Can you glue edges of an octagon to make a two-holed torus!

(This week we are working on this idea in a geometry session at Cheenta.)

2

We want to take a 672 degree polynomial with integer coefficients. Suppose we plugin 673th root of 1 in that polynomial and get 0.

What can you say about the integer coefficients of this polynomial?

(This problem will come up in a Complex Number and Geometry session this week.)

3

We want to partition the number of shortest paths from (0,0) to (2019, 2019). You are allowed to walk only on the grid (lines with either x coordinate an integer or y coordinate an integer between 0 to 2019)

Can you take the vertical line x = 673 to partition these set of shortest paths into mutually exclusive and exhaustive subsets of paths?

(This problem will come up in a Combinatorics session this week.)