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This problem caught many people out. Well done Thomas, from Bristol Grammar School, for this correct solution:

Bilbo lives at any point 100 miles north of the line of latitude near to the south pole which has a 100 miles circumference. He therefore goes 100 miles south onto that line, travels east around the world, and then travels 100 miles north back home.

Of course, Bilbo could also live at any point 100 miles north of the line of latitude near to the south pole which has a 50 miles circumference. He would go 100 miles south onto that line, travels east around the world twice(!), and then travels 100 miles north back home...

... or 100 miles north of the line of latitude near to the south pole which has a 25 miles circumference (and travel east around the world four times ...

... or 100 miles north of the line of latitude near to the south pole which has a 20 miles circumference (and travel east around the world five times ...

... or 100 miles north of the line of latitude near to the south pole which has a 10 miles circumference (and travel east around the world ten times ...

... or 100 miles north of the line of latitude near to the south pole which has a 100/n miles circumference (and travel east around the world n times.