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programming - How to improve this code for solving the "Mr.S and Mr.P" puzzle?




Mr.S and Mr. P puzzle — "Formalization of two Puzzles Involving Knowledge", McCarthy, John (1987)


We pick two numbers $a$ and $b$, such that $a\geq b$ and both numbers are within the range $(2,99)$. We give Mr.P the product $a b$ and give Mr.S the sum $a+b$. Then following dialog takes place:


Mr.P: I don't know the numbers
Mr.S: I knew you didn't know. I don'tknow either.
Mr.P: Now I know the numbers
Mr.S: Now I know them too


Can we find the numbers $a$ and $b$?



I tried to do this, but it is very slow. I am sure there must be a simpler way.


Clear[pool, f1, f2, f3];

pool = Join @@ Table[{i, j}, {i, 2, 99}, {j, 2, i}];

f1[x_] := Length@Select[pool, Times @@ # == x &] != 1
f2[x_] := Length@Select[pool, Plus @@ # == x &] != 1
f3[x_] := And @@ (f1 /@ (Times @@@ Select[pool, Plus @@ # == x &]))
f4[x_] := Length@Select[Select[pool, Times @@ # == x &], f3[#[[1]] + #[[2]]] &] == 1
f5[x_] := Length@Select[Select[pool, Plus @@ # == x &], f4[#[[1]]*#[[2]]] &] == 1

Select[pool, 
  f1[#[[1]]*#[[2]]] && f2[#[[1]] + #[[2]]] && f3[#[[1]] + #[[2]]] && 

    f4[#[[1]]*#[[2]]] && f5[#[[1]] + #[[2]]] &] // Timing

Answer



I tried to understand the other two solutions, but honestly, I couldn't. So I tried to write a version that is easier to understand.


EDIT: I've refactored the code a little, primarily pulling out the "knowledge operators" personKnowsSolution and personKnowsProperty. I'm not really following McCarthy's axiomatization of knowledge, this is just my ad-hoc way of expressing "knowledge" in Mathematica.


Clear[personKnowsSolution, personKnowsProperty, mrP, mrS]
personKnowsSolution[informationFilter_, possibilities_] := 
 Join @@ Select[GatherBy[possibilities, informationFilter], Length[#] == 1 &]
personKnowsProperty[informationFilter_, possibilities_, property_] := 
 Select[possibilities, property[informationFilter[#]] &]
mrP[{a_, b_}] := a*b

mrS[{a_, b_}] := a + b
(
  allPossibilities = Join @@ Table[{i, j}, {i, 2, 99}, {j, 2, i}];

  (* Mr. P doesn't know the solution *)
  mrPWouldKnowSolution = personKnowsSolution[mrP, allPossibilities];
  mrPDoesntKnowSolution = Complement[allPossibilities, mrPWouldKnowSolution];

  (* Mr. S doesn't know the solution *) 
  mrSWouldKnowSolution = personKnowsSolution[mrS, allPossibilities];

  mrSDoesntKnowSolution = Complement[allPossibilities, mrSWouldKnowSolution];

  (* Mr. S knows Mr. P doesn't know the solution *)
  sumsWhereMrPWouldKnowTheSolution = Union[mrS /@ mrPWouldKnowSolution];
  mrSKnowsMrPDoesntKnow = 
   personKnowsProperty[mrS, mrSDoesntKnowSolution, 
    Not[MemberQ[sumsWhereMrPWouldKnowTheSolution, #]] &];

  (* Given that, Mr. P knows the solution *)
  mrPKnowsTheSolution = personKnowsSolution[mrP, mrSKnowsMrPDoesntKnow];


  (* Given that, Mr. S knows the solution *)
  mrSKnowsTheSolution = personKnowsSolution[mrS, mrPKnowsTheSolution]
) // Timing

Output: {0.063, {{13, 4}}}


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