This clever puzzle combines weights, coin counting, and ternary encoding:

You have 5 bags, each with 100 coins, and each bag’s coins are uniform in weight, but either:

  • 9 g, or
  • 10 g, or
  • 11 g

You may use a digital scale that reports the total weight of any group of coins.

Goal: Determine, with as few weighings as possible, the exact weight of coins in each bag.

Step 1: Strategy Overview

We’re told that each bag could be light (9g), normal (10g), or heavy (11g).

That’s three possible states per bag → a ternary system!

With 5 bags, the total number of weight configurations is:

[ 3^5 = 243 ]

So, we need a strategy that can distinguish between 243 different outcomes—and we can encode all 243 in a single ternary number.

Step 2: Use Weighted Coin Counts

Take coins from each bag in these amounts:

  • Bag 1 → 1 coin
  • Bag 2 → 3 coins
  • Bag 3 → 9 coins
  • Bag 4 → 27 coins
  • Bag 5 → 81 coins

These counts correspond to the powers of 3: (3^0, 3^1, 3^2, 3^3, 3^4)

Step 3: Calculate Expected Weight

If all coins were 10g, then total weight would be:

[ 1×10 + 3×10 + 9×10 + 27×10 + 81×10 = 1210 \text{ g} ]

Now suppose you perform a single weighing and get total weight (W).

Let the difference (d = W - 1210). Each 1g deviation corresponds to:

  • -1 per light coin (9g)
  • 0 per normal coin (10g)
  • +1 per heavy coin (11g)

Because you took different multiples of coins from each bag, each weight deviation from 1210 reflects a ternary digit in base 3 centered at 10g.

Step 4: Decode the Deviation

Write (d) as a number in balanced ternary (digits = -1, 0, +1), using the weights: [ d = a×1 + b×3 + c×9 + d×27 + e×81 ]

Each coefficient tells you whether that bag was:

  • -1 → 9g coins
  • 0 → 10g coins
  • +1 → 11g coins

The powers (1, 3, 9, …) uniquely identify which bag’s coins contributed which deviation.

Final Answer

One weighing is enough.
Take 1, 3, 9, 27, 81 coins from the 5 bags.
Compute the total.
Compare to 1210g, and convert the difference into balanced ternary to identify each bag’s coin weight (9g, 10g, or 11g).

Reference