Fake gold
Problem
A merchant brings four bars of gold to a jeweller. The jeweller suspects that one bar is a fake, and knows that it will be slightly lighter than the real ones.
He only has to use the balance twice before he finds the fake bar. How does he do it?
Now, can you do it with nine bars of gold (one fake light one), again only using the balance twice?
Getting Started
What would happen to the balance if the weights were eqaul?
What would happen if they weren't?
Student Solutions
The solutions to both parts of the problem were explained very well by Robert (Moorgate C P School, Staffordshire), Jack and Jesse (Tattingstone School), Daniel (Anglo-Chinese School, Singapore), Samantha, Julie (The Abbey) and Ashley.
This solution to the four bar problem is from Laura (St Ives School, Haslemere)
My solution to the four bars of gold Find the fake is:
The man weights two bars of gold in either of the scales - one side will be lighter, that will have the fake in it. Split the two bars and weigh it again, the lighter will be the fake.
This answer for the 9 bar problem comes from Caroline, Sarah and Ellie (The Mount School, York),
- Place 3 bars on each side. If they balance then the fake is in the other 3. If they don't balance then the fake is in the lighter side.
- Take the 3 bars which the fake is in and weigh 2 of them against each other. If they balance it's the one left over and if they don't balance it is the lighter one.
Teachers' Resources
Why do this problem?
This problem is an exercise in logical thinking. It is a good opportunity to insist on a precise explanation with justification.Possible approach
For each subsequent action, ensure that a reason is given. If incorrect suggestions are made, do not correct them but carry out the actions anyway. If children challenge them, ask for justifications.
When a satisfactory solution has been reached and everyone is happy, pose the question about nine bars and give the children some time to work on it in pairs. You might choose to give each pair a large piece of paper to record their thinking, which they can then share with the rest of the class.
You could then confirm a correct solution practically.
Key questions
What would happen to the balance if the weights were eqaul?What would happen if they weren't?