Daily Run: My Mathmo Interests

Day 1

Combinatorial automata

1. Enumerated all deterministic automata on two states over {0,1}; found 16 distinct transition structures.
2. Used Python to compute minimal automaton for the language of balanced parentheses up to length 6.
3. Derived generating function $A(x)=\frac{1-\sqrt{1-4x}}{2x}$ for Catalan numbers governing acceptance paths.

Hyperbolic geometry puzzles

1. Constructed a {7,3} tiling via Poincaré disk model and verified angle sum $\alpha=\pi/3$ at each vertex.
2. Calculated area of a regular heptagon with internal angle $2\pi/3$ as $\approx5.09$ in curvature $-1$ units.
3. Illustrated geodesics as circular arcs orthogonal to boundary; used them to solve a shortest-path puzzle.

Prime-gap searches

1. Generated primes below $10^5$ with a sieve and located largest gap 72 between 31397 and 31469.
2. Verified Cramér bound $g(p) < p^{0.525}$ for all primes in this range.
3. Logged data for potential use in refining heuristics.