Task[edit]
Solve the "Impossible Puzzle":


X and Y are two different whole numbers greater than 1. Their sum is no greater than 100, and Y is greater than X. S and P are two mathematicians (and consequently perfect logicians); S knows the sum X+Y and P knows the product X*Y. Both S and P know all the information in this paragraph.
The following conversation occurs:

 S says "P does not know X and Y."
 P says "Now I know X and Y."
 S says "Now I also know X and Y!"
What are X and Y?


Guidance
It can be hard to wrap one's head around what the three lines of dialog between S (the "sum guy") and P (the "product guy") convey about the values of X and Y.

So for your convenience, here's a break-down:





 Quote

 Implied fact


 1)

 S says "P does not know X and Y."

 For every possible sum decomposition of the number X+Y, the product has in turn more than one product decomposition.


 2)

 P says "Now I know X and Y."

 The number X*Y has only one product decomposition for which fact 1 is true.


 3)

 S says "Now I also know X and Y."

 The number X+Y has only one sum decomposition for which fact 2 is true.

Terminology:

 "sum decomposition" of a number = Any pair of positive integers (A, B) so that A+B equals the number. Here, with the additional constraint 2 ≤ A < B.
 "product decomposition" of a number = Any pair of positive integers (A, B) so that A*B equals the number. Here, with the additional constraint 2 ≤ A < B.


Your program can solve the puzzle by considering all possible pairs (X, Y) in the range 2 ≤ X < Y ≤ 98, and then successively eliminating candidates based on the three facts. It turns out only one solution remains!

See the Python example for an implementation that uses this approach with a few optimizations.

See also
   Wikipedia:   Sum and Product Puzzle

mul:=Utils.Helpers.summer.fp1('*,1); //-->list.reduce('*,1), multiply list items
var allPairs=[[(a,b); [2..100]; { [a+1..100] },{ a+b<100 }; ROList]]; // 2,304 pairs
 
sxys,pxys:=Dictionary(),Dictionary();  // hashes of allPairs sums and products: 95,1155
foreach xy in (allPairs){ sxys.appendV(xy.sum(),xy); pxys.appendV(xy:mul(_),xy) }
 
sOK:= 'wrap(s){ (not sxys[s].filter1('wrap(xy){ pxys[xy:mul(_)].len()<2 })) };
pOK:= 'wrap(p){ 1==pxys[p].filter('wrap([(x,y)]){ sOK(x+y) }).len() };
sOK2:='wrap(s){ 1==sxys[s].filter('wrap(xy){ pOK(xy:mul(_)) }).len() };
allPairs.filter('wrap([(x,y)]){ sOK(x+y) and pOK(x*y) and sOK2(x+y) })
.println();