Daniel Fortunov

Five Of Diamonds Assumptions:

"number" refers to one of {2, 3, 4, 5, 6, 7, 8, J, Q, K, A} (ie. 'Picture cards' have a "number")
"suit" refers to one of {H, S, C, D}
Let U be the set of all (number,suit) card tuples.
U = {
(A,H), (Q,H), (4,H),
(J,S),
(8,C), (4,C), (2,C), (7,C), (3,C)
(A,D), (5,D)
}
Mr P knows the number.
Mr Q knows the suit.

Phase 1:

P cannot ascertain what card it is.
Therefore it must be a card which is not uniquely identifiable by its number (which P already knows).

Let A be the set of cards in U which share a number.
A = {
(A,H),(Q,H),(4,H)
(A,D),(5,D),
}

Phase 2:

Q knew that P wouldn't be able to uniquely identify the card, based on knowing it's suit.
Therefore the suit of the card must be one for which all numbers of that suit are duplicated (if a card is not one whose number is duplicated then P would have known what card it was, given that he already knows the number)
So, the suit of the card must be a suit which appears in set A. Namely, hearts or diamonds.

Phase 3:

Given Q's statement, P makes the deduction in Phase 2.
P knows the number of the card, and knows that it is in set A, and from this he concludes the solution.
Therefore the number of the card must be one which uinquely identifies a card from set A, namely, the number must be one of {Q, 4, 5}.

Let C be the set of cards in A for which this holds true.
C = {
(Q,H),(4,H)
(5,D),
}

Phase 4:

Given P's statement, Q makes the deduction in Phase 3.
Q knows the suit of the card, and now knows that it is in set C, and this allows him to discover the solution.
So the solution is a card in set C which is uniquely identifiable by its number, namely (5,D).

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