Blind man's cards (solution)

The probability of success is 100%. Hard to believe? Here's how.

First, split the 9 cards into two stacks, one with 4 cards and the other with 5 cards. Let 1 and 2 represent the first and second stack respectively. "b" represents the face down cards, "w" represents the face up cards, and "T" is the total number of cards in that stack.

Since we don't know how many face down cards are in each stack, let us denote them with x and y. Therefore, the number of face up cards in each stack is 4-x and 5-y respectively.

  1    2
b x    y
w 4-x  5-y
T 4    5

Now, x+y=4, therefore:

  1    2
b x    4-x
w 4-x  5-(4-x)
T 4    5

Flip all the cards in stack 1:

  1    2
b 4-x  4-x
w x    5-(4-x)
T 4    5

And there you are! The number of face down cards is equal in both stacks. Since there are 4 face down cards in total, each stack must have 2 face down cards.

Bonus: In fact, this procedure can be generalized to any arbitrary number of cards on the table with an even number of face down cards. Try it out!

P.S.: Saw this question on the net and I'll admit I didn't manage to solve it. The original solution was purely the procedure, so I decided to work out the "explanation" above. I learnt: Sometimes, once you spell out your problems properly, they may turn out to be easily solvable, after all!