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 2000 cards (Posted on 2009-01-04)
There are cards labeled from 1 to 2000. The cards are arranged and placed in a pile.

The top card is placed on the table, then the next card at the bottom of the pile.

Then the next card is placed on the table to the right of the first card, and the next card is placed at the bottom of the pile.

This process is continued until all the cards are on the table.

The final order (from left to right) is 1, 2, 3, ... , 2000.

In the original pile, how many cards were above card labeled 1999?

 See The Solution Submitted by pcbouhid No Rating

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 re: non-computer solution | Comment 8 of 9 |
(In reply to non-computer solution by Paul)

Here are some samples of the original positions of the last two cards on the table, for random deck sizes between 1000 and 2000.

For example, the first set indicates that for a deck size of 1473, the last two on the table (the 1472nd and 1473rd) came from positions 1410 and 898 in the original deck. The binary equivalents are to the right.  There's a pattern, but not exactly the same at the left side in each instance.

`1473 101110000011410 10110000010 898 01110000010`
`1089 10001000001 642 01010000010 130 00010000010`
`1332 101001101001128 10001101000 616 01001101000`
`1765 11011100101 458 001110010101482 10111001010`
`1671 11010000111 270 001000011101294 10100001110`
`1274 100111110101012 01111110100 500 00111110100`
`1687 11010010111 302 001001011101326 10100101110`
`1224 10011001000 912 01110010000 400 00110010000`
`1970 11110110010 868 011011001001892 11101100100`
`1600 11001000000 128 000100000001152 10010000000`
`1291 101000010111046 10000010110 534 01000010110`
`1908 11101110100 744 010111010001768 11011101000`
`1002 01111101010 468 00111010100 980 01111010100`
`1737 11011001001 402 001100100101426 10110010010`
`1685 11010010101 298 001001010101322 10100101010`
`1815 11100010111 558 010001011101582 11000101110`
`1191 10010100111 846 01101001110 334 00101001110`
`1699 11010100011 326 001010001101350 10101000110`
`1459 101101100111382 10101100110 870 01101100110`
`1768 11011101000 464 001110100001488 10111010000`
`1758 11011011110 444 001101111001468 10110111100`
`1970 11110110010 868 011011001001892 11101100100`
`1978 11110111010 884 011011101001908 11101110100`
`1458 101101100101380 10101100100 868 01101100100`

 Posted by Charlie on 2009-01-05 19:27:42

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