Route to Infinity

Look at the route the arrows follow in this diagram. Look away from the screen and try to describe their path.

Will the route pass through the point (18,17)?
If so, which point will be visited next? Explain how you found out.

How many points does the route pass through before it reaches the point (9,4)?

Explain how you found out.

If this diagram is extended to the whole first quadrant, and the route continues in the same way, what can you say about the lattice points it goes through? (A lattice point is a point with integer coordinates.)

Can you draw a route that goes through all the lattice points in all 4 quadrants?

See the Notes for Teachers.

This problem is adapted from the NRICH task Route to Infinity with permission of the University of Cambridge. All rights reserved.

Download the NRICH Poster here.