Some Smullyan-style puzzles

I mentioned Raymond Smullyan’s logic puzzle books as a source of “tricks” for dungeons and never really gave any examples. I probably wouldn’t use them exactly as written, but the template is pretty easy to understand if you have a very basic understanding of propositional logic.

Portia’s caskets

One species of problem I like very much are the “Portia’s Caskets” problems from What is the name of this book? (the puzzles being inspired by Portia’s test for her suitors in The merchant of Venice).

The general form is to have two or more boxes, each inscribed with a statement that is either true or false.*

I think this sort of puzzle can be reasoned through fairly easily, though so I’d add a time limit to increase tension. I have a three minute hourglass I used to time riddles earlier and that worked pretty well.

So here’s an example you might use somewhere in a dungeon:

There are three boxes, one lead, one silver, and one gold. They are inscribed with statements, each of which is either true or false. One contains the McGuffin (key to get out of the room, treasure, map, etc.) and one or both of the others contain some peril or setback (an evil spirit, poison gas, alarm, curse, etc.), so you don’t want to guess blindly or open them all. The caskets may be inscribed:

Lead: The treasure is in the gold casket.

Silver: At least one of these statements is false.

Gold: The treasure is not in the silver casket.

As presented, there is only one casket the treasure can be in. The important thing to remember when creating these is that you need to account for every possible truth-function (state of being true or false) for each statement … but some combinations of truth-functions are not possible. For example, it is not possible that the Lead has a true statement and the Gold has a false statement at the same time, given the rest of the information from the set-up.

One variation that you can use is to say that the caskets were created by one of two smiths, one of whom always puts false inscriptions on his caskets and one of whom always puts true statements on his caskets.

The lady or the tiger?

Another way to present what is essentially the same the puzzle is to put the statements on doors (which Smullyan describes in increasing complexity in one chapter of The lady or the tiger?, after the famous short story).

There is a beholder behind one door and the statements are either true or false.

Door one: There is a beholder behind this door.

Door two: There is no beholder behind this door.

Door three: At most two of these statements are true.

Comment: this is actually a much easier problem, since two of the three doors must not have beholders, so a random door-opener has a 2/3 chance of opening a “safe” door. But you really, really don’t want to be surprised by a beholder, huh? Happily the way the problem is set up one door must be safe and you can tell which one it is.

A better version has some reward behind at most one door and a peril behind one or two doors. On the other hand you can increase the number of doors…Smullyan had a variation with nine doors!

The most interesting thing about these problems, for me anyway, is that while you need to plan ahead to make sure they are solvable, you can pretty much pick the statements out of the air and end up with a solvable problem. I actually wrote both of these problems and then checked and luckily, both were solvable as written, but if not I could have changed a “most” to “least” or added or removed a “not” to create a fairly vexing problem. <UPDATE: as you can see from the comments, I sort of messed up these examples.  You might be better off stealing the problems verbatim from Smullyan or at least having someone else read them to make sure you are presenting something solvable!  How embarrassing — I try to be all clever with some logic problems and screw them up…>

*A sadistic DM might set the players up by giving a series of such puzzles and then having a last puzzle that does NOT stipulate that the statements are either true or false. Logically of course a statement can be indeterminate or meaningless. Or think of it this way: are the statements necessarily about these caskets or doors? Of course not; we need to stipulate that they are self-referential. The players can be rather cruelly reminded that unless they know the statements on the caskets/doors are either true or false, and about the caskets/doors & statements themselves, they could have no bearing on what is inside.

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Published in: on November 15, 2010 at 6:00 am  Comments (5)  
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  1. I can’t actually figure out the three casket puzzle. It could be in the gold casket, if all three statements are true. It could be in the lead casket if two of the statements are false (if lead and silver give false statements but gold gives true). It could be in the silver casket if all three statements are false.

    The problem lies in the silver casket, which doesn’t narrow down th possibilities at all. The term “at most” gives the possibility of one or zero statements being false if the silver statement is true, while also allowing the possibility of two or three statements being false if the silver statement is false. So unless there’s a piece of the set-up which has been left out (like “at least one statement is true and at least one is false”), there’s no way to determine the answer as presented.

  2. Egads, I’m an idiot. I kept thinking “At least one of these statement is false” for the Silver. I’ll correct the post.

    Thanks Tony! I guess I need to be a little more careful about checking my work, huh?

  3. I went back and looked at some of Smullyan’s actual puzzles and he more often has all three inscribed with “the [treasure] is in/not in X” and just gives the meta-statment about how many are true or false separately. That is probably the better way to go.

  4. Yeah, I also don’t get the Beholder one. Why aren’t both of these possible?

    It’s door two:

    Door one: There is a beholder behind this door. FALSE

    Door two: There is no beholder behind this door. FALSE

    Door three: At most two of these statements are true. TRUE

    (or does statement 3 being true mean there MUST be a way to construct it so that 2 statements are true? if so, then it can just be false and then it’s still in Door 2)

    It’s door three

    Door one: There is a beholder behind this door.FALSE

    Door two: There is no beholder behind this door.TRUE

    Door three: At most two of these statements are true. TRUE

    • Jumping Jesus on a Pogo Stick. Did I manage to fark up both problems?

      With the door problem, though, the beholder can’t be behind door one. The “solution” here is we know door one must be “safe”. It’s OK if don’t which of the other two is safe, because we only need to open one door.
      I edited the text after the problem to clarify that — “Only one possible solution” did not necessarily mean only one place the behold was; it just meant only one place he definitely wasn’t.
      I may have to start proofreading this stuff.


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