What’s your favorite math

hmmm.... i'm a pascal's triangle enjoyer. lots of fun identities, and i dont come across binomial expansions that often but i always use pascal's triangle for that. if you highlight all the odd numbers, you get sierpinski's triangle. the diagonals give you the triangular numbers, the tetrahedral numbers, and so on. the rows sum to powers of 2. the hockeystick lemma. probably some other stuff that i dont know.

this is a hard question because the word "favorite" makes me think about picking the absolute best one and having a passionate explanation for why i like it so much. pascal's triangle is not groundbreakingly cool or advanced, but it's kind of fun.

other math party tricks that i like:

the dropping sticks approximation of pi (buffon's needle)

the Long Line

the well ordering theorem

axiom of choice proofs

55, 165, 330, and 462 are all divisible by 11. This has to do with the binomial coefficient definition, in which you'll see that if unless we choose either all the elements, or none of them, the factor of our prime number is guaranteed not to cancel out from the numerator's factorial.

We can then use the fact that the sum of all the elements in the nth row of pascal's triangle adds up to 2^n. Once we remove the 1's on each end of the triangle (hence the minus 2) we know that for prime numbers p that all elements are divisible by p, their sum is as well!

I started to craft a visualization model for a Pascal"s 4-simplex.

Now I need to insert the 12 triangle disks. (I need to puzzle them into each other like a kind of interwebbing. )

Multinomial coefficients and Pascal's Simplex

For binomials [ (a+b)ⁿ ] the use of Pascal's triangle is helpful. In Pascal's triangle each single row in the triangle defines the coefficients of binomials of each n-value.

For trinomials [ (a+b+c)ⁿ ] this pattern can be extended to a 3-dimensional Pascal's tetrehedron, where each level (and hence a complete triangle) in that tetrehedron defines the coefficients of trinomials of each value of n.

MATH POSITIVITY YES THANK YOU I LOVE IT HERE

I agree, Pascal's triangle is super cool!! It's math and it's a combinatorics thing and it's A TRIANGLE WHAT MORE COULD YOU POSSIBLY EVER WANT????

Thank you so much for sharing this with me.

On a related note, let's take a moment to appreciate Sierpinski's Triangle, which is what you get when you colour in all the odd numbers in Pascal's Triangle :)