KA-0094Weighing and balance puzzles
5 points, difficulty 3 of 3Level 5–6about 135sOne of nine identical-looking coins is slightly heavier. Using only a balance, what is the smallest number of weighings that is certain to find it?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
A balance has three outcomes, not two.
2The strategy
Left heavier, right heavier, or level. One weighing can therefore separate three groups and two weighings can separate nine, so split into three groups rather than two.
3The full solution
Weigh 3 against 3. Whichever side is heavier holds the coin, and if they balance it is in the third group. Then weigh 1 against 1 from that group.
Solution
The reliable way
A balance has three outcomes, so a good weighing splits the coins into three groups, not two. Weigh three coins against three. If one side sinks the heavy coin is there; if they balance it is in the untouched three. Either way you are down to three coins. Now weigh one of those against another: if one sinks it is the heavy one, and if they balance it is the third. Two weighings, always. The transferable idea: use all three outcomes, so each weighing divides the possibilities by three rather than by two.
The elegant way
Two weighings distinguish 3 x 3 = 9 cases, exactly the number of coins, so two is both enough and necessary.
Why this is on the test: Forgetting that level is a useful third outcome is what turns a two-weighing problem into a three-weighing one.