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deck o' cards

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I just edited this one to add:

YAY! ANOTHER PUZZLE I CREATED...

I did this one with a normal 52 card deck of cards.

I got three different answers, but the final one I triple checked so i'm fairly confident....

Draw an X on the back of one of your cards, it'll be easier to do this.

put the card with the X on it on the top of the deck.

On your first turn, move the top card to the bottom of the deck.

On you second turn, move the top two cards to the bottom of the deck

On your third turn, move the top three cards to the bottom of the deck and etc. etc.

After how many turns will the card with the X on it be back on top of the deck?

Be sure to put the cards to the bottom of the deck in the same order.

So when you're done, the cards will be in the same order that you started in...

Now I can cheat when I play cards!

The 7 of hearts has an X on the back...

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Envision an infinite stack of identical decks of cards.

That way we don't have to move cards to the bottom; we just remove cards from the top and discard them.

When we've discarded a multiple of 52 cards - some number of complete decks - the next card will then show an X.

On each turn, we discard a number of cards, beginning with 1, that increases by 1 card each turn.

So the number of discards after turn N is SUM [1, 2, 3, 4 , ..., N] = the Nth triangular number.

The first 39 triangular numbers are

1 3 6 10 15 21 28 36 45 55

66 78 91 105 120 136 153 171 190 210

231 253 276 300 325 351 378 406 435 465

496 528 561 595 630 666 703 741 780

None of these numbers are multiples of 52 until we get to 780; 780 = 15 * 52.

So after 39 turns, we'll have discarded 15 complete decks.

We'll then be staring at card number 781, the top card of the 16th deck, which will have an X.

So the answer seems to be ... 39 turns of the cards.

Your fingers must be sore if you verified this three times ... <!-- s:? --><!-- s:? -->

Nice one!

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