Pascal's Triangle — Build a Pyramid of Numbers by Adding

Pascal's triangle starts with a 1 at the top, and every number below is the sum of the two just above it. It is simple to build but packed with hidden patterns.

On a large sheet, write a 1 at the top. In the next row write 1 and 1. For every row after, put a 1 at each end, and in each gap write the SUM of the two numbers diagonally above it. Fill in eight or nine rows. (You can lay out coins or counters and combine piles instead of writing, if you like.)

Add up each row: 1, 2, 4, 8, 16 -- the powers of two! Now shade in only the ODD numbers and step back: a triangular fractal pattern appears. Add the numbers along shallow diagonals and you get the Fibonacci sequence. Mark these discoveries on your triangle.

What the pyramid hides. (1) Each row is the answer to 'how many ways can you choose k things from n?' -- the numbers are the 'combinations'. (2) The rows are also the coefficients you get expanding (a+b) to a power, the binomial theorem. (3) Shading the odd numbers draws the Sierpinski fractal -- pure pattern out of pure adding. (4) Because the triangle counts how many ways things can happen, Pascal and Fermat used it in 1654 to work out the odds in games of chance, launching the whole of probability and, from it, modern statistics. Though named for Pascal, it was known centuries earlier in India, Persia and China.