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?

(In reply to re: computer solution (spoiler) by elementofsurprize)

I see what happened in your calculations.  When you cycle to the top of the deck, your calculations always take the first card available in the whole deck; but that fails to skip a card in some instances. I've recalculated the numbers in a way that matches yours, and note that the first divergence is after placing the originally 2000th card onto the 1938th position on the table. Your calculation places the originally 32nd card next.  But that's the very next card; you haven't skipped one.  There's no demarcation given at the top of the deck to indicate you should start over, as the old cards were just placed at the bottom, so in the actual physical embodyment, there's no warning to start fresh and disregard the fact that you're not skipping over a card.

After placing card 2000 into position 1938, card 32 should be skipped over and card 64 placed into position 1939. 

. . .
1847: 1544 1560 1576 1592 1608 1624 1640 1656 1672 1688 1704 1720 1736 1752 1768 1784 
1863: 1800 1816 1832 1848 1864 1880 1896 1912 1928 1944 1960 1976 1992 
1876:   16   48   80  112  144  176  208  240  272  304  336  368  400  432  464  496 
1892:  528  560  592  624  656  688  720  752  784  816  848  880  912  944  976 1008 
1908: 1040 1072 1104 1136 1168 1200 1232 1264 1296 1328 1360 1392 1424 1456 1488 1520 
1924: 1552 1584 1616 1648 1680 1712 1744 1776 1808 1840 1872 1904 1936 1968 2000 
1939:   32   96  160  224  288  352  416  480  544  608  672  736  800  864  928  992 
1955: 1056 1120 1184 1248 1312 1376 1440 1504 1568 1632 1696 1760 1824 1888 1952 
1970:   64  192  320  448  576  704  832  960 1088 1216 1344 1472 1600 1728 1856 1984 
1986:  128  384  640  896 1152 1408 1664 1920 
1994:  256  768 1280 1792 
1998:  512 1536 
2000: 1024 

Posted by Charlie on 2009-01-05 10:05:50