I really like these kinds of puzzles. I'm not sure if they have a name, but I like to think of them as "invariant puzzles". The key to understanding them is fin

I really like these kinds of puzzles. I'm not sure if they have a name, but I like to think of them as "invariant puzzles". The key to understanding them is finding an invariant - something that doesn't change when you manipulate the objects in the puzzle. You can tell it's an invariant puzzle because they seem to be missing one vital piece of information. This is because one of the factors (in this case, whether a card is face up or face down) has been canceled out due to the invariant. So what's the invariant? Well, let's start with what we're trying to find. We want the number of face up cards in the new deck (let's call it Deck 2) to equal the number of face up cards in the old deck (Deck 1), so Up1 = Up2. We know we can flip cards, so that's equivalent to Up1 = Down2. But we're still missing a bit of information. How can we add more face down cards to the second deck when we don't know which way they're facing? Here comes the invariant... it doesn't matter! We can make Up1 and Down2 closer to equal by increasing Down2 (by moving a Down card) or decreasing Up1 (by moving an Up card). Essentially, the value of (Down2 - Up1) decreases by 1 when we move a card no matter what. So we want to keep moving cards until (Down2 - Up1) = 0. (Down2 - Up1) starts at 10 in this case, so we do it 10 times. We flip Deck 2; Up becomes Down, Down becomes Up, and we're left with Up1 = Up2. QED! Another cool invariant puzzle is the wine mixing problem (also known as the water/wine problem, the red/white marble problem): http://www.smart-kit.com/s330/red-wine-and-white-wine-brain-teaser/ http://www.smart-kit.com/s330/red-wine-and-white-wine-brain-... (Oh, and incidentally there are some problems that seem to be missing an important piece of information, but aren't invariant puzzles, like xkcd's Blue Eyes (http://xkcd.com/blue_eyes.html http://xkcd.com/blue_eyes.html). Puzzles that include language like "everyone behaves rationally" are almost always recursion problems.)