This blog has moved to www.mscroggs.co.uk.

## Sunday, 9 March 2014

### Sunday Afternoon Maths III

Here's this week's collection. Answers & extensions tomorrow. Why not discuss the problems on Twitter using #SundayAfternoonMaths

#### Colliding Parallel People

If two people stand 1km apart and walk in the same direction, how far will the have to walk until they collide due to the curvature of the Earth? (diameter of Earth = 12,742km)

#### Twenty

How many three digit integers are there for which the product of the digits is 20?

1. Colliding Parallel People: It depends on where they start and what direction they go in. If they both start 0.5km from the North pole so that they are 1km apart, and they both walk North, they will each walk 0.5km. If they start 1km apart on the equator and both walk North it will be much longer. If one starts 1km North of the other somewhere near the equator and they both walk East then they will never meet.

Twenty: 20 = 1 x 2 x 2 x 5. Going through it systematically (I don't know if there is a better way) 541, 522, 514, 451, 415, 252, 225, 154, 145. So 9 ways.

1. For the parallel people, I meant that they face in a parallel direction. If they both walk North, they'll be facing towards each other... I think. I'll check before the answers are up

2. As I said before, what if one of them is South of the other and they both walk East?

1. Then they won't be walking in straight lines?

2. Agreed. Latitude east-west lines as non-linear, as they are not greater circles and need more than a point and a direction to define their path.

3. "Same direction" is not a well-defined concept on a sphere, since there are no parallel lines. However, I think the gist of what you're saying is this:

Two people stand facing directions perpendicular to the line which connects them (each 90 degrees clockwise from some greater circle with defined direction, standing on that circle).

This definition of "same direction" certainly makes one think of Euclid's fifth, but of course, that postulate doesn't hold in spherical geometry.

With this definition, let's assume, without loss of generality, that they're both on the equator facing north (note that north is only truly perpendicular to east/west at the equator). They will walk along their respective meridians until reaching the north pole, 1/4 of the way around the world. Thus, the first meeting will be after a 12,742 * (2pi/4) kilometer walk for both.