certain unintended solutions by rewording, I
>inadvertently introduced new unintended solutions.
>A. Answer the two original questions, with the protocol changed as follows:
>instead of bringing two tablets, each brings a small bottle of liquid
>poison, and each must drink from the other's bottle first and then from his
>own bottle. This is closer to the original problem statement.
>B. Analyze the differences between the two protocols.
>To clarify some questions that have been raised. By "cooking" I mean
>finding alternative sequences of events that lead to the same result.
>The behavior of the two Royal Servants must be consistent with their
>desire to save themselves.
The problem is still horrendously unspecified.
If a person is in the state "poisoned by X" and takes a poison Y which is
weaker, are they now in the state "poisoned by the strongest (X)", "poisoned
by the most recent (Y)", or "poisoned by both X and Y"?
Similarly, what happens if someone takes two poisons simultaneously?
If it is possible to be in a state of being simultaneously poisoned by two
poisons, what happens when someone takes a dose of a stronger poison; does it
cancel everything or just one of the poisons within their system? If so,
which one? What if the poison is stronger than one and weaker than another of
the poisons already in your system? What if you're poisoned by X and Y, X
stronger, and you take X? Do you now end up poisoned by X or by nothing?
Now that that's done, you have to figure out what the puzzle statement is
letting each servant do. For instance, it is not clear whether the servants
are, according to the rules of the puzzle, permitted to not bring their
strongest poison.
Next is the problem of answering the question. You want a solution for a
Royal Servant which guarantees he's safe. You then want an explanation of how
the servant was nevertheless not safe. This is contradictory--if his solution
_really_ left him safe, it's impossible for him to die no matter what happens,
and if it's possible for him to die, his solution must not have really left
him safe.
Presumably what is meant is that each servant _thinks_ he has a guaranteed
safe solution, but doesn't really have one. Unfortunately, this is extremely
unspecified too; it is now possible to choose _any_ unsafe strategy for the
servant and solve the problem.
Next try at interpretation: you want each servant to think he has a solution
without having one... but his error has to be a _reasonable_ error. (For
instance, perhaps he made the bad assumption that the other servant would
not attempt a solution). The trouble is that "reasonable error" is undefined.
--
This is a newer version of the memetic .signature infection. Now that's an
idea. Copy it into your .signature today!
Kenneth Arromdee (UUCP: ....!jhunix!arromdee; BITNET: arromdee@jhuvm;
INTERNET: arro...@jyusenkyou.cs.jhu.edu)
Stein Kulseth FBA
|> [PUZZLE DELETED]
|>
|> 1. Explain what happened in the way the person who originally devised the
|> puzzle intended.
|> 2. Cook the puzzle by finding three alternative explanations.
A number as already posted good solutions to 1.
Here is an alternative solution:
[SPOILER ALERT]
After finding out that the smart thing to do is to ingest a weak poison
first and then bring two placebo tablets to the test, the Royal Servants
both know that the other may also try this strategy (and as we know, this
will result in the death of both). They could decide to ingest the weak
poison and then bring their strongest poison to the test, but this would
fail if the other plays fair and brings along a real poison.
So they decide to ingest their secondmost powerful poison before the test,
and then bring two tablets of the most powerful.
Now they will survive if either the other brings placebo
tablets, or if his #1 poison is less powerful than their own #2.
Alas, as it turns out, both Servants' #1 poisons are more powerful than
their opponent's #2 poison, offsetting its effect. Then they both die
from the last poison tablet (their own #1 poison)
--
Stein Kulseth
Norwegian Telecom Research, Box 83, N-2007 KJELLER, NORWAY
internet: st...@hal.nta.no X400: stein....@forskning.teledir.no
Phone: + 47 6 80 98 05 Fax: + 47 6 81 00 76
Tore Mengshoel
|> University. I believe his name was Michael Rabin; perhaps someone at
|> CMU can verify or correct this. I have modified the original statement
|> of the puzzle slightly to eliminate ambiguities.
1. Both could be stupid, having one strong and one weak tablet each
(both guys having tablets of their own kind only),
and each of them taking the other guys strong tablet and then
their own weak one --- this is obviously ot the intended solution.
2 Both could have had their own strongest tablet and the other
guys strongest tablet, hoping that the other guy has two of his
own strongest tablet, thus causing the other guy to eat the same
tablet twice and die. As both is doing the same thinking, they
both die. This is probably the intended 'model'.
3. They could both have being doing some research on their own and
found the same strongest drug, magical or medical. If there are
four tablets of the same strength, both would die.
4. If one of them has the strongest of each type of poisons,
while the other one has two tablets who are the strongest of
one kind of poison, the following could happen:
The guy with similar tablets takes the tablet that the other
guy has that is similar to his own tablets => he dies.
The other guy takes one of the three similar tablets first,
and then the unique one, which happens to be weaker => he dies.
Tore
Stein Kulseth FBA
(Tore Mengshoel) gives the following answers to the 'Cooked poisons' puzzle:
|> 1. Both could be stupid, having one strong and one weak tablet each
|> (both guys having tablets of their own kind only),
|> and each of them taking the other guys strong tablet and then
|> their own weak one --- this is obviously ot the intended solution.
This solution is OK, there is no law against stupidity
|> 2 Both could have had their own strongest tablet and the other
|> guys strongest tablet, hoping that the other guy has two of his
|> own strongest tablet, thus causing the other guy to eat the same
|> tablet twice and die. As both is doing the same thinking, they
|> both die. This is probably the intended 'model'.
|> 3. They could both have being doing some research on their own and
|> found the same strongest drug, magical or medical. If there are
|> four tablets of the same strength, both would die.
|> 4. If one of them has the strongest of each type of poisons,
|> while the other one has two tablets who are the strongest of
|> one kind of poison, the following could happen:
|> The guy with similar tablets takes the tablet that the other
|> guy has that is similar to his own tablets => he dies.
|> The other guy takes one of the three similar tablets first,
|> and then the unique one, which happens to be weaker => he dies.
These aren't solutions, as all of them requires the Royal Physician to
have access to a magical poison and/or the Royal Magician to have access
to a medical poison. The problem explicitly states:
... Nor does either have any way of getting access to the other's poisons.
^^^
Timothy Y. Chow
>A. Answer the two original questions, with the protocol changed as follows:
>instead of bringing two tablets, each brings a small bottle of liquid
>poison, and each must drink from the other's bottle first and then from his
>own bottle. This is closer to the original problem statement.
Seems like people have had a good chance to work on this puzzle, so for what
it's worth, here's what I had in mind as the solution. I have not fully
analyzed the boo-boo I made by changing liquids to tablets, so I will treat
only the above case here.
One's first impulse is to say that the two Royal Servants brought the same
poison. This possibility was not clearly eliminated in Rabin's problem
statement. It is ruled out here because of the distinction between magical
poisons and medical poisons and the explicit statement that neither has any
way of getting access to the other's poisons.
The second idea is that they both brought water, hoping the other would do
so, so that they would both survive. (Some kind of Hofstadterian "super-
rationality principle" or something like that.) This also was not ruled out
in Rabin's problem statement, but is ruled out here because after an hour
the king would realize that they had cheated and would execute them both.
The key to the problem is to observe that nothing prevents the Royal
Servants from ingesting a poison just before the ordeal. A bit of thought
yields the solution originally intended (which has already been suggested by
some): both drink a weak poison ahead of time and bring water, each hoping
that the other would not come up with the same trick.
Two possible objections to this:
(1) The king did not allow them to bring anything but their strongest
poison. However, this rule was stated by the king, and not by the narrator
of the problem, so the possibility cannot be ruled out, since there is
nothing said about the king's making any attempt to enforce this.
(2) The king would discover after the ordeal that the winner had cheated.
This possibility I tried to eliminate by replacing the liquid with tablets.
However, even with liquids we can eliminate this possibility by observing
that both Royal Servants could drink up all their own poison since they get
to drink from their own bottle last. (One can quibble that trace amounts
are all that is needed for testing; this is why I thought of switching to
tablets.)
However, if we accept the above explanation, there are at least three other
explanations. These three other explanations do not require any appeal to
carelessness or stupidity. (Nothing in the problem statement rules out
carelessness or stupidity, but I think it is more aesthetically pleasing not
to appeal to them.)
There are at least four possible strategies.
1. Conventional strategy. Ingest nothing beforehand, bring poison.
2. Weak poison strategy. Ingest weak poison beforehand, bring water.
3. Water strategy. Ingest nothing beforehand, bring water.
4. Double dose strategy. Ingest poison beforehand, bring stronger poison.
It is technically possible, in 4, to bring a poison equal to or weaker than
the poison ingested beforehand, but since this guarantees one's own death we
have to appeal to carelessness or stupidity to support it. (Note that the
careful wording of the problem shows that if one drinks a poison, then one
or more weaker poisons, and then a poison stronger than the first, the
person becomes healthy again.) Hence I do not consider it.
Which of these are "reasonable" strategies? I claim that a reasonable
strategy is either (1) a conventional strategy or (2) a strategy that wins
against some reasonable strategy (interpret this definition recursively).
(I cannot defend this claim rigorously, but it seems to me to be within
the spirit of the problem.) Hence 2 is reasonable because it wins against
1, and 3 is reasonable because it wins against 2, and 4 is reasonable
because it wins against 3. Thus all four strategies are reasonable and
might be employed by either one of the Servants, depending on what they
think their opponent will do.
Given this the three other scenarios are pretty much obvious.
A. 2 versus 3
B. 1 versus 4 where the conventional strategist's poison is intermediate
in strength between the two poisons of the double doser.
C. 4 versus 4 where each one's stronger poison is stronger than the other's
weaker poison.
One other scenario is 1 versus 2 where the latter Servant's weakest poison
is stronger than the other's strongest. This was not ruled out in Rabin's
problem statement but is ruled out here by the clause about the experience
each Servant has of the other's poisons.
Note that scenario A has been suggested by some already but in my opinion
with insufficient justification. Note also that although 4 wins against 1
in certain circumstances, we need 3 to justify 4, since strategy 4 is always
WORSE than strategy 1 if the opponent adopts strategy 1.
Finally, for the smart-alecks: see if you can do anything with the fact
that the sex of the Royal Servants was unspecified but that the action of
the poison was described using a masculine pronoun!
Ken Arromdee
>The second idea is that they both brought water, hoping the other would do
>so, so that they would both survive. (Some kind of Hofstadterian "super-
>rationality principle" or something like that.) This also was not ruled out
>in Rabin's problem statement, but is ruled out here because after an hour
>the king would realize that they had cheated and would execute them both.
>(2) The king would discover after the ordeal that the winner had cheated.
>This possibility I tried to eliminate by replacing the liquid with tablets.
>However, even with liquids we can eliminate this possibility by observing
>that both Royal Servants could drink up all their own poison since they get
>to drink from their own bottle last.
These contradict. If in (2) it is possible to escape the king seeing you
cheated by drinking all the liquid, then it is possible to similarly escape
it for the case when both brought water, by the same method: drinking all the
liquid.
>(Note that the
>careful wording of the problem shows that if one drinks a poison, then one
>or more weaker poisons, and then a poison stronger than the first, the
>person becomes healthy again.)
The careful wording of the problem indicates something a bit different,
however, in cases where the third poison is intermediate between the first two.
Call the poisons Z, X, and Y, Z the strongest and X the weakest. Suppose
someone drinks Z and then X, and then drinks a third poison Y.
The problem says that if someone drinks X and then Y, they are cured. It also
says that if someone drinks Z and then Y, they die. You thus have the
contradiction that the same person is both cured and dies.
This contradiction may be resolved by distinguishing the concepts "being
poisoned by X" and "being poisoned by Y". In that case, the result is
"cured of X" and "poisoned by Y", which isn't contradictory.
But once you have separated the two concepts, it is now reasonable to ask the
question of what happens in the Y, X, Z case: after someone drinks Y and X,
they are "poisoned by X" _and_ "poisoned by Y". Does drinking Z cancel both
of these or one? Now that we recognize these as two separate concepts, it's
no longer clear that Z cancels both of them. And if Z _doesn't_ cancel both
of them, bot only one, there is a _new_ solution: each servant takes a weak
potion Y and a weaker potion X beforehand. They both honestly bring their
strongest poisons. The first poison they take in front of the king (the
other guy's poison) cancels out one of X and Y, and the second poison they
take cancels out the other one, leaving them both safe. (And, what happened
in the puzzle is that it so happened that one servant's weakest poison was
stronger than the other servant's strongest poison, in which case this
cancellation does not work.)
Timothy Y. Chow
should not be 2 against 3 but 1 against 3.
In article <1992Mar27.0...@jyusenkyou.cs.jhu.edu> arro...@jyusenkyou.cs.jhu.edu (Ken Arromdee) writes:
<In article <1992Mar26.2...@galois.mit.edu> tyc...@riesz.mit.edu (Timothy Y. Chow) writes:
<>The second idea is that they both brought water, hoping the other would do
<>so, so that they would both survive. (Some kind of Hofstadterian "super-
<>rationality principle" or something like that.) This also was not ruled out
<>in Rabin's problem statement, but is ruled out here because after an hour
<>the king would realize that they had cheated and would execute them both.
<>...
<>(2) The king would discover after the ordeal that the winner had cheated.
<>This possibility I tried to eliminate by replacing the liquid with tablets.
<>However, even with liquids we can eliminate this possibility by observing
<>that both Royal Servants could drink up all their own poison since they get
<>to drink from their own bottle last.
<
<These contradict. If in (2) it is possible to escape the king seeing you
<cheated by drinking all the liquid, then it is possible to similarly escape
<it for the case when both brought water, by the same method: drinking all the
<liquid.
Come on...this is obviously fallacious reasoning. The reason the king can
infer cheating in the case when they both bring water is that they are both
STILL ALIVE at the end of the hour, and it was explicitly stated in the
problem that when a healthy person drinks a poison he dies WITHIN AN HOUR;
moreover, this fact is COMMON KNOWLEDGE (and therefore known to the king).
The only way the king could detect cheating in (2) is by testing the
residue.
<>(Note that the
<>careful wording of the problem shows that if one drinks a poison, then one
<>or more weaker poisons, and then a poison stronger than the first, the
<>person becomes healthy again.)
<
<The careful wording of the problem indicates something a bit different,
<however, in cases where the third poison is intermediate between the first two.
<
<Call the poisons Z, X, and Y, Z the strongest and X the weakest. Suppose
<someone drinks Z and then X, and then drinks a third poison Y.
<
<The problem says that if someone drinks X and then Y, they are cured. It also
<says that if someone drinks Z and then Y, they die. You thus have the
<contradiction that the same person is both cured and dies.
Excuse my impatience, but it is clear that you did NOT read the problem
statement carefully. Here is what I said:
>If a healthy person ingests a poison, he will die within an hour unless
^^^^^^^
>he ingests a stronger poison, in which case he becomes healthy again.
The problem does NOT say that if someone drinks X and then Y, they are
cured. It says that if a HEALTHY person drinks X and then Y, they are
cured. It does NOT say that if someone drinks Z and then Y, they die.
It says that if a HEALTHY person drinks Z and then Y but does NOT DRINK
ANY POISON STRONGER THAN Z after drinking Z then they die.
I stand by my statement. I was fully aware of this ambiguity when I stated
the problem; hence the careful wording.
SH...@slacvm.slac.stanford.edu
will toss in my two cents worth. I found the question to have been
stated quite clearly and I'm fairly sure that they each secretly
took their own weakest poison just before showing up with placebos
figuring that by taking the others "strong" poison they would be
made well again (while the other died of the poison he had brought
himself). They each died by their own deception. Quite fitting.
As for other plausable explainations, I'm working on it (fun).
Ken Arromdee
><>The second idea is that they both brought water, hoping the other would do
><>so, so that they would both survive. (Some kind of Hofstadterian "super-
><>rationality principle" or something like that.) This also was not ruled out
><>in Rabin's problem statement, but is ruled out here because after an hour
><>the king would realize that they had cheated and would execute them both.
><>...
><>(2) The king would discover after the ordeal that the winner had cheated.
><>This possibility I tried to eliminate by replacing the liquid with tablets.
><>However, even with liquids we can eliminate this possibility by observing
><>that both Royal Servants could drink up all their own poison since they get
><>to drink from their own bottle last.
><These contradict. If in (2) it is possible to escape the king seeing you
><cheated by drinking all the liquid, then it is possible to similarly escape
><it for the case when both brought water, by the same method: drinking all the
><liquid.
>Come on...this is obviously fallacious reasoning. The reason the king can
>infer cheating in the case when they both bring water is that they are both
>STILL ALIVE at the end of the hour, and it was explicitly stated in the
>problem that when a healthy person drinks a poison he dies WITHIN AN HOUR;
OK, that's another underspecification in the problem then. Under exactly what
circumstances does the king infer cheating? The "obvious" answer is that the
king can infer cheating when both servants survive. "obvious", in a problem
such as this, is a sure way to get nowhere; it needs to be specified.
>>If a healthy person ingests a poison, he will die within an hour unless
> ^^^^^^^
>>he ingests a stronger poison, in which case he becomes healthy again.
>The problem does NOT say that if someone drinks X and then Y, they are
>cured. It says that if a HEALTHY person drinks X and then Y, they are
>cured.
So consider a person who takes in succession a strong, weak, and intermediate
poison. Also consider the case where a person takes in succession only a
weak and intermediate poison.
Are the two cases, just before taking the intermediate poison, equivalent?
If yes, then you survive in both cases, and the first is an example of a
non-healthy person who drinks X and then Y and is cured. If the two cases
are _not_ equivalent, then one must still distinguish the cases "poisoned by
X" and "poisoned by both X and Y", leading to the problems I described in a
previous article....