You are in pitch darkness. Someone hands you a pack of cards. They tell you that 11 (as an example) of them are face up.
How can you separate the pack into two piles so there are equal numbers face up in each pile? Remember you can't see the cards.
You are in pitch darkness. Someone hands you a pack of cards. They tell you that 11 (as an example) of them are face up.
How can you separate the pack into two piles so there are equal numbers face up in each pile? Remember you can't see the cards.
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I think we've suffered long enough and I'm getting to the point of wanting to cheat and look it up.
I have got as far as working out one needs to turn over the topmost card. That results in an even number of face-up-cards, either 10 or 12. But I cannot see how to get them all into two piles with equal numbers face up.
I have considered turning over every other card in each pile, turning over one pile, having piles of equal numbers of cards and different sized piles.
I am sure you need two different size piles. I suspect one of the piles needs to be equal to the number of face-up cards or a multiple of it. But...
If we have no knowledge of where in the deck the face-up cards are, I cannot see how we can ensure there are an equal number of face-up in each pile.
I give up.
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Usually I cheat with these puzzles, my lazy.
But when I read this one originally, I wanted to use it in something I was writing (about problem solving you see), and so honour compelled me to work it out.
My thinking:
all the information we have is the number of cards face up in the original pack
we can see nothing (and the cards are not Braille ones),
so what can we do with our hands alone? Imagine the pack is in your hands, as you sit their in darkness.
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What actions can we carry out?
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Tear all the cards in two, leaving them the same way up as they currently are
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Suppose the pack is very small, just 3 cards and I tell you 1 is face up.
Deal off 1 card. Now we have a pile of 1 and a pile of 2.
If the first pile has 1 card face up, how many does the second have?
If the first pile has 0 cards face up, how many does the second have?
How could we equalise the number of face-up cards either way?
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Ok, take out the number of cards equal to the number that you are told are face up, turn them all over.
Cards that were not face-up add 1 to the face-ups in the new pile and leave the number of face-ups in the original pile unchanged.
Cards that were face-up leave new pile unchanged but subtract 1 face-up from the old pile.
So
Pick 11 face-down cards - New pile 11 face-ups, old pile still 11 face-ups
Pick 6 face-ups, 5 face-downs - New pile 5 face-ups and old pile 5 face-ups
Pick 5 face-ups, 6 face-downs - New pile 6 face-ups and old pile 6 face-ups
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Mary,
How did you work that out?
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Yes ah yes, that is the solution
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Simon, I followed Richard's very clear hint
Mind you, tearing the cards in half also works....