Little math problems July 2019

[Albert Chan]

If only orthogonal and 45 degree diagonal moves allowed:

total distance = k*(X³ - X), where k = (1+√2)/3 ≈ 80.47%

Compare against orthogonal moves only, distance saving = (2-√2)/3 ≈ 19.53%

Prove:
Trivia 1: total orthogonal moves = total diagonal moves
Trivia 2: diagonal move equivalent to 2 orthogonal moves
Trivia 3: from post #7, f(n) = (X³ - X)/8, divisible by 3

→ k = (1+√2) / (1+2) = (1+√2)/3