Ooh, I get to reply to one of your additions before justlooking. That's unusual.

Anyway, it seems very good. I can't see any mistakes and you've covered it fairly comprehensively. All I can suggest is maybe some pictures of patterns you can get for different multiples (or if you're feeling adventurous, maybe even some kind of program that will show the multiple patterns of a number that we can specify), and also to include the fibonacci pattern that you can get from knight's move, because that's just uber-cool.

Edit: That page links to the Tetrahedral Numbers page and there's a mistake there. The illustration shows a tetrahedron of height 5.

Like always, brilliant work.

Pascal's triangle is so cool! There are so many interesting things you can find hidden in it!

You covered quite a lot but there's far too many to list...
• Probability - rows represent the number of ways coins can be tossed up.
• Each row is 11^n, n being the row number
• In the 'triangular numbers' row you have highlighted, if you add two numbers next to each other you get the sequence of square numbers.
• If you separate the numbers into rows (like in the picture below) and add the numbers in each row... you get the Fibonacci sequence.

Oh, and mathsy - you beat me to it!

Uploaded Images

Great!
ryos is right, nowhere in the page has it been mentioned that the numbers in the Pascal's triangle are coefficients of the first, second, third,......terms in the binomial expansion. Illustrations of (a+b)², (a+b)³, (a+b)^4, (a+b)^5 etc. may also be given in the page. The numbers in the rows of Pascal's triangle can be compared to nC0, nC1, nC2, nC3,....nCr etc. (like 3C0, 3C1, 3C2, 3C3 for (a+b)³) to explain their being equivalent.

Lots of great ideas there. Just one thing:

justlookingforthemoment wrote:

• Each row is 11^n, n being the row number.

Only for the first few rows. The pattern is ruined when you get rows that contain numbers more than 9.