You face a classic logic-and-weight puzzle:

You have 10 bags, each with 100 coins.
All coins look identical.

  • 9 bags contain only 10g coins.
  • 1 bag contains only 9g or 11g coins (unknown which).

You may use a digital scale once to determine which bag is counterfeit.

Step-by-Step Solution

Step 1: Assign a Unique Sample Pattern

Take coins from each bag in a distinct count:

  • Take 1 coin from Bag 1
  • 2 coins from Bag 2
  • …
  • 10 coins from Bag 10

Now you have 55 coins total: [ 1 + 2 + \cdots + 10 = \frac{10 × 11}{2} = 55 ]

Step 2: Expected Weight if All Bags Are Genuine

If all coins were 10g:

[ 55 × 10 = 550 \text{ grams} ]

Now place your 55 coins on the scale and record the actual weight.

Step 3: Interpret the Difference

Let the measured weight be (W). Then:

  • If one bag has 9g coins, the total will be less than 550.
  • If one bag has 11g coins, the total will be more than 550.

Let the difference be (|W - 550|).
This difference tells you how many grams off you are—i.e., how many coins you took from the counterfeit bag.

Because you took a unique number of coins from each bag, this difference directly identifies the counterfeit bag.

  • If (W = 548), you’re 2g short → Bag 2 is counterfeit (with 9g coins).
  • If (W = 562), you’re 12g over → Bag 6 is counterfeit (with 11g coins).

Final Answer

Take (n) coins from Bag (n) (1–10), weigh them all once.
Compare total to 550g.
The difference (in grams) tells you the bag number and whether it’s lighter or heavier.

Reference