I'm a hobbyist working on a mechanical sorting machine to sort magic the gathering cards. I'm by no means a mathematician though, and I was wondering if you all wouldn't mind helping me out with a practical math problem to determine the best route to go with my machine.
The average Magic The Gathering set of cards contains 250 unique cards. My machine will sort those cards alphabetically placing the cards into multiple piles repeatedly to filter the cards down into the correct position. My question is, how many piles would be needed to sort 250 cards with only three passes. Here is some further info more clarification of the question.
Example, a card passes through the scanner, and the computer determines that the cards position in the alphabetized set is within the first 50 positions of the alphabetized set. 50 positions is %20 percent of a 250 card set, which I've chosen that number. It just seems like an easy place to start.
Then the next card passes through, and the scanner determines that the cards position is in the 3rd 50 positions of the alphabetized set (positions 101-150) so it places this in pile three. It keeps doing this until it has 250 cards broken down into 5 piles.
The cards are then scanned again for further refinement until they are in the correct order. So the question is, how many piles would make it possible to on average sort a 250 card set in 3 passes. I only chose 5 piles to illustrate the question, so I hope that's not confusing.
Any help on this would be highly appreciated.
Kind Regards Spencer