Coins of Three Different Weights

We discuss several coin-weighing problems in which coins are known to be of three different weights and only a balance scale can be used. We start with the task of sorting coins when the pans of the scale can fit only one coin. We prove that the optimal number of weighings for $n$ coins is $\lceil 3n/2\rceil -2$. When the pans have an unlimited capacity, we can sort the coins in $n+1$ weighings. We also discuss variations of this problem, when there is exactly one coin of the middle weight.Comment: 18 page

Quantum counterfeit coin problems

AbstractThe counterfeit coin problem requires us to find all false coins from a given bunch of coins using a balance scale. We assume that the balance scale gives us only “balanced” or “tilted” information and that we know the number k of false coins in advance. The balance scale can be modeled by a certain type of oracle and its query complexity is a measure for the cost of weighing algorithms (the number of weighings). In this paper, we study the quantum query complexity for this problem. Let Q(k,N) be the quantum query complexity of finding all k false coins from the N given coins. We show that for any k and N such that k<N/2, Q(k,N)=O(k1/4), contrasting with the classical query complexity, Ω(klog(N/k)), that depends on N. So our quantum algorithm achieves a quartic speed-up for this problem. We do not have a matching lower bound, but we show some evidence that the upper bound is tight: any algorithm, including our algorithm, that satisfies certain properties needs Ω(k1/4) queries

Baron Munchhausen Redeems Himself: Bounds for a Coin-Weighing Puzzle

We investigate a coin-weighing puzzle that appeared in the Moscow Math Olympiad in 1991. We generalize the puzzle by varying the number of participating coins, and deduce an upper bound on the number of weighings needed to solve the puzzle that is noticeably better than the trivial upper bound. In particular, we show that logarithmically-many weighings on a balance suffice.Comment: 19 page

Optimal detection of two counterfeit coins with two-arms balance

AbstractWe consider the following coin-weighing problem: suppose among the given n coins there are two counterfeit coins, which are either heavier or lighter than other n−2 good coins, this is not known beforehand. The weighing device is a two-arms balance. Let NA(k) be the number of coins from which k weighings suffice to identify the two counterfeit coins by algorithm A and U(k)=max{n|n(n−1)⩽3k} be the information-theoretic upper bound of the number of coins then NA(k)⩽U(k). We establish a new method of reducing the above original problem to another identity problem of more simple configurations. It is proved that the information-theoretic upper bound U(k) are always achievable for all even integer k⩾1. For odd integer k⩾1, our general results can be used to approximate arbitrarily the information-theoretic upper bound. The ideas and techniques of this paper can be easily employed to settle other models of two counterfeit coins

Two counterfeit coins

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

Inflation (mis)perceptions in the euro area

There has been a growing interest on inflation perceptions in the euro area, in particular, following the euro cash changeover. It has been pointed out that a gap emerged between observed and perceived inflation since the introduction of the euro notes and coins. Such a statement relies on the fact that inflation perceptions, measured by the well-known balance statistic from the European Commission's consumer survey, hiked after January 2002 and remained high thereafter, as opposed to the observed inflation, which has remained fairly stable. In this paper, we discuss the issue of inflation perceptions measurement, by comparing the balance statistic with an alternative refined measure, which is computed using the probability method. We argue that the balance statistic should be used carefully, as it can induce to misleading conclusions. In fact, we provide, for both euro area and country level, evidence showing that, using the proposed alternative measure, the breakdown in the relationship between observed and perceived inflation did not occur at the time of the euro cash changeover.