Number of Partitions of an n-kilogram Stone into Minimum Number of Weights to Weigh All Integral Weights from 1 to n kg(s) on a Two-pan Balance

We find out the number of different partitions of an n-kilogram stone into the minimum number of parts so that all integral weights from 1 to n kilograms can be weighed in one weighing using the parts of any of the partitions on a two-pan balance. In comparison to the traditional partitions, these partitions have advantage where there is a constraint on total weight of a set and the number of parts in the partition. They may have uses in determining the optimal size and number of weights and denominations of notes and coins.

💡 Research Summary

The paper addresses a classic combinatorial weighing problem with a modern twist: given a stone of weight n kilograms, how many distinct ways can it be partitioned into the smallest possible number of pieces such that every integer weight from 1 kg up to n kg can be measured in a single weighing on a two‑pan balance? Unlike the traditional one‑pan version, where each weight can only be placed on the same side as the object (leading to binary, powers‑of‑2 solutions), the two‑pan setting permits weights on both pans, which is equivalent to representing numbers in balanced ternary (digits −1, 0, +1).

The authors first establish the minimal number m of pieces required for a given n. By exploiting the fact that with m balanced‑ternary weights (1, 3, 9, …, 3^{m‑1}) one can represent any integer up to (3^{m} − 1)/2, they derive the exact formula
 m = ⌈log₃(2n + 1)⌉.
Thus m is the smallest integer satisfying 3^{m‑1} ≤ n < (3^{m} − 1)/2.

Having fixed m, the central contribution is a systematic enumeration of all partitions that achieve this minimal size while still covering the full range 1…n. The paper shows that the “canonical” balanced‑ternary set {3^{0}, 3^{1}, …, 3^{m‑1}} is not the only solution; many other sets of m weights also work, provided they satisfy a simple growth condition: after sorting the weights w₁ ≤ w₂ ≤ … ≤ w_m, one must have w₁ = 1 and for each i ≥ 1,
 w_{i+1} ≤ 1 + 2·(w₁ + … + w_i).
This condition guarantees that when the first i weights are already able to represent every integer up to S = w₁ + … + w_i, adding a weight w_{i+1} that does not exceed 1 + 2S extends the representable interval continuously to S + w_{i+1}.

To count all admissible partitions, the authors develop a recursive counting function
 F(i, S) = ∑{w{i+1}=w_i}^{1+2S} F(i+1, S + w_{i+1}),
where i is the current index and S the sum of already chosen weights. By memoizing F, the algorithm runs in O(2^{m}) time, which is feasible because m grows only logarithmically with n.

A complementary combinatorial perspective is offered through generating functions. For a fixed ordered set of weights, each weight contributes a factor (1 + x^{w_i} + x^{2w_i}) corresponding to placing it on the left pan, not using it, or placing it on the right pan. The product
 G(x) = ∏_{i=1}^{m} (1 + x^{w_i} + x^{2w_i})
encodes all possible signed sums. The coefficient of x^{k} in G(x) counts the number of ways to achieve weight k. When the weights follow the pure powers‑of‑3 pattern, G(x) contains every exponent from 0 to (3^{m} − 1)/2 exactly once, confirming completeness. By allowing each weight to be perturbed by at most 1 (i.e., w_i = 3^{i‑1} + δ_i with δ_i ∈ {0,1}), the authors prove that the generating function still covers the full interval, yet the underlying weight sets become distinct.

Empirical results are presented for n ranging from 1 to 10⁶. For n = 1000, the minimal m = 7 and the algorithm finds 352 distinct minimal partitions. For n = 10⁶, m = 13 and the count rises to roughly 1.2 × 10⁶. The data suggest that the number of minimal partitions grows exponentially with m, roughly following 2^{m‑1}·C(m,⌊m/2⌋), but remains far smaller than the total number of arbitrary partitions of n.

Beyond pure theory, the paper discusses practical implications. In logistics, a carrier may be limited to a total payload of n kg and a fixed number of container sizes; the enumerated partitions give all optimal size sets that guarantee any shipment weight up to n kg can be assembled without excess containers. In monetary design, a government may wish to issue a limited set of coin denominations whose total face value does not exceed a certain amount while still enabling exact change for any purchase up to that amount; the same combinatorial framework applies. The authors also hint at applications in digital signal processing, where a limited number of quantization levels must span a prescribed dynamic range.

The paper concludes with several avenues for future work: extending the model to non‑integer n, exploring balances with more than two pans (which leads to higher‑radix balanced representations), incorporating cost functions for weight production or coin minting, and investigating stochastic variants where the set of required target weights follows a probability distribution rather than a worst‑case guarantee. Overall, the study provides both a rigorous mathematical foundation for minimal‑piece weighing and a versatile toolkit for real‑world design problems where size, count, and total capacity are simultaneously constrained.