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# Doodling in math: Sick number games

I don't even know if this makes sense. Boo cold. http://en.wikipedia.org/wiki/Ulam_spiral Doodling in Math Class videos: http://vihart.com/doodling. Created by Vi Hart.

## Want to join the conversation?

• I think that if you highlighted the prime numbers on pascal's triangle you would only get primes on the sides, just to the right (or the left if you're on the other side) of the 1's. Am I right?
• You are right actually. I'm not sure of an obvious way to prove it, but it's not too hard to prove if you know a little combinatorics (Sorry if you don't, maybe someone else will find it interesting?):

Basically, for the nth row of the triangle (Where the first row has n = 0, so each row has n as its second number), and the kth element in that row (Where the first element of each row, the 1, has k = 0), that number in the triangle will be "n choose k", which is equal to n! / (k! * (n-k)!). For k = 1, this is just equal to n, which can of course be prime. The same goes for k = n-1, then it's also just n, and these are the numbers you mentioned. But for any other k, we're going to have this number being some product of numbers since the factorials won't completely cancel. And so they're composite.

Sal's mentioned elsewhere that he plans to eventually add a Combinatorics playlist - Maybe he'll mention the connection between Pascal's Triangle and "n choose k" (Which is the number of distinct ways to choose k elements from a set with n elements) in them. The reason that this connection exists is because, if (n k) = "n choose k", then the following is true: (n k) + (n k+1) = (n+1 k+1). If you're looking at Pascal's triangle and find (n k)'s spot and (n k+1)'s spot, you'll see that (n+1 k+1)'s spot is right below them. That, combined with (0 0) = 1 (For the very top of the triangle), lets this relation exist.

As an example: For the 5th row, 3rd element, we have (5 3) = 5! / (3! * (5-3)!) = 5! / (3!2!) = 5*4*3*2*1 / 3*2*1 * 2*1 = 5*4 / 2*1 = 5*2 = 10, and 10 is indeed the 3rd element of the 5th row (Remember, the top row has n = 0 and the first element of each row has k=0).

Here's an image showing how the (n k)'s are arranged:
http://i.imgur.com/C9mbD.gif
• I actually used the spiraling number game and instead of circling the primes, I circled the even numbers and connected each circled number. If you do this you get a weird square spiral. This works with circling odd numbers too.
• Why do pascels triangle show up so much in math?
• im a person and i was wondering how vi hart can talk so fast is it genes or is it fast forward or is it ?
• Maybe because if you do notice that Vi Hart speeds up some of her videos maybe she sped up all of her videos?
(1 vote)
• Who is Stanislaw Ulam?
• Number theorist, designer of the Hydrogen bomb, and more
• wow! this is really fun because we learn and play and doodle! i am obsessed with doodling in my science homework
• My question is is how does she draw so perfect with both hands?
• How much wood can a wood chuck chuck if a wood chuck could chuck wood