Part of a toy is made of two rectangular blocks that are both 6 cm by 12 cm. The top block turns about the other by a pin that goes through the centre of each block.
Suppose the top block turns exactly once. How many points are there on the top block that are never over an edge of the bottom block?
Solution
First of all it is clear that there has to be at least one point of the top block that doesn’t cross an edge of the bottom block. The centre point X is such a point.
If we look at a small enough circle with centre X, then no point in that circle will cross an edge of the bottom block. Even a small circle contains an infinite number of points. So the answer to the question is an infinite number.
But let’s be more precise about how small a small enough circle will be. If we expand our circles with centre X, the first time we cut an edge of the bottom block is when we reach the point U that just sits on the edge PQ. (Or the corresponding point on the edge of the bottom block that is parallel to PQ.) At this point the radius of the circle centre X and having U on its circumference is 3 cm. So every point on the top block that is inside a circle whose radius is less than 3 cm will not touch an edge of the bottom block.
Extension
Is it possible for a point on the top block to be over an edge of the bottom block:
- Once
- Twice
- Three times
- Four times
- An odd number of times
- Eight times
- More than eight times
Explain your answers.