Kvant Math Problem 466

Consider first a smaller version of the problem.

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Problem

Among 1977 coins, 50 are counterfeit. Each counterfeit coin differs from a genuine one by 1 g (either heavier or lighter). There is a pan balance with a pointer indicating the difference in mass between the two pans. With a single weighing, one must determine whether a selected coin is counterfeit or genuine. How can this be done?

S. V. Fomin

Exploration

Consider first a smaller version of the problem. Suppose we have $7$ coins among which $2$ are counterfeit, each differing from the genuine weight by $1$ gram, and we are allowed a single weighing on a pan balance with a pointer indicating the difference. The challenge is to design a weighing such that the difference unambiguously identifies whether a specific coin is genuine or counterfeit. If we simply place the selected coin on one pan and fill the other pan with an arbitrary subset of the remaining coins, the difference could vary depending on which coins are counterfeit. Clearly, the weighing must exploit the known total number of counterfeits to calibrate the difference so that it uniquely reveals the status of the chosen coin.

One approach is to assign each coin a weight in the pan proportional to its index or position in a coding sequence and arrange the remaining coins on both pans so that the possible outcomes map bijectively to the set of subsets of counterfeits. This resembles a binary-weight coding technique often used in information theory. For small cases, listing all configurations quickly shows that naive equal distribution fails; a systematic weighted arrangement is required to encode the necessary information in a single weighing.

The crucial insight is that the total number of counterfeits is small compared to the total number of coins, and each differs by exactly $1$ gram. Therefore, the difference in mass produced by a weighing depends linearly on which coins are counterfeit, and if the weighing is cleverly designed, this linear combination can be made injective with respect to the status of the selected coin.

Problem Understanding

We are asked to identify a single coin as genuine or counterfeit among $1977$ coins, knowing that exactly $50$ are counterfeit and that a single pan balance weighing indicates the total difference in grams between the pans. This is a Type D problem: we must construct a weighing scheme and verify that it works. The core difficulty lies in encoding the identity of the selected coin into the difference of the pan weights in such a way that no other arrangement of the 50 counterfeit coins could produce the same measurement.

Intuitively, the solution should exploit a combinatorial or binary-coding scheme: assign the coins to positions with weights corresponding to powers of $3$ (or another sufficiently large base), place the chosen coin on one pan and distribute the remaining coins according to a carefully designed pattern so that the total mass difference uniquely identifies whether the chosen coin is genuine or counterfeit.

Proof Architecture

Lemma 1: There exists an assignment of integer coefficients $a_1,\dots,a_{1977}$ to the coins such that the sum $\sum a_i \delta_i$ uniquely determines $\delta_j$ for any selected coin $j$, where $\delta_i$ is $0$ for genuine and $\pm 1$ for counterfeit. This is true because the number of possible patterns of $50$ counterfeit coins is much smaller than $3^{1977}$, so a weighted sum with sufficiently large coefficients can encode each pattern injectively.

Lemma 2: A single weighing using coefficients $a_i$ as the number of copies of each coin on the pans (or as weights in grams) produces a difference $\Delta$ equal to the linear combination $\sum a_i \delta_i$. This is true because the pointer measures the algebraic sum of weight differences, which is linear in the deviations $\delta_i$.

Lemma 3: By choosing $a_i = 3^{i-1}$ for $i=1,\dots,1977$, the difference $\Delta$ uniquely identifies $\delta_j$ for the selected coin. This follows from the representation of integers in base $3$, allowing every combination of $50$ nonzero $\delta_i$ to yield a unique sum.

The hardest part is verifying that no two distinct subsets of $50$ counterfeit coins could produce the same sum in absolute value for the selected coin's presence or absence. Lemma 3 is the most delicate, as it depends on the base-3 expansion being sufficiently large to separate all subsets of 50 elements.

Solution

Label the coins as $C_1, C_2, \dots, C_{1977}$, and let $C_j$ be the coin whose status we wish to determine. Assign each coin a weight coefficient $a_i = 3^{i-1}$ in grams for the weighing. Place $C_j$ on the left pan. For the remaining coins, place $|a_i|$ copies on the left pan if $a_i > 0$ and on the right pan if $a_i < 0$, or equivalently, distribute them according to the sign convention in the coding. Since each counterfeit coin differs by $\pm 1$ gram, let $\delta_i = 0$ if $C_i$ is genuine, $\delta_i = 1$ if $C_i$ is heavier, and $\delta_i = -1$ if $C_i$ is lighter.

The total difference indicated by the pointer is

$$\Delta = \sum_{i=1}^{1977} a_i \delta_i.$$

Because $a_i = 3^{i-1}$, the sum $\sum_{i\ne j} a_i \delta_i$ cannot reach $a_j = 3^{j-1}$ in absolute value unless $\delta_j \ne 0$. More formally, the maximum possible contribution of the remaining $1976$ coins is

$$\sum_{i\ne j} 3^{i-1} \le 3^{1977}-1,$$

whereas $a_j = 3^{j-1}$ is isolated in the base-3 representation. Therefore, observing $\Delta$ allows us to determine whether $\delta_j = 0$ or $\delta_j = \pm 1$ unambiguously. If $\Delta$ includes the term $3^{j-1}$, then $C_j$ is counterfeit; otherwise, it is genuine.

Thus, a single weighing constructed with base-3 weighting of the coins enables the identification of any selected coin among the 1977, knowing that exactly 50 are counterfeit. This completes the construction.

The required object is the distribution of coins on the pans according to the coefficients $a_i = 3^{i-1}$, with $C_j$ on the left pan and the remaining coins placed according to the sign convention.

$$\boxed{\text{Place the selected coin on one pan; assign each coin weight }3^{i-1}\text{ and distribute the remaining coins so that the pointer difference equals the weighted sum. The measurement reveals whether the selected coin is genuine or counterfeit.}}$$

Verification of Key Steps

Step one is to confirm that the sum of contributions from the remaining 1976 coins cannot cancel or mimic the contribution of $C_j$. Compute explicitly for small $n$ that the sum of the largest $k$ powers of $3$ is strictly less than the next higher power of $3$, ensuring uniqueness in base-3 representation. For instance, $1 + 3 + 9 + 27 = 40 < 81 = 3^4$, confirming the principle holds in general.

Step two is to verify that the 50 counterfeits cannot combine to produce ambiguity. The sum of the 50 largest coefficients is

$$\sum_{i=1928}^{1977} 3^{i-1} = \frac{3^{1977}-3^{1927}}{2} < 3^{1977},$$

so no subset of 50 coins can generate a sum exceeding $3^{1977}$, preserving the injectivity of the map from coin patterns to $\Delta$.

Step three is to check the linearity of the weighing. Adding or subtracting a coin of $\pm 1$ gram contributes exactly $a_i \delta_i$ to $\Delta$, confirming that the pointer difference equals the weighted sum.

Alternative Approaches

A similar approach could employ binary coding, placing coins on the left pan corresponding to 1-bits and on the right pan to 0-bits, and using larger weights to separate sums. However, base-3 coding is preferable because it accommodates the three possibilities for each coin (genuine, lighter counterfeit, heavier counterfeit) in a single weighing. Direct combinatorial distribution without a positional weight assignment is infeasible due to the enormous number of subsets of 50 counterfeits, which would create unavoidable ambiguities. The weighted pan strategy efficiently encodes all information linearly and requires only a single measurement, making it optimal.