Labs / Mathematics
Pascal's Triangle
Every number is just the sum of the two above it, yet that tiny rule hides binomial coefficients, powers of two, and a fractal. Click any cell to light up the paths that build it, then colour the triangle by divisibility to watch a pattern emerge.
C(5, 2) = 10row sum 2⁵ = 32term 10·a³b²
What to try
- Click a cell and count the highlighted paths from the tip. Does the number of routes match the value inside?
- Add up a whole row in your head. Why does every row total land on a power of two?
- Switch on Mod 2. Which cells stay coloured, and why does the leftover shape look like a Sierpinski triangle?
- Compare the Mod 3 and Mod 5 patterns. Does a bigger divisor make the coloured triangles bigger or smaller?
- Pick a cell near the edge versus one in the middle. Where do the largest coefficients — and the most paths — live?