A080342 Number of weighings required to identify a single bad coin out of n coins, using a two-pan balance.

0, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Author

Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), Mar 19 2003

Keywords

• nonn
• easy
• changed

Comments

It is known that there is exactly one bad coin, which is heavier than the others. No weights are used in the weighings.

0 appears once, 1 twice, 2 6 times, 3 18 times, 4 54 times, ... which is the same as the number of base-3 numbers of length n; see A007089. - Jonathan Vos Post, Apr 20 2011

Records appear at positions 3^n+1 (=A034472(n)). - Robert G. Wilson v, Aug 06 2012

The "Heavy Marble" section of "Brainteaser Problems" in the Mongan et al. reference describes the n = 8 case in detail and then derives the general formula given below. Of course this sequence applies also when the single, unlike object is lighter than all the others. If the unlike object is only known to have a different weight (that is, to be lighter than all the others or heavier than all the others), use A064099. - Rick L. Shepherd, Sep 05 2013

If it is unknown whether the bad coin is heavier or lighter, then the minimum number of weighings is A029837(n) and the number of coins that must be used in the first weighing is A004526(n), for n > 2. - Ivan N. Ianakiev, Apr 13 2017

Examples

			a(1) = 0 since no weighings are needed - the coin is bad. a(2) = 1 since one weighing is needed.
		

References

• J. Mongan, N. Suojanen, and E. Giguère, Programming Interviews Exposed: Secrets to Landing Your Next Job, 2nd Edition, Wiley Publishing, Inc., 2007, pp. 169-172.

Links

• Rick L. Shepherd, Table of n, a(n) for n = 1..10000
• R. K. Guy, R. J. Nowakowski, Coin-weighing problems, Am. Math. Monthly 102 (2) (1995) 164-167.
• B. Manvel, Counterfeit coin problems, Math. Mag. 50 (2) (1977) 90-92, theorem 1.

Crossrefs

Cf. A000244, A007089, A025192, A064235.

Cf. also A004526, A029837, A064099.

Cf. A081604.

Programs

Haskell

import Data.List (transpose)
a080342 n = genericIndex a080342_list (n - 1)
a080342_list = 0 : zs where
   zs = 1 : 1 : (map (+ 1) $ concat $ transpose [zs, zs, zs])
-- Reinhard Zumkeller, Sep 02 2015

Mathematica

f[n_] := Floor[ Log[3, n]] - Floor[2^-FractionalPart[ Log[3, n]]] + 1; Array[f, 105] (* Robert G. Wilson v, Aug 05 2012 *)

PARI

a(n) = ceil(log(n)/log(3)) \\ Rick L. Shepherd, Sep 05 2013

Formula

a(n) = floor(L) - floor(2^(-f(L))) + 1, where L = log_3(n) and f() = fractional part.

a(n) = ceiling(log_3(n)). - Rick L. Shepherd, Sep 05 2013

A064235(n) = 3 ^ a(n). - Reinhard Zumkeller, Sep 02 2015