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What colour is the bear? Solution

What colour is the bear?

Have a go at the problem. Where does the bear start?

To repeat the problem: A bear walks ten miles south, then ten miles west, then ten miles north, and ends where it started. What colour is the bear? Or rather, where does the bear start? The hint says "Think about the difference between travelling north/south and travelling east/west."

Track of bear

The easiest way to solve this problem is to say that there is one place on Earth that is guaranteed only has one colour of bear, so the answer is that the bear is in the Arctic, probably the North Pole, and so is white!

That is the right answer, but why? Well, if the bear starts at the North Pole and walks for ten miles, then it is walking south whatever direction that it's travelling in. When it turns and travels due west, then it won't travel in a straight line. It will travel in a circle round the North Pole. If it carried on walking, it would get back to where it started. However, it only travels ten miles. Then it turns again, then time north, towards the North Pole. It ends up back at the North Pole.

This happens because travelling north or south is different to travelling east or west. If you travel north, you eventually get to the North Pole, no matter where you start on the Earth's surface. If the whole world travelled north, we'd all end up in the same spot! If we carried on travelling, we'd be heading south rather than north, and we would all end up at the South Pole. Travelling east and west is different. If I started at the UK and travelled east, I would travel through Europe, then the Middle east, then the Far east or Asia. I would cross the Pacific, and through the Americas, across the Atlantic, then I would arrive back at home. I would do this despite travelling west all the way. If I travelled west, I would meet America first, then the Far East, despite travelling west. What's more, if everyone travelled west, then we wouldn't meet each other. Some would stay ahead of me, some behind, and some on a completely different track, but we would never find a point which is West, the way that the North Pole is North.

In fact, if you travel south ten miles, west ten miles, north ten miles and east ten miles, you are unlikely to end up exactly where you started. If you start north of the equator, then the final side of the square, travelling east, takes you further round the world, then the second side of the square, travelling west. This is because the southern-most side of the square is closer to the equator. The only way to end up where you started is to start five miles north of the equator, and cross the equator by travelling south. That way, both east and west journeys are the same length. If you don't understand any of this, try looking at a globe and running your fingers along the longitudes (north/south) and latitudes (east/west). Mapmakers who try to make flat maps of what is, after all, a sphere, have to play various tricks. Sometimes they elongate the north and south poles (which are points) to be as wide as the equator. This makes Canada look enormous (yes, I know it's big, but not THAT big!) Other mapmakers do the same, but squash up the latitudes near the poles, so Canada is the right area, but looks a very odd shape. Some make the map so that Alaska curls round.

The problems happen because we are used to thinking of two-dimensional surfaces as flat, like a piece of paper. The surface of the globe is two-dimensional, as you can define any point of it with two numbers, the longitude and latitude. But it is curved, and so 'normal', or Euclidean, geometry doesn't apply. These are called non-Euclidean surfaces.

I did ask "Where else on Earth could you make this journey? (No bears live here!)" That was a clue, and no doubt someone guessed the South Pole (which has penguins and no Polar Bears). That's not quite right. If you start at the South Pole, then you can't travel south to start with, as there's nowhere to go! Concentrate on the travelling west. It is possible to travel west for ten miles and end up where you started. You need to be 5/π miles from the South Pole for this (since this radius gives a circumference of ten miles). But you need to begin by travelling ten miles south. So start (10 + 5/π) from the south pole. Then you can travel ten miles south, then ten miles in a complete circle round the south pole, then ten miles north on your own tracks, which will take you back to your starting point.