In preparation for a card trick, a certain number of cards are taken from a standard deck of cards.

The trick goes as follows: Cards are taken from the top of the deck and transferred to the bottom, each time calling out a letter in the name of a card, starting with A, then C, then E, spelling out ACE; the next, fourth, card transferred is then shown to be the ace of hearts. This continues, saying "T" for the fifth card, "W" for the sixth card and "O" for the seventh card, and showing the eighth card to be the two of hearts.

Since letters are transferred to the bottom, this continues as many times through this modified deck as needed to have shown all the hearts through the king, and then all the spades, from ace through king.

How many cards from the full deck were used to make this modified deck so as to let the trick work?

Edited for format errors

 

Number of cards = 47

 

I started with a brute force method in excel by aligning columns of "a" "c" "e" "A" "t","w","o" 2 etc.  I used 52 cards as the maximum and 26 cards as the minimum.   I quickly found 47 as an answer and then 42.  I realized that on the third round through the deck that a 42 card deck wouldn't work, due to the very last card having to be the Ace of Hearts and the King of Spades. But I didn't really like the brute force method.

 

I then made a matrix of 26 by 26 where the values were the placement of the Hearts and Spades based on spelling each word and then positioning the card. Values (4, 8, 14, 19, 24, 28, 34, 40, 45, 49, 54, 60, 65, 69, 73, 79, 84, 89, 93, 99, 105, 110, 114, 119, 125, 130) were used and the matrix was filled in with the difference between values.  All values between 26 and 52 were found in the matrix with the exception of 27, 38, 42, 43, and 47.

 

New matrix, 26 x 26, but I made one set of values twice the original set to represent passing through the deck a second time.  The values 27, 38, 42 and 43 were found in the second matrix.

 

This left 47.

 

Cards are:

1          a          n          e

2          c           10-h      v

3          e          j           e

4          A-h       a          n

5          t           c           7-s

6          w          k          e

7          o          J-h        i

8          2-h       q          g

9          t           u          h

10         h          e          t

11         r           e          8-s

12         e          n          n

13         e          Q-h       i

14         3-h       k          n

15         f           I           e

16         o          n          9-s

17         u          g          t

18         r           K-h       e

19         4-h       a          n

20         f           c           10-s

21         I           e          j

22         v          A-s       a

23         e          t           c

24         5-h       w          k

25         s           o          J-s

26         i           2-s        q

27         x          t           u

28         6-h       h          e

29         s           r           e

30         e          e          n

31         v          e          Q-s

32         e          3-s        k

33         n          f           i

34         7-h       o          n

35         e          u          g

36         I           r           K-s

37         g          4-s

38         h          f

39         t           i

40         8-h       v

41         n          e

42         I           5-s

43         n          s

44         e          i

45         9-h       x

46         t           6-s

47         e          s

Edited on September 29, 2006, 9:00 am

Posted by Leming on 2006-09-29 08:55:31