Fox 320

Bleaug has a solution for this general case.

http://www.8foxes.com/

Fox 319

http://www.8foxes.com/

Fox 318

http://www.8foxes.com/

Dervish Fox: I see some beauty between this one and Fox 316.

Red Fox: Would you please help me see that beauty?

Dervish Fox: The term "sin(B)+sin(C)-sin(A)" has evolved into "sin(B+C-A)" in this one. And the rest is identical.

Red Fox: And you found that beautiful?

Dervish Fox: Both answers are very different and very similar at the same time. Don't you feel anything about that?

Red Fox: I am a rational guy. I depend on my intelligence only. I can easily prove both and there is nothing magical or irrational about that. Everything's explainable. Pure and simple!

Dervish Fox: And is that all you can be? Is that the only direction you can grow? Is that who you are, or are you more than that? Why walk when you can fly? Why mutter when you can sing? Why scribble when you can paint?

Red Fox: I think I am starting to feel little bit of annoyance.

Dervish Fox: Not bad for a start my friend, not bad at all !

Fox 317

Based on our earlier analysis, the sample picture below will never happen.

By "equal-separation" we mean 120-120-120 split.
Otherwise, some may consider 12:00:00 as "equally-separated".

Peace out!

http://www.8foxes.com/

Fox 316

http://www.8foxes.com/

Fox 315

http://www.8foxes.com/

Polar Fox: I have a problem with this one!

Red Fox: What's wrong with it?

Polar Fox: Nothing moves continuously?

Red Fox: Come again?!

Polar Fox: The universe is discrete! THERE IS NO CONTINUUM.

Red Fox: Then how do the things move, flow, or slide?

Polar Fox: Nothing moves! Absolutely nothing moves!!

Red Fox: But the time ticks away, no? Clock arms DO advance.

Polar Fox: No, they don't! But they die in one instance and resurrect in the next one. In between no measurable time passes. Matter oscillates between existence and non-existence continuously.

Red Fox: Continuously? Isn't that ironic?

Polar Fox: Between any two existence, there is nothing but emptiness.

Red Fox: I am having the feeling that your intelligence fall into non-existing state just now.

Polar Fox: Hold on. I think I am about to jump back into the reality. Wait a sec.

Red Fox: Take your time dude. Just take your time!

Fox 312 - Solutions

Two solutions for Fox 312 are identified below.

Bleaug:
As in Fox 262 (see its solution here) A area (= OQR area below) extremal/constant implies P midpoint of QR. Because:
A constant <=> dA = 0 <=> area(PRR') = area(PQQ') <=> area(P'RR') = area(P'QQ') => PR = PQ when dP tends towards 0.

Hence A = 1 = 2xy or y = 1/2x i.e. answer (D).

Migue:
If area is A the answer is f(x)=y=A/2x.
f(x) is the envelope of all straight lines that close in with axes X-Y constant area equals to A.
Equation of one of this lines is: x/a + y/b = 1 (a,0) in X & (0,b) in Y.
Notice that a & b are two variable parameters.Another equation is: ab/2 = A (area of triangle that straight line draws with axes).
(1) Let f(x,y,a,b)= bx+ay-ab = 0
(2) Let g(a,b)=ab-2A = 0
(3) f'a · g'b - f'b · g'a = 0 (Jacobian of derivatives respect to a and b parameters).

f'a is the derivative of f respect a, ...

y/b - x/a = 0 (3)

If we resolve this system of equations (1),(2) & (3) we obtain:
a = 2x & b = 2y -> ab = 4xy = 2A -> xy = A/2 (equilateral hyperbola)
and y = f(x) = A/2x.

Answer is D) for A =1.

Fox 314

Bleaug adds new meaning to "laziness". Hmmm, do your lazy tricks lazy scientists to solve this one, huh? Although not confirmed, the similarity of the terms with Fox 3 is striking. Enjoy...

http://www.8foxes.com/

Fox 310

http://www.8foxes.com/

Fox 313

Bleaug submitted the following fox, saying: "a problem which can be expressed in geometric terms and that is elegantly solved using pure algebraic arguments, still giving deep insight."

General case of this problem can be found in literature (with at least 14 proofs :)
But let's try this smaller version here.

http://www.8foxes.com/

Fox 311 - Solution

This good geometric solution to Fox 311 is by Newzad.

http://www.8foxes.com/