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"To Mock a Mockingbird" by Raymond Smullyan. From Wikipedia: > To Mock a Mockingbird and Other Logic Puzzles: Including an Amazing Adventure in Combinatory Log

"To Mock a Mockingbird" by Raymond Smullyan. From Wikipedia: > To Mock a Mockingbird and Other Logic Puzzles: Including an Amazing Adventure in Combinatory Logic (1985, ISBN 0-19-280142-2) is a book by the mathematician and logician Raymond Smullyan. It contains many nontrivial recreational puzzles of the sort for which Smullyan is well known. It is also a gentle and humorous introduction to combinatory logic and the associated metamathematics, built on an elaborate ornithological metaphor. Here's an example puzzle from the first half (the second half deals with combinatory logic): > Suppose I offer to give you one of three prizes-Prize A, Prize B, or Prize C. Prize A is the best of the three, Prize B is middling, and Prize C is the booby prize. You are to make a statement; if the statement is true, then I promise to award you either Prize A or Prize B, but if your statement is false, then you get Prize C-the booby prize. > Of course it is easy for you to be sure to win either Prize A or Prize B; all you need say is: "Two plus two is four." But suppose you have your heart set on Prize A-what statement could you make which would force me to give you Prize A? Fun book.

javajosh 6y ago
Apart from playing twice, I can't think of a statement that could exclude you from choosing arbitrarily between A and B, since within the conceit of the problem there is no way to exclude B with a mere statement.
- xyzzyz 6y ago
  You probably need to come up with a statement that’s true, but is made false by the act of getting B.
  - javajosh 6y ago
    So, "I don't have B"? But the system is mutable, and we can easily see that the proposition is true, and then (possibly) false. Then you have to say "I will never have B". But that's not necessarily true! You would need either an oracle, some sort of iterative evaluation, or a perspective from which the entire interaction is a single static object, something like a tree of outcomes, which you allow to be used in the evaluation (e.g. one possible implementation of an oracle). I find all of this highly unsatisfying.
    - karatestomp 6y ago
      "You will not award me prize B for this response." Good enough. All this sort of thing becomes a Genie problem if you think about it too much. We all get what the solution means.
- qznc 6y ago
  For this statement I will receive prize A or C.
  - carapace 6y ago
    You're not supposed to spoil it. The whole point of these puzzles is the joy and knowledge you gets from solving them on your own. Publishing the answers is selfish.
    - javajosh 6y ago
      Well, I for one don't consider this a satisfying solution (you could argue it does not count as a solution at all). How is the host supposed to evaluate the future? As I said in another comment, for that the host either needs iteration or an oracle, and both of those are missing from the premise of the question. It's like asking riddle where the solution is "Use a time-machine" when no time-machine is mentioned in the riddle: it's indicative of a riddle so weak it's almost an insult.
      - qznc 6y ago
        I don't consider it a statement about the future but rather about the inevitable consequence whenever this game is played. Maybe it should be rephrased as: For this statement one will receive prize A or C. @Carapace I find joy in discussing the validity of answers like this and such discussion is not possible without posting the answer first. I do feel sorry if I spoiled someone the joy of puzzling themselves though.
        javajosh 6y ago
        >I don't consider it a statement about the future Okay, so what does one will receive mean, if it doesn't reference the future???
      - carapace 6y ago
        Yeah, I haven't quite got the hang of it myself. The encoding of formal logic in English sentences is fraught with ambiguity until you grok the specific style used. Here's an example (SPOILER ALERT: with solution) of what I mean. The English version is just a verbose translation of a simple Boolean table. ================================== Black-Hat Chuck ================================== :date: 2014-02-01 10:56 :summary: A solution to the Three Hats puzzle. I recently ran across a `cool puzzle site`_. Here is one of the puzzles and a solution: There are 3 black hats and 2 white hats in a box. Three men (we will call them A, B, & C) each reach into the box and place one of the hats on his own head. They cannot see what color hat they have chosen. The men are situated in a way that A can see the hats on B & C's heads, B can only see the hat on C's head and C cannot see any hats. When A is asked if he knows the color of the hat he is wearing, he says no. When B is asked if he knows the color of the hat he is wearing he says no. When C is asked if he knows the color of the hat he is wearing he says yes and he is correct. What color hat and how can this be? .. class:: attribution caption ~ `Three Hats`_ Solution ------------ I'm going to call the three men Alan, Bob, and Chuck. First, let's imagine what Alan might be able to see. There are four possibilities. =========== ============= Bob's Hat Chuck's Hat =========== ============= White White Black White White Black Black Black =========== ============= Since there are only two white hats, if Alan sees that both Bob's and Chuck's hats are white his own hat would have to be black. In other words, by admitting that he can't tell which hat he is wearing Alan is saying that either or both of Bob's and Chuck's hats are black. If we eliminate the case of both hats being white we are left with three possibilities. =========== ============= Bob's Hat Chuck's Hat =========== ============= Black White White Black Black Black =========== ============= From this it should be easy to see that if Bob sees that Chuck's hat is white his own hat would have to be black. Bob would be uncertain of the color of his own hat only if Chuck's hat is black. So Chuck, being no dummy, can conclude that his own hat is black. It's an elegant puzzle with a very simple and satisfying solution. .. _cool puzzle site: http://wuriddles.com/ .. _Three Hats: http://wuriddles.com/easy.shtml#3hats
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davidgould 6y ago
“I have a gun.”

javajosh 6y ago

Apart from playing twice, I can't think of a statement that could exclude you from choosing arbitrarily between A and B, since within the conceit of the problem there is no way to exclude B with a mere statement.

xyzzyz 6y ago
You probably need to come up with a statement that’s true, but is made false by the act of getting B.
- javajosh 6y ago
  So, "I don't have B"? But the system is mutable, and we can easily see that the proposition is true, and then (possibly) false. Then you have to say "I will never have B". But that's not necessarily true! You would need either an oracle, some sort of iterative evaluation, or a perspective from which the entire interaction is a single static object, something like a tree of outcomes, which you allow to be used in the evaluation (e.g. one possible implementation of an oracle). I find all of this highly unsatisfying.
  - karatestomp 6y ago
    "You will not award me prize B for this response." Good enough. All this sort of thing becomes a Genie problem if you think about it too much. We all get what the solution means.
qznc 6y ago
For this statement I will receive prize A or C.
- carapace 6y ago
  You're not supposed to spoil it. The whole point of these puzzles is the joy and knowledge you gets from solving them on your own. Publishing the answers is selfish.
  - javajosh 6y ago
    Well, I for one don't consider this a satisfying solution (you could argue it does not count as a solution at all). How is the host supposed to evaluate the future? As I said in another comment, for that the host either needs iteration or an oracle, and both of those are missing from the premise of the question. It's like asking riddle where the solution is "Use a time-machine" when no time-machine is mentioned in the riddle: it's indicative of a riddle so weak it's almost an insult.
    - qznc 6y ago
      I don't consider it a statement about the future but rather about the inevitable consequence whenever this game is played. Maybe it should be rephrased as: For this statement one will receive prize A or C. @Carapace I find joy in discussing the validity of answers like this and such discussion is not possible without posting the answer first. I do feel sorry if I spoiled someone the joy of puzzling themselves though.
      - javajosh 6y ago
        >I don't consider it a statement about the future Okay, so what does one will receive mean, if it doesn't reference the future???
    - carapace 6y ago
      Yeah, I haven't quite got the hang of it myself. The encoding of formal logic in English sentences is fraught with ambiguity until you grok the specific style used. Here's an example (SPOILER ALERT: with solution) of what I mean. The English version is just a verbose translation of a simple Boolean table. ================================== Black-Hat Chuck ================================== :date: 2014-02-01 10:56 :summary: A solution to the Three Hats puzzle. I recently ran across a `cool puzzle site`_. Here is one of the puzzles and a solution: There are 3 black hats and 2 white hats in a box. Three men (we will call them A, B, & C) each reach into the box and place one of the hats on his own head. They cannot see what color hat they have chosen. The men are situated in a way that A can see the hats on B & C's heads, B can only see the hat on C's head and C cannot see any hats. When A is asked if he knows the color of the hat he is wearing, he says no. When B is asked if he knows the color of the hat he is wearing he says no. When C is asked if he knows the color of the hat he is wearing he says yes and he is correct. What color hat and how can this be? .. class:: attribution caption ~ `Three Hats`_ Solution ------------ I'm going to call the three men Alan, Bob, and Chuck. First, let's imagine what Alan might be able to see. There are four possibilities. =========== ============= Bob's Hat Chuck's Hat =========== ============= White White Black White White Black Black Black =========== ============= Since there are only two white hats, if Alan sees that both Bob's and Chuck's hats are white his own hat would have to be black. In other words, by admitting that he can't tell which hat he is wearing Alan is saying that either or both of Bob's and Chuck's hats are black. If we eliminate the case of both hats being white we are left with three possibilities. =========== ============= Bob's Hat Chuck's Hat =========== ============= Black White White Black Black Black =========== ============= From this it should be easy to see that if Bob sees that Chuck's hat is white his own hat would have to be black. Bob would be uncertain of the color of his own hat only if Chuck's hat is black. So Chuck, being no dummy, can conclude that his own hat is black. It's an elegant puzzle with a very simple and satisfying solution. .. _cool puzzle site: http://wuriddles.com/ .. _Three Hats: http://wuriddles.com/easy.shtml#3hats

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davidgould 6y ago

“I have a gun.”