Reasoning · Grade 3-2 Weighing Applications

Problem

Find the counterfeit coin by weighing

Exactly one of coins 1 to 8 is a heavier fake. Each weighing puts equal counts on both pans, so without the fake they would balance. The pan that dips holds the fake; a level weighing rules its coins out. Find the fake from the three weighings.
1 3 6 2 4 5 Scale 1: tips left 2 5 3 8 Scale 2: level (balanced) 5 6 8 1 3 7 Scale 3: tips right
Your answer
How to solve
Strategy Eliminate Possibilities — There is a genuinely finite set of 8 candidate coins. Each weighing tells us which group must contain the fake (or, if level, which coins are cleared). By keeping only coins consistent with all three weighings, we narrow eight candidates down to one.
1STEP 1

Use weighing 1 to keep a heavy group

The first tip puts the fake among 1, 3, 6.

fake ∈ {1, 3, 6}
2STEP 2

Use weighing 3 to keep another heavy group

The third tip narrows it to 1, 3, 7.

fake ∈ {1, 3, 7}
3STEP 3

Combine weighings 1 and 3

Only 1 and 3 appear in both.

{1, 3, 6} ∩ {1, 3, 7} = {1, 3}
4STEP 4

Use weighing 2 to decide between 1 and 3

The level second weighing shows 3 is not the fake.

level → 3 is not the fake → fake = 1
5STEP 5

Check coin 1 against all three weighings

That leaves coin 1, consistent with all three.

Answer
coin 1
Coin 1 makes weighing 1 sink on the left (1 is there), weighing 2 stay level (1 is absent), and weighing 3 sink on the right (1 is there) — all three results match, and no other coin fits all three, so the answer is unique and sensible.
Takeaway

When the scale tips, the fake is on the heavy side; when it balances, the fake is somewhere else — cross out groups until one coin is left!

  • Use weighing 1 to keep a heavy group
  • Use weighing 3 to keep another heavy group
  • Combine weighings 1 and 3
  • Use weighing 2 to decide between 1 and 3
  • Check coin 1 against all three weighings