# Seven on an asteroid – a mathematical puzzle

I happened to see this puzzle on Facebook. Really nice puzzle (if rather hard).

Seven astronauts are exploring a small spherical asteroid. All seven started out in the same point.

Astronaut A went 30 kilometers in one direction, then turned 90 degrees to the left, then moved another 30 kilometers, then turned 90 degrees to the left again, and then moved for 30 kilometers more.

Astronaut B moved in the same pattern, except his movements between the 90 degree turns were 40 kilometers long instead of 30.

Astronauts C, D, E, F and G also moved in the same pattern, but went in sections of 50, 60, 70, 80, and 90 kilometers, respectively.

The astronauts started moving (from the same starting point) in several different directions, but, as it happened, after the movements described above, all but one ended up in the same place.

1. Which astronaut ended up in a different place?
2. How large is the asteroid?

(Post 172 in the inner system, incidentally. Post 4 in reality.)

## 2 thoughts on “Seven on an asteroid – a mathematical puzzle”

1. toafan says:

My first thought was that one of them must have traveled past the equator. The circumference of the asteroid would be between (four times eighty) and (four times ninety) kilometers, but I couldn’t tell you the exact number.

I got my globe out and played around with some rubber bands. This had me thinking that they must have all hit the equator but traveled around the equator before heading back to base. But that was in error, because due east is always 90 degrees from due north.

Then I realized that nothing in the statement of the problem says that the path of a given astronaut did not cross. For example, one of the astronauts could have traveled the full circumference of the asteroid and made it back to the start point before making their first 90-degree turn. With this on the table, I topologically have no clue what is going on and am setting this aside for now to go sleep.

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1. Finally – a comment on my blog! It had hardly been a year!
(There had technically been other comments before – one or two spams, some self-pingbacks, and at least one misplaced private message. Nothing remotely worthy of approving, however.)

On the problem – I’m not going to spoil much, but yes, in the intended solution, at least one of the paths crossed itself at least once, and the circumference is less than 4*90 km. (Note: as it happens, those two statements are equivalent.)
If it helps (not a spoiler, just a clarification), all the sections are intended to be great circles (and not, say, circles of latitude, other than perhaps the equator).

(Sorry for the “4:03 am at 4:03 am” thing, by the way – it’s apparently an inevitable consequence of the way I have my dates set up. I’ll try to ask on the forum if you want, but I’m not sure there’s a way to change it and only it, unfortunately.)

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