Thursday, July 17, 2008

Mathmatical Rice Krispie Treats

A square plate and a coincidental cutting and stacking of (approximately) square Rice Krispie treats produced a tasty square pyramid with some interesting mathematical qualities. The layers of the pyramid are each a square, 1, 4, 9, 16 (Integer sequence A000290), they form a pyramid and summing each subsequent layer yields the (obviously called) square pyramid numbers, 1, 5, 14, 30 (Integer sequence A000330).

Two other interesting properties of the square pyramid numbers:
  • The sequence contains exactly one square greater than 1, namely 4900.
  • Gives number of squares formed from an n X n square. In a 1 X 1 square, one is formed. In a 2 X 2 square, five squares are formed. In a 3 X 3 square, 14 squares are formed, and so on.
Compare the suare pyramid number sequence with the triangular numbers, 1, 3, 6, 10 (sequence A000217). These are like the bowling pin two dimensional pyramid, but each level has fewer treats than the square pyramid stacking above. I think I like the quantity from the square pyramid better.

The On-Line Encyclopedia of Integer Sequences is fun for looking for interesting sequences of integers, if you like that kind of thing. They even have the Lost numbers (4, 8, 15, 16, 23, 42, sequence A104101). The encyclopedia generates lists and graphs of number sequences, but even more interestingly, though less useful, it will generate a sound file of the sequence as played on an instrument of your choice.

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