

A264386


Gergonne's 27card trick with three piles: finding a card after three dealings with pile information.


1



1, 10, 19, 4, 13, 22, 7, 16, 25, 2, 11, 20, 5, 14, 23, 8, 17, 26, 3, 12, 21, 6, 15, 24, 9, 18, 27
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OFFSET

0,2


COMMENTS

See the links for J. D. Gergonne's 27card trick with three piles each of 9 cards. Putting the told pile (the one with the card to be found) at the top (t), the middle (m) or the bottom (b) at each of the three dealings with three piles allows 3^3 = 27 possibilities. They are ordered lexicographically using t = 0, m = 1 and b = 2. The a(n)th card from the top of the 27card pile at the end is the card to be found for these 27 possible shufflings. E.g., a(2) gives the number for the shuffling (2)_3 = 002 (in the threeposition base3 version): the told 9pile is first put on top, then again on top and finally at the bottom, denoted by ttb. Then the searched card is the 19th from the top of the 27card pile.
In the Gardner reference the numbers to be added to obtain a(n) are for t, m, b for the first dealing 1, 2, 3, for the second one 0, 3, 6 and the third one 0, 9, 18, respectively. Hence for a(2) corresponding to ttb one finds 1 + 0 + 18 = 19.
This sequence (with offset 1) is the following element of the symmetric group S_27 (in cycle notation of type 1^9 2^8): (1) (4) (7) (11) (14) (17) (21) (24) (27) (2,10) (3,19) (5,13) (6,22) (8,16) (9,25) (12,20) (15,23) (18,26).
a(0)..a(17) coincides with A030102(9)..A030102(26).


REFERENCES

M. Gardner, Mathematische Zaubereien, Dumont, 2004, pp. 5052. Original: Mathematics, Magic and Mystery, Dover, 1956.


LINKS

Table of n, a(n) for n=0..26.
Ethan D. Bolker, Gergonne's Card Trick, Positional Notation and Radix Sort, Mathematics Magazine Vol. 83, No. 1 (February 2010), pp. 4649.
MacTutor History of Mathematics archive, Joseph Diaz Gergonne .
Wikipedia, Joseph Diaz Gergonne.


FORMULA

a(n) = (reversed((n)_3))_10 + 1, n = 0 .. 26, where (n)_3 is the three position version of n in base 3. E.g., (4)_3 = 011, reversed 110, as decimal 9+3+0 = 12, adding 1 gives a(4) = 13.
a(n) = n_1 + n_2 + n_3 with n_1 = 1, 2, 3, n_2 = 0, 3, 6 and n_3 = 0, 9, 18, for t, m, b, respectively, at the ith dealing, i = 1, 2, 3.
E.g., tmm (or 011): a(4) = 1 + 3 + 9 = 13. (Gardner, p. 51.)


EXAMPLE

The 27 possible positions for the told pile of 9 cards after the three dealings are ordered like
ttt, ttm, ttb, tmt, tmm, tmb, tbt, tbm, tbb,
mtt, mtm, mtb, mmt, mmm, mmb, mbt, mbm, mbb,
btt, btm, btb, bmt, bmm, bmb, bbt, bbm, bbb.
They correspond to the threeposition version of n in base 3, for n=0..26.
The Gardner counting for mmb (n=14) is 2 + 3 + 18 = 23 = a(14). The formula uses (14)_3 = 112, reversed 211, written as decimal 2*9 + 1*3 + 1*1 = 18 + 3 + 1 = 22, adding 1 gives a(14) = 23.


CROSSREFS

Cf. A030102.
Sequence in context: A171767 A153689 A240022 * A173822 A122607 A113703
Adjacent sequences: A264383 A264384 A264385 * A264387 A264388 A264389


KEYWORD

nonn,base,fini,full


AUTHOR

Wolfdieter Lang, Dec 22 2015


STATUS

approved



