A Logical Puzzle, Part 3

Note: A username and password may be required to access certain documents. Please contact Bill Rapaport.

Only one of the following is true about a particular hand of cards:

1. There is a King in the hand or there is an Ace in the hand, or both.

2. There is a Queen in the hand or there is an Ace in the hand, or both.

3. There is a Jack in the hand or there is a 10 in the hand, or both.

Question: Is it possible that there is an Ace in the hand?

The correct answer is…

There is no Ace in the hand !!

If (a) is the case, then (b) cannot be the case
(because only one of (a) and (b) is true!).

∴ ¬Queen ¬Ace are both true.

∴ There is no Ace.

(I.e., if (a) is the case, then there is only a King in the hand!)

Similar reasoning works in the other cases:

• If (a), then ¬(b); ∴ there is no Ace.
  (as we just saw)

• If (b), then ¬(a); ∴ there is no Ace.
  

• If (c) & there is no Ace, then there is no Ace.

So, is it possible for (c) to be true and for there to be an Ace?

• If there is a Jack and an Ace, then (a);
  ∴ ¬(b)
  ∴ ¬Ace

• If there is a 10 and an Ace, then (a), hence no Ace!

• If there is a 10, a Jack and an Ace, then (a), hence no Ace!

For more information, link to:

Philip N. Johnson-Laird's Theory of Mental Models (and related issues)