|
%I #20 Oct 29 2017 13:31:00
%S 2,3,9,11,24,35,28,31,80,60,121,119,116,195,75,79,204,323,228,199,146,
%T 264,529,504,200,675,540,251,840,899,186,191,1088,748,1225,324,740,
%U 1140,1521,1079,1680,336,1204,484,540,460,1692,1151,734,2499
%N Penny Flipping, or Flipping Coins: Given a stack of n coins, flip the top coin, then the stack of the top two coins, then the stack of the top three etc... starting again with the top coin after flipping all n coins. A flip of m coins reverses their order and inverts their state. This is the number of flips required to restore the stack to its original configuration.
%C Here "original configuration" seems to mean each coin in original orientation and either original or reverse order; for the original order and either original or reverse orientation n*A002326(n) flips required, while for both original order and original orientation n*A003558(n) required. - _Henry Bottomley_, Jan 19 2007
%C Birtwistle (1973) attributes the problem to Iain Bride and John Gilder of UMIST (University of Manchester Institute of Science and Technology).
%D G. M. Birtwistle, Simula Begin, Auerbach Publishers, Philadelphia, 1973 [Uses this problem to illustrate the power of the Simula language]
%D Popular Computing (Calabasas, CA), Penny Flipping, Vol. 3 (No. 23, Feb 1973), pages PC23-10 to PC23-13) and Vol. 3 (No. 29, Aug 1975), pages PC29-6 to PC29-8. Gives first 32 terms.
%H B. B. Newman, <a href="http://www.jstor.org/stable/2690434">The Flippin' Coins Problem</a>, Mathematics Magazine, Vol. 54 (1981), pp. 51-59.
%F For n>1, if A002326(n)=A003558(n) then a(n)=n*A002326(n), otherwise a(n)=n*A002326(n)-1. - _Henry Bottomley_, Jan 19 2007
%e For 3 coins (starting with HHH) the flips move the stack through the sequence: HHH -1-> THH -2-> THH -3-> TTH -1-> HTH -2-> HTH -3-> THT -1-> HHT -2-> TTT -3-> HHH. (-n-> indicates n coins are flipped)
%t b[n_] := MultiplicativeOrder[2, 2n+1];
%t c[n_] := MultiplicativeOrder[2, 2n+1, {-1, 1}];
%t a[1] = 2; a[n_] := If[b[n] == c[n], n*b[n], n*b[n]/2 - 1];
%t Array[a, 50] (* _Jean-François Alcover_, Oct 29 2017, after _Henry Bottomley_ *)
%Y Cf. A002326, A003558, A056951.
%K easy,nonn,nice
%O 1,1
%A Richard Forster (gbrl01(AT)yahoo.co.uk), Jan 02 2004
|
