AbstractWe consider the problem of ascertaining the minimum number of weighings which suffice to determine the counterfeit (heavier) coins in a set of n coins of the same appearance, given a balance scale and the information that there are exactly two heavier coins present. An optimal procedure is constructed for infinitely many n's, and for all other n's a lower bound and an upper bound for the maximum number of steps of an optimal precedure are determined which differ by just one unit. Some results of Cairns are improved, and his conjecture at the end of [3] is proved in a slightly modified form

Tošić, Ratko

Elsevier - Publisher Connector 

Two counterfeit coins 

Discrete Mathematics 46 (1983) 295-298 
North-Holbd 
295 
Ratko T’O!iI~ 
Institute cf ibfuttitics, U&e&y of Nooi sad, 21001 Na.ri sad, P.0. Box 224, Yugoslavia 
ReceiveG My 1981 
We consider the problem of ascertaining the minimum number of weighings which sufike to 
determine the counterfeit (heavier) coins ia a set of n coins of the same appearance, given a 
balance scale and the information that there are exactly two heavier coins present. An optimal 
procedure is constructed for infinitely many n’s, and for all other n’s a lower bound and an 
upper bound for the maximum number of steps of an optimal procedure are determined which 
differ by just one unit. Some results of Cairns are improved, and his conjecture at the end of [33 
is proved in a slightly modified form. 
1. Induction 
Consider the following problem. Let X = {c,, c2,, . . . , c,,} be a set of n coins, 
indistinguishable xcept that exactly two of them are slightly heavier than the vest 
(in the sense specified below). Given a balance scale, we want to find an optimal 
weighing procedure, i.e., a procedure which minimizes the maximum number of 
steps (weighings) which are required to identify b<:bth heavier coins. 
We suppose that both heavier coins are of equal weight, and so are all light 
coins. If A is the weight of a light (good) coin, then the weight of a heavy 
(defective) coin is less than $A, so that the larger of two numerically unequal 
subsets of X is always the heavier. It means that no information is gained by 
balancing two numerically unequal sets. We also suppose that the scale reveals 
which, if either, of two subsets of X is heavier but not by how much. 
Consider a pair of numerically equal disjoint subsets (A, B) of X. Step (A, B) 
will mean the balancing of A against B. The following outcomes are possible: 
(a) The sets balance, symbolized by A = B. 
(b) The sets do not balance, symbolized by A# B. We use the notation, if 
necessary, A > B, B > A, where > between two sets means ‘is heavier than’. 
Let P:(1) denote any procedure which enables us to identify both heavier coins 
in the set X of n coins, I being the maximum number of weighings to be required. 
Similarly, PA(r) denotes any procedure which enables us to identify the heavier 
coin, if there is exactly one in the set of R coins, 1~ being the maximum number of 
weighings to be required. It is well known that Pi(r) is optimal if and only if 
3’-’ < n g 3’. 
We write p2(n) = 1 if P:(1) is optimal. It follows by information-theoretical 
0()12-365x/83/$3.00 (Q 1983, Elsevier Science Publishers B.V. (North-Holland) 
2% 7h counterfeit c&s 
reasonings that 
(1) 
where [xl denotes the least integer ~:x. 
In this paper we determine an infinite set of n’s for which this lower bound is 
reached, and the upper bound which differs by just one unit from the lower 
bound. The corresponding procedures are constructed inductively. 
Theorem 1. 
i 
Proof. Ir is P’ + to check that the following statements hold: 
rr>3& =$ 2k-1 :>3 , 
n>2*3” =, 
i.e.. 
22/c, 
~tb2-3~ + log, 
n I 01 2 32k+ 1. 
(2) 
(34 
(3b) 
(44 
Mb) 
From <4a), (4b) and ( I), we have 
n > 3k * p2(n) S2k, @a) 
nX*3k =$ p2(n)S2k+1. (5b) 
Now, (2) will be proved if we prove the folIowing two statements: 
ns2-3k 3 p2(n)S2k+1 (k =o, 8,2,. . .), (64 
?I S3k+’ + k(n)~2k+2 (k = 0, 1.2, . . .). W) 
The proof of (6a) and (6b) uses mathematical induction. The statements are 
obviousfy true for k = 0. In fact, we have p2(:Tb,) = 0 and ~~(3) = 1. Suppose now 
ahat k 2 1 and (6a) and (6b) are true for all k G m - 1 and that the corresponding 
procedures are constructed. Then the procedures Pi(2m -t 1) for 3” c n ~2 9 3” 
and Pz(2m+2) for 2-3”<n<3”” can be constructed according to the follow- 
ing schemes: 
R Tolriir 297 
(a) Constructjon of Pi(2m + 1) fbr n s2 l 3”. The first step is (A, B), where 
A={cIrcz,..., ck~2J) and B =&/~J+I, . . .s c2[&J}. (Here, -[xl denotes the 
greatest integer GX.) If A = B, then each of the sets A and I3 contains exactly one 
heavier coin. Continue now by successively applying the two independent proce- 
dures &l(m) and I$&~). These procedures exist because I&rj = IAl = II31 ~3”. 
If A # B we may. suppose A >B by symmetry. A X3 implies that both heavier 
coins are in X\B, where IX\BI= [in’1 <[2.3”/21= 3’Y CIsntinue now the 
procedure by applying P&Bj(2rn). It can be constructed by the induction 
hypothesis. 
A procedure Pz(2m + 1) for n ~2 = 3” is constructed. 
(b) Construction of Fz(2m +2) for n s3m+*. Let us define the sets A, B, C, fi 
and c in the following way: 
Put a=B, c=C if n#3”“‘-1 and ~~=BU(C~~+L_~}, ~=CU{C~~+L_~} if 
n= 3 m+l -1. 
Step (bl). (A, B). 
Step (b2). @, c). 
Several cases are distinguished according to the results off the above steps. 
If A <B and fi = c, then each of the sets 6 and e contains exactly one heavier 
coin. Continue now by successive application of the two independent procedures 
Phi(m) and &l(m) to the sets fi and 6, respectively. These procedures exist 
since IBI = ICI S 3”. 
If A < B and B > c, then both heavier coins are in X\(A U c), where 
IX\@ U C)I s 3”. Continue now by applying P&,(Au&2m) which can be con- 
structed by the induction hypothesis. 
The case A c B, B < c is impossible. 
If B < A and fi = c, then both heavier coins are in X - (fi U t?) where IX - 
(B U e)l s 3m. Continue now by applying P&4,&2m) which exists by the 
induction hypothesis. 
If B <A and B < c, then each of the sets A and c contains exactly one heavier 
coin. As IAl s ICI 6 3” we may apply the procedure:, P/&(m) and P$$m). 
If B <A and B > c, then fi and B must be different and C3m+l_2 is one of the 
heavier coins. The other one is in A. Here IAl G S’“, therefore P$,,!rn) compl@.es 
our procedure. 
If A = B and 15 c c, then both heavier coins are in X\ (A U 8), where 
IX\(A U&I G 3”. Continue now by applying P&\(AUGJ1(2m) which can be con- 
structed by the induction hypothesis. 
If A = B and 6 = c, this is possible only if n - 3 [in] = 2. If yt C 3m+1 - 1, then 
the heavier coins are c~~,,,~J +1 and c3 [n/3] +2; if PI= 3 W’ - 1, then one heavier coin 
is in e, the other is ~3m+l-2~ apply P&(m). 
If A = B and s > c, then each of the sets A and fi contains exactly one heavier 
coin. Continue now by successive application of the two independent procedures 
P,‘*,(m) and @,(m). These procedures exist because IAl s@( ~3”. 
A procedure Pf(2m + 2) for n S3”+l is constructed. The theorem is proved. 
Theorem 2. There a~ infinitely many n’s for which the procedure described in the 
proof of Theorem 1 is optimal. More precisely, the pnxedure is optimal at least for 
all n’s beionging to the set 
U ([[3”&+ 1‘1,2= 3k]U[[3k&+ 11,3k+‘]), 
k-l 
where [p, y] denotes the set of ait integers n such that p s n Gq. 
Pro&. It is easy to cheek the following inequalities: 
Va,b) 
it follows from (7a) and (7b) that 
n 0 2 > Fk forall n~[[3~*+1].2=3~] 
and 
n 0 >332k+l 2 for all n E[r3%+ 11. 3k+“], 
which together with (2) implies the statement.. 
E.g. our procedure is optimal for all n’s from the intervals [9,9], [ 14, 181, 
[X,27]. [40,54], [68,81]. [ 116.1623 etc. 
Our results are stronger than those of Cairns [3, Corollary of Lemma 8.3 and 
‘Tsaeorem M(n, 2)]. Also, his conjecture that F&I) has one of the values 2k - 1, 
2k. 2k + 1, dependiq on n, for 3k-’ < n ~3~, is shown to be slightly incorrect. It 
follows from Theorem 1 and 2 that Fl(n) has one of the values 2k - 2,2k - 1,2k, 
I \r 3’ ’ < n ~3~, and the corresponding procedures are constructed. 
References 
f 1 i H. Hc1lm-w.. Dynamic Programming (Princeton Univ. Press, Princeton, 1957). 
[Z] K. klellman and B. Gluss. On various versions of the defective coin problem, Information and 
(‘onlrol -I (19611 118-131. 
131 S.S. Cairns, Bakance scale sorting, Amer. Math, Monthly 70 (1963) 136-148. 
[G‘l C.A.B. Smith, Th e counterfeit coin problem, Math. Gazette 31 (1947) 31-39. 
!yl 6 O.H. Katona. Combinatorial search problems, in: J. N. Srivastava, ed., A Survey of Com- 
bll tutorial Theory (North-Holland. Amsterdam, 19?3] 285-308. 


file:///data/remote/core/dit/data/elsevier/pdf/d4a/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wMDEyMzY1eDgzOTAxMjMx.pdf

Two counterfeit coins

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Elsevier - Publisher Connector