Top 10 Secrets of Pascal’s Triangle
Brett Berry
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They’re all good — and getting students to figure out why they work is a good source of thinking exercises.

Sierpinski Triangle example is also a nice spreadsheet exercise — and you can look at Pascal’s triangle modulo numbers other than 2 to get other interesting fractals.

One of my favourites that you didn’t mention is that you can extend Pascal’s triangle upwards too. First add the normally invisible zeros.

... 0 1  0  0  0  0 ...
... 0 1 1 0 0 0 ...
... 0 1 2 1 0 0 ...
...

Then, since column 0 is zero, you can work out row by row what all of the other columns must be.

...
... 0 1 -2 3 -4 5
... 0 1 -1 1 -1 1 ...
... 0 1 -1 1 -1 1 ...
... 0 1 0 0 0 0 ...
... 0 1 1 0 0 0 ...
... 0 1 2 1 0 0 ...
...

Then the connection is made to the expansion of binomials and hence to the geometric series, calculus and all sorts of other fun. Of course, since Newton played with generalised binomials before inventing calculus, this is a well trodden path.