By Burkard Polster and Marty Ross
The Age, 1 December 2008
Here is a famous puzzle:
A bear leaves his den. He travels 10km South, then 10km East, and finally 10km North. He finds that he is back in his den. What colour is the bear?
This is a great puzzle for so many reasons. First of all, it's funny and engaging. What can maths possibly say about the colour of a bear?
Secondly, it's a genuinely puzzling puzzle. At first glance the scenario seems impossible, since compass directions are all about right angles. Surely, we just have three sides of a square?
Thirdly, the solution (which we give below) is delightfully simple. And, the puzzle offers a sneaky double-punch. Even many people who know this puzzle are surprised to learn there is more than one solution.
Fourthly, the puzzle provides an engaging entry to beautiful and deep mathematics. As we shall try to make clear this is no accident: at heart mathematicians are simply happy children, being paid to solve puzzles.
Mathematical Games
Undoubtedly, the greatest mathematics popularizer of all time is Martin Gardner. His wonderful Mathematical Games column ran in Scientific American for 25 years. Through it, Gardner introduced non-mathematicians to a wealth of beautiful mathematics, and always with clarity, humour and elegance. At the age of 94, Gardner is still writing his beautiful books. For we “Maths Masters”, Martin Gardner is a God.
Gardner's columns have been collected in a series of books, beginning in 1959 with Mathematical Puzzles and Diversions. Our bear problem above is Gardner's very first puzzle in this very first collection.
In his exposition of the puzzle, Gardner begins with the well-known solution: the bear is white. Why? Because the bear is a polar bear, with his den at the North Pole. So, he travels south from the North Pole (it doesn’t matter along which line of longitude), then east along a circle of latitude. And, on the last leg he travels north, back to the North Pole.
After presenting this solution, Gardner challenges the reader to find another solution. Even with the proffered solution in mind, it is not easy to think of another. In fact, there are many!
Here is the idea for a new solution. The "bear" starts very near the South Pole: the bear is now a penguin! If he starts at just the right latitude, then going South the 10 kilometers will take him so close to the South Pole that traveling 10km East will take him on a full loop of the circle of latitude. Then, going North exactly retraces the first leg, and he ends up back where he started. We'll leave as homework for you to sort out the details, and to complete the puzzle: there are yet more solutions.
Bears in Space
The bear puzzle is fun, with a beautiful solution. But this puzzle also points to a serious geographical problem. We like to think of the compass directions as providing natural rectangular coordinates. But, the bear's paths show that this cannot actually be the case. The geometry of the Earth must be fundamentally different to that of a flat plane.
This issue was of huge historical importance, for sailors wanting accurate maps to aid their navigation. The bear puzzle shows that any such map of the Earth’s surface must unavoidably be distorted. The question then becomes, what's the least troubling distortion? This is a fascinating story, culminating in the 16th Century with (but my no means ended by) the map projection of Gerardus Mercator.
We shall use the bear puzzle as an excuse to ask a similar question in a wholly different context: what shape is Space? That this question can even be asked often surprises people. Isn’t Space just space-shaped? There’s an X direction and a Y direction and a Z direction, and that’s it? Well, maybe not.
Imagine we place our bear (now an astronaut) into a rocket and shoot him out into Space. He travels a million kilometers, takes a right angle and travels another million kilometers. He then takes another right angle and travels a final million kilometers.
Could our bear possibly be back on Earth? Yes, depending upon the shape of our Universe, he just might. It is possible that the Universe is fundamentally curved, just like the surface of the Earth, and that the rectangular X-Y-Z “compass” directions simply don’t work.
Mathematically, the way to determine whether the Universe is curved is exactly by shooting (theoretical) bears into space and considering their triangular paths. The amount the Universe is curved is then indicated by the sum of the angles in the triangles. In a rectangular Euclidean world the sum of the angles must be 180 degrees. But, in other imaginable worlds, and perhaps in our actual Universe, the sum may be greater or smaller.
Torturing Triangles
This is exactly what happens on the surface of the Earth. The sum of the angles in a spherical triangle is always greater than 180 degrees. How much greater? The precise answer is given by the following beautiful formula. If the Earth has area E and the triangle has area A, then the sum of the angles is
So, a tiny triangle has sum very close to 180 degrees. But if the triangle grows to consume almost all of the Earth then A will be very close to E, and so the sum approaches 180+720=900 degrees! What does such a large triangle look like? Exactly like the tiny equilateral triangle, except we declare the "inside" to be the "outside" and vice versa. That is, instead of three angles of a little more than 60 degrees (the orange arcs pictured), we picture the triangle as having three (white) angles each a little less than 300 degrees!
Notice that there is another, intriguing way to view the above formula. Suppose we take a small triangle on the Earth, and we measure the triangle’s area and the sum of its angles. Then, the formula can be rearranged to tell us the Earth’s area. That is, as long as we assume the Earth is uniformly spherical, a little triangle exactly tells us the radius of that sphere. We have obtained global information from a tiny portion of the sphere.
(At this stage, we must confess that there are technical details with the above formula: a triangle should be made up of straight lines, but what exactly do we mean by “straight line” on a curvy surface? In brief, we mean “as straight as possible”. On the Earth, this amounts to straight lines being parts of circles of greatest possible radius, such as the Equator and lines of Longitude.)
The Earth is Round!
Of course it is no surprise that the surface of the Earth is curved: people have pretty much always known that. Yes, even Columbus. And yes, even the early Christian church. But what we are discussing here is a much more subtle notion. It’s not just that the Earth looks curved to an astronaut in Space. The Earth is unavoidably, intrinsically curved.
Taking a cylinder, it will obviously look curved. But we can cut the cylinder open and lay it flat on the ground. In fact, the reverse procedure is exactly how anyone would make a cylinder: take a rectangular piece of paper and roll it up. What this demonstrates is that compass directions work perfectly well on a cylinder. It is in this manner, in the failure of compass directions to work, which shows the Earth is curved in a much more subtle and fundamental way.
The first person to successfully investigate these concepts was the great mathematician Carl Friedrich Gauss. In the early 1800’s Gauss considered formulas akin to our angle formula above. He came up with a precise quantity to characterize the intrinsic curving of surfaces, a quantity which now goes by the name Gauss curvature.
The Gauss curvature of a surface can be either positive or negative, and can vary from point to point. Positive curvature corresponds to angle sums of triangles exceeding 180 degrees, such as for the Earth. Negative curvature arises when angle sums are below 180 degrees. From the outside, such negatively curved surfaces look saddle-shaped.
Is Space Round As Well?
Now, what about Space? The first person to imagine three dimensional worlds and how they might be curved was Gauss’s brilliant student, Bernhard Riemann. Riemann came up with a corresponding notion, now known as Riemann curvature, of how 3D worlds could be curved. It is complicated, but again the heart of it can be captured by triangles: sums of angles greater than 180 degrees indicate positive curvature, and sums below 180 degrees indicate negative curvature.
Can we actually detect this curvature of Space? Well, we can at least try. If we replace overworked bears with light rays, and imagine the rays connecting three distant points. (There is a famous story that Gauss was the first to actually try to detect such the curvature of space, though historians hotly dispute what Gauss was actually testing for).
In fact, mathematicians and astrophysicists are actively involved in trying to measure the curvature of the Universe (or at least the parts of the Universe that we can see). It is much more subtle than we have indicated, and the details and the techniques are very complicated. After all, there’s that whole fourth dimension business to worry about. And there’s the fact that the matter in the Universe, the ultimate source of Space’s curvature, is clumped here and there rather than being uniformly spread around. But in principle, it is all possible. And at its heart it is the same as the bear puzzle: what’s the sum of the angles in a triangle?
A Loopy Calculation
There’s another aspect of the bear puzzle worth pondering. If the bear leaves the North Pole and keeps on going in the one direction, he will eventually get to the South Pole, pass through and return to the North Pole. This fact, that any such path loops upon itself reflects the fact that the Earth is a finite world.
Three dimensional worlds can be finite in exactly the same manner, though they are much more complicated to analyse, and there is still much that mathematicians don’t know. A huge breakthrough came in 2002, when the Russian mathematician Grigori Perelman solved what is known as the Poincaré Conjecture. This problem had exactly to do with characterizing the 3D analogue of the Earth’s surface. For his incredible work, Perelman was awarded the Fields Medal, mathematics’ highest honour.
And what of our Universe? No one knows, but it is possible that if we head straight out into Space, we’ll loop back and return to our starting point. And in fact, if the Universe is positively curved then the Universe actually must be finite, and something like this must happen. (If the Universe is negatively curved then it may still be finite, but an infinite Universe is also a possibility).
And how large would such a finite Universe be? If the Universe is uniform, with the same positive curvature at each point, then we could actually answer this. All we would have to do is take a “little” triangle, say one with sides a million kilometers long. We then measure the sum of the angles in this triangle, along with the triangle’s area. Then, an analogue of our angle formula above would tell us exactly the volume of the Universe. William Blake waxed poetic of seeing the world in a grain of sand: mathematicians actually scheme to do it!
The Summer Puzzle Pages
We have strayed a long way from our playful bear puzzle. And with such wanderings in mind, we welcome you to our Summer Puzzle Pages. The puzzles have many sources. Some of the puzzles are from Gardner's collections, some from the great puzzle inventors Sam Loyd and H. E. Dudeney. Some have come from our columns and other writings, and some motivated by fun and silliness from the movies. And some, they’re from where we don’t quite know.
We now give you fair warning. Some of the puzzles are quite easy, but some are decidedly evil! All answers will appear on our website www.qedcat.com on 8 December, though the desperate and impatient may contact us earlier for hints. As always, all correspondence will be entered into. If you disagree with any of our solutions, or feel you have a better solution, please let us know.
And will you be doing serious mathematics in the pondering of these puzzles? Yes, perhaps. And we’ll. But we prefer to think of it from the other point of view. To quote Martin Gardner:
There is not much difference between the delight a novice experiences in cracking a clever brain teaser and the delight a mathematician experiences in mastering a more advanced problem. Both look on beauty bare.
We hope you agree. Good luck!
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