The first pair, made up S1 and S2, are told sum of two integers (x + y). While the second pair, made up of P1 and P2, are told the product (xy).

At the outset, none of the participants know the identity of the knave in the other pair, although they are aware that each group is mixed.

Further, all participants have been told that 1 < x < y < 100.

Interaction between the four participants takes place as follows. In the initial phase, each participant writes a statement, initially hiding it from all other participants. The four statements are then revealed to all participants simultaneously

Initial phase:

•S1 wrote: “I deduce that 64 < xy < 196”

•S2 wrote: “It is impossible for P1 and P2 to deduce x and y from xy at this point”

•P1 wrote: “It is impossible for S1 and S2 to deduce x and y from x + y at this point”

•P2 wrote: “I deduce that x+y = 33”

Following these disclosures, a sequence of remarks are made by the participants in the following order:

Follow-up conversation:

•S1 says “It is impossible to determine which P is the knave from the above statements alone”

•P1 says “Now I know x and y”

•S2 says “Now I know x and y”

•P2 says “P1 and S2 are knaves”

Identify the two knaves and determine x and y, if you can!