There and back again
Problem
Bilbo decides to leave his hobbit-hole and go on an adventure. He walks 100 miles South, then 100 miles East, then finally 100 miles North, at which point he is surprised to find that he has arrived back home!
Many people would think that because of this Bilbo must live at the North Pole.
However, Bilbo doesn't live at the North Pole!
Can you describe where you think his home might be?
Have you found all the possible locations?
Getting Started
You could use some trial and improvement to help you visualise what is happening:
Student Solutions
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.
Teachers' Resources
An intriguing problem that challenges students to visualise different possibilities.
And when they have found one solution they can be encouraged to find more...
Older students who have been introduced to trigonometry might like to use the relationship between the circumference of a circle and its radius to estimate/calculate how far Bilbo's home is from the poles?
For this they may find it useful to know that the circumference of the earth is about 24,900 miles, and the radius of the earth is about 3,960 miles.