With this in mind, let's solve a couple of last week's questions. First of all, into what number of equilateral triangles can an equilateral triangle be cut? Reflection shows that you can't cut such a triangle into two equilateral triangles. How do we know? Think of the opposite. After all, if you have managed to dissect such a triangle into two similar ones, reassembling them is no problem.
This is a typical and often useful mathematical move: stand the problem on its head and get a completely different, but not unrelated, question. Suppose I can order up any two equilateral triangles that I like. Can I ever join them up to make a bigger equilateral triangle? Well, the only way they can fit together is edge to edge, which results in a diamond if sides are the same, and a more angular beast if they are not.
Using this reverse assembly method (a), it is easy to see that 3 is also not possible. On the other hand, 4 is, though only just, to quote the old joke. Now we can always divide an equilateral triangle into an even number of triangles greater than 4 by cutting off a strip with an odd number of triangles in it (b). We can also add 3 more triangles by dividing any of the component triangles into 4 equal triangles (c). So we can manage any even number greater or equal to 4, and any odd number greater than or equal to 4 + 3 = 7. So 2, 3 or 5 apart, any number is possible.
What of the climber who leaves base camp at noon and reaches the summit by an ever so circuitous route at midnight. Next day he leaves at noon and arrives back at camp at midnight. Is there a time on the two days when he was at the same height above sea level?
How can we look at this differently? One trick is to imagine two mountaineers, one starting from the top and one from the bottom. No matter by what routes and rates they proceed, so long as the one that started at the top ends at the bottom, and the one that started at the bottom ends at the top, there must be at least one time on the two successive days at which they were both at the same height.
And the anagram of MOUNTAINEER? That was ENUMERATION.
Points to ponder
1 Hmm. What old joke might that be?
2 Divide an equilateral triangle into 11 equilateral triangles.
3 How many different flat shapes can be made by joining 4 identical equilateral triangles edge to edge?
4 Divide the shape shown (d) into a number of pieces identical in shape with the original.
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