Little math problems July 2019

[Albert Chan]

(08-07-2019 01:23 PM)John Keith Wrote:  Brute-force program for the diagonal case, odd sizes only. Exact mode for symbolic results:

First few results:

3: '8+8*√2' ~19.314
5: '24+24*√2+16*√5' ~93.718
7: '48+48*√2+(16+16*√2)*√5+16*√13' ~259.94
9: '160+80*√2+(48+16*√2)*√5+16*√13+16*√17' ~554.72

Compared against X^3-X, direct-route distance savings:

3: 19.53%
5: 21.90%
7: 22.64%
9: 22.96%

Distance saving plateaued after that, with maximum savings of 23.48% for huge X

Update: maximum saving rate calculated by switching sums to integral.
XCas> d := 8*int(int(sqrt(a*a+b*b), a= -1/2 .. x/2), b= 1/2 .. x/2)
XCas> limit(1 - d/(x^3-x), x=inf) → ln(√(2)-1)/3 + (-√(2)+3)/3 ≈ 23.48%