Smallest Set of Weights for Two-Pan Balance/Examples/40

Examples of Smallest Set of Weights for Two-Pan Balance

Consider a balance such that weights may be placed in either or both of the pans.

Let $S$ be the smallest set of weights needed to weigh any given integer weight up to $40$ units.

Then $\size S = 4$.

Proof

From Smallest Set of Weights for Two-Pan Balance, a set of $4$ weights in the sequence $\sequence {3^n}$:

$1, 3, 9, 27$

allows one to weigh any given integer weight up to $\dfrac {3^4 - 1} 2 = 40$.

Hence the result.

$\blacksquare$

Sources

• 1612: Claude-Gaspar Bachet: Problèmes Plaisans et Delectables qui se font par les Nombres
• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {1-2}$ The Basis Representation Theorem: Exercise $5$
• 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Bachet: $108$