The Spice Merchant's Weights

The four weights are 1, 3, 9, and 27 grams. Placing weights on either pan of the balance scale creates a ternary system covering all integers from 1 to 40. Formula: k weights (powers of 3) cover 1 to (3^k - 1)/2.

When a weight can be on the left pan (positive), the right pan (negative), or not used (zero), it contributes three states. This is exactly base-3 representation. To cover 1 through N with k weights, use powers of 3: 1, 3, 9, 27, ... The total coverage is (3^k - 1)/2. Step 1: With one side only (binary): weights 1, 2, 4, 8 cover 1-15. Each weight is double the previous. Step 2: With both sides (ternary): each weight w can contribute +w (on opposite pan), -w (on same pan as item), or 0 (not used). This is a balanced ternary system. Step 3: Weight 1: can measure 0 or 1. Weight 3: combined with 1, can measure 1+3=4, 3-1=2, 3 alone, 1 alone. Covers 1, 2, 3, 4. Step 4: Weight 9: combined with {1,3}, can measure 9-3-1=5, 9-3=6, 9-3+1=7, 9-1=8, 9, 9+1=10, 9+3-1=11, 9+3=12, 9+3+1=13. Covers 1-13. Step 5: Weight 27: combined with {1,3,9}, covers up to 27+9+3+1=40. Covers 1-40. Step 6: The formula: k weights of values 1, 3, 9, ... 3^(k-1) cover all integers from 1 to (3^k - 1)/2. For k=4: (81-1)/2 = 40. Perfect. Key Insight: The ternary system doubles the efficiency of binary for physical weights. While binary (one-sided) weights grow as powers of 2, ternary (two-sided) weights grow as powers of 3 — a much faster expansion. The balance scale's two pans are the physical realisation of positive, negative, and zero in balanced ternary.