Skip to main content

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≥b and both numbers are within the range (2,99). We give Mr.P the product ab 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}}}


Comments

Post a Comment

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...
Post a Comment
Read more

How to remap graph properties?

Graph objects support both custom properties, which do not have special meanings, and standard properties, which may be used by some functions. When importing from formats such as GraphML, we usually get a result with custom properties. What is the simplest way to remap one property to another, e.g. to remap a custom property to a standard one so it can be used with various functions? Example: Let's get Zachary's karate club network with edge weights and vertex names from here: http://nexus.igraph.org/api/dataset_info?id=1&format=html g = Import[ "http://nexus.igraph.org/api/dataset?id=1&format=GraphML", {"ZIP", "karate.GraphML"}] I can remap "name" to VertexLabels and "weights" to EdgeWeight like this: sp[prop_][g_] := SetProperty[g, prop] g2 = g // sp[EdgeWeight -> (PropertyValue[{g, #}, "weight"] & /@ EdgeList[g])] // sp[VertexLabels -> (# -> PropertyValue[{g, #}, "name"]...
Post a Comment
Read more