Wednesday, March 10, 2010

Thinking Caps On!

On our coach journeyfrom Liverpool to London, the game of 'I Spy' having palled, my elderly friend Dr Isaacs propounded the following divertissement.
You are to imagine a staircase of an hundred steps. On the bottom step sits a pigeon; on the next, two pigeons; on the next, three, and so forth, all the way to the top, where sit an hundred pigeons.  How many pigeons are there in toto?
Well! I began by adding one, plus two, plus three &c, until my antique companion laughed me quite out of countenance and told me that there was a much easier way, supposedly discovered by the great mathematician Carl Gauss when but a boy (Though this may be a fable- see this link for someone who thinks so: http://www.americanscientist.org/issues/pub/gausss-day-of-reckoning/1).
I had much to do with this problem, and required many hints, but at last I took a ribbon from my hair and wrote the numbers along it.  I then folded the ribbon in half so that the first number lay upon the last and what do you think? I had fifty pairs of numbers, of course, and each pair added to one hundred and one.  If you multiply one hundred and one by the number of pairs (fifty) you come to five thousand and fifty. I happened to remark to Dr Isaacs that this was all very well, but how can one know that this will always happen without adding the numbers all together in the way I first tried.  He was quite ecstatic and declared he liked my question better than my answer, for I demanded proof, and that is the first step to being a mathematician! For my proof, he simply showed me this picture, which is done for ten, not an hundred, but you can see the way of it readily enough.

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