Monday - Friday, July 16, 2004
For the first few days, Brian led us through diverse examples where a certain series of numbers arose : 1, 2, 5, 14. Arranging the numbers from 1-8 in a 2x4 grid in increasing order, "planting" binary trees with three internal vertices, triangulating the hexagon, lattice paths of a 4x4 grid that stay below the diagonal, groupings of a nonassociative binary operation on four letters -- all of them were numbered by this sequence. He then led us through a demonstration of how there was a bijection (a one-to-one and onto mapping) between them, so that the representation in one example corresponded to a representation in another. A direct formula was finally developed from the lattice paths. Other topics included partitions, figurate numbers, and hexagonal automata (a.k.a. snowflakes).
We've also done a number of kinestetic exercises in combinatorics, one involved passing a string amongst a circle of people to create patterns in the circle and the other, introduced by Peter, that had a circle of people blinding extending their hands into the centre, locking hands with whomever they found and seeing (upon opening our eyes) how many rings of people we had holding hands. [View photos]. Many conjectures resulted but no proofs have yet been posted.
Finally, by Friday, we'd beeen introduced to enough ideas that we began to settle into topics : three main groups arose : one on developing worksheets for that curious sequence, one on on-line applets for describing the sequence, and the other on more general combinatoric ideas.
Oh, and that sequence? If you're curious, the next number is 42, the answer to life, the universe & everything. For more information, why not visit Sloane's The On-Line Encyclopedia of Integer Sequences or research it yourself! Stay posted, otherwise!
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This material is based upon work supported by the National Science Foundation under Grant No. 0314808.