The hockey stick theorem: an animated proof

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:

$\displaystyle\sum_{i=0}^{k} \binom{n+1}{i} = \binom{n+k+1}{k}$

But this works in two ways, considering the symmetry of Pascal’s triangle. The flipped version would be:

$\displaystyle\sum_{i=0}^{k} \binom{n+1}{n} = \binom{n+k+1}{n+1}$

Using Pascal’s identity, it is fairly trivial to prove either identity by induction. Instead I present an intuitive, animated version of the proof: