I randomly made this thing out of boredom:
Cute, but it’s probably a sign that I need some better ideas for hobby programming projects xD
An interesting theorem related to Pascal’s triangle is the hockey stick theorem or the christmas stocking theorem. This theorem states that the sum of a diagonal in Pascal’s triangle is the element perpendicularly under it (image from Wolfram Mathworld):
So here the sum of the blue numbers , 1+4+10+20+35+56, is equal to the red number, 126. It is easy to see why it’s called the hockey stick theorem – the shape looks like a hockey stick!
An alternative, algebraic formulation of the hockey stick theorem is follows:
But this works in two ways, considering the symmetry of Pascal’s triangle. The flipped version would be:
Using Pascal’s identity, it is fairly trivial to prove either identity by induction. Instead I present an intuitive, animated version of the proof: