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Solving a logic puzzle the right way™

22 Jun 2020 in Blog

I take a break from my internship with Capital One to solve a puzzle they posed: for a chance to win!

Today, the intern group sent out a logic puzzle—correct submissions are entered into a raffle to win prizes. What I really want to do, though, is show how I solved it (and discuss some pitfalls along the way).

Without further ado, the puzzle:

Valero’s 5k Fun Run was held yesterday in the downtown district. Determine the shirt color and hometown of each of the top runners, and match each to their finishing time.

Clues

  1. Mathew finished 1 minute after Anthony.
  2. The contestant in the white shirt was either Anthony or the runner who finished in 22 minutes.
  3. The competitor from Kamrar finished sometime after the runner from Pierson.
  4. Salvador finished 1 minute after the runner in the maroon shirt.
  5. Greg finished 1 minute before the competitor in the white shirt.
  6. The contestant from Corinth didn’t wear the pink shirt.
  7. The contestant who finished in 22 minutes was either the contestant from Kamrar or the contestant in the pink shirt.
  8. The competitor who finished in 23 minutes wasn’t from Corinth.

What’s not clear from the description is that there is a fourth color (aquamarine), that no two runners have the same color shirt, nor are any two from the same town. (This becomes somewhat more obvious in the grid we were given as an aid.)

The most egregious ambiguity, however, I will save for later.

Cracking the puzzle in prolog

Prolog is a logic programming language: source code consists of facts and predicates (also called functors; not to be confused with category theory). Predicates are often notated name/3 to mean that the predicate name has 3 parameters. A simple example:

man(socrates). % socrates is a man
mortal(X) :- man(X). % X is mortal if X is a man

Firing up the interpreter, we can ask questions about the universe, given a set of facts and queries:

?- [socrates]. % This is how you load files (omit the .pl extension)
?- mortal(socrates).
true.
?- man(X).
X = socrates.
?- mortal(X).
X = socrates.

Different interpreters give slight different truth/false values.

Note that not only can we ask if a predicate is true for a value, but we can ask which values satisfy a predicate! With properly coded predicates, this can be used to generate solutions to a problem. (The process by which this occurs is called unification, an algorithm that also lies at the heart of the Hindley-Milner type system and inference algorithm used in Standard ML, the precursor to OCaml and Haskell.)

A quick example of a built-in functor, member/2 (in some prologs, this requires loading a library, but only when used in source-code):

?- member(1, [1,2,3]).
true.
?- member(X, [1,2,3]).
X = 1 ;
X = 2 ;
X = 3.
?- member(1, [X,Y]).
X = 1 ;
Y = 1.

Note that lower-case words are atoms (along with numbers and strings and such), while upper-case words are variables for unification. And lists have the somewhat universal [] syntax.

Onto the puzzle. We first decide on a representation of the solution: since we care about the order the 4 people finished in, we’ll use a list to hold the runners in order of finish time. Each item will be a list of their name, shirt color, and town:

% Sol = [[Name, Color, Town] times 4]
% the list is ordered by finish time, which is then mapped to 21,22,23,24
solve(Sol) :-

    Sol = [[Name1, Color1, Town1],
           [Name2, Color2, Town2],
           [Name3, Color3, Town3],
           [Name4, Color4, Town4]],

Then, we add some uniqueness constraints:

    % puzzle constraints
    permutation([Name1, Name2, Name3, Name4], [anthony, greg, mathew, salvador]),
    permutation([Color1, Color2, Color3, Color4], [aquamarine, maroon, pink, white]),
    permutation([Town1, Town2, Town3, Town4], [corinth, janesville, kamrar, pierson]),

Then we iterate the puzzle constraints:

    %1
    nextto([anthony, _, _], [mathew, _, _], Sol),
    %2
    (member([anthony, white, _], Sol) ; nth1(2, Sol, [_, white, _])),
    %3
    (append(Left, Right, Sol),
        member([_, _, pierson], Left),
        member([_, _, kamrar], Right)),
    %4
    nextto([_, maroon, _], [salvador, _, _], Sol),
    %5
    nextto([greg, _, _], [_, white, _], Sol),
    %6
    \+member([_, pink, corinth], Sol),
    %7
    (nth1(2, Sol, [_, _, kamrar]) ; nth1(2, Sol, [_, pink, _])),
    %8
    \+nth1(3, Sol, [_, _, corinth]).

Or at least, so we think.

In one early version of this solution, I had the argument order for nth1/3 backwards, giving me bogus answers. In another, I had forgotten to encode the uniqueness of the shirt colors and towns, so my solutions had more than one person in the same shirt or from the same town! (Aside: I would love to know if there’s a more elegant way to encode those than completely destructing the solution.)

But the really, truly, frustrating thing is that the interpreter gives two solutions to these constraints!

?- [race].
?- solve(S).
S = [[greg, pink, pierson], [anthony, white, kamrar], [mathew, maroon, janesville], [salvador, aquamarine, corinth]] ;
S = [[greg, maroon, pierson], [salvador, white, kamrar], [anthony, pink, janesville], [mathew, aquamarine, corinth]] ;

Worse, working through the constraints, both are entirely valid—unless you read or as exclusive or! Then, Anthony can no longer be wearing the white shirt, and the second solution is the only valid one. Some claim ; is supposed to mean exclusive or, others that is logical or (which isn’t exclusive). Alas, I found no easy way to rectify the final code (below) to account for this oddity short of mapping a ; b to (a, \+b ; b, \+a), which I find ugly.

% vim: ft=prolog

:- use_module(library(lists)).

% Sol = [[Name, Color, Town] times 4]
% the list is ordered by finish time, which is then mapped to 21,22,23,24
solve(Sol) :-

    Sol = [[Name1, Color1, Town1],
           [Name2, Color2, Town2],
           [Name3, Color3, Town3],
           [Name4, Color4, Town4]],

    % puzzle constraints
    permutation([Name1, Name2, Name3, Name4], [anthony, greg, mathew, salvador]),
    permutation([Color1, Color2, Color3, Color4], [aquamarine, maroon, pink, white]),
    permutation([Town1, Town2, Town3, Town4], [corinth, janesville, kamrar, pierson]),

    %1
    nextto([anthony, _, _], [mathew, _, _], Sol),
    %2
    (member([anthony, white, _], Sol) ; nth1(2, Sol, [_, white, _])),
    %3
    (append(Left, Right, Sol),
        member([_, _, pierson], Left),
        member([_, _, kamrar], Right)),
    %4
    nextto([_, maroon, _], [salvador, _, _], Sol),
    %5
    nextto([greg, _, _], [_, white, _], Sol),
    %6
    \+member([_, pink, corinth], Sol),
    %7
    (nth1(2, Sol, [_, _, kamrar]) ; nth1(2, Sol, [_, pink, _])),
    %8
    \+nth1(3, Sol, [_, _, corinth]).

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