mystery calculator

[Jeremy Gardiner]

This set of cards called the "mystery calculator" (made in china) came out
of a Christmas cracker:

Card 0
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39,
41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63

Card 1
2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 38, 39,
42, 43, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63

Card 2
4, 5, 6, 7, 12, 13, 14, 15, 20, 21, 22, 23, 28, 29, 30, 31, 36, 37, 38, 39,
44, 45, 46, 47, 52, 53, 54, 55, 60, 61, 62, 63

Card 3
8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 40, 41, 42,
43, 44, 45, 46, 47, 56, 57, 58, 59, 60, 61, 62, 63

Card 4
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 48, 49, 50,
51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63

Card 5
32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63

The trick: You secretly select a number from one of the cards. You indicate
to me the cards on which that number appears; I tell you the number you
selected!

Example: You indicate cards 1, 3 and 5. Your selected number is 42!

Solution below.

The "mystery calculator" appears on a web site:
http://www.mathsyear2000.org/explorer/mathsmagic/realmystery/

There is also a computer program version of the "mystery calculator"
although I haven't tried it: http://www.tucows.com/preview/297643

The first three sequences are A005408, A042964, and A047566. The other three
do not appear to be in the OEIS; I have submitted them as A115419, A115420,
and A115421.

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The solution: I add together the first term from each of the indicated
cards. The total equals the selected number. The numbers in each sequence
all have a "1" in the same position in their binary expansion.

Example: You indicate cards 1, 3 and 5. Your selected number is 2+8+32 = 42.