Fox 332

Won't you come and see?
Some fly away,
some remain lonely.
-- Himalayan Fox

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Fox 54 - Discussion

Bob Ryden provided a solution below. This has been the first attempt to solve this fox. His solution has not been confirmed yet, but it is published here to start the discussion. Wanna say somethin', comment it out!

Let the radius of the spheres = 1.
Inside is a tetrahedron whose vertices are the centers
of the spheres. Its edge length = 2
and its volume = (1/8) √3

On each face of the tetrahedron, build a triangular prism.
Volume of each prism = (1/2) 2 (√3) 1
Total volume of four prisms = 4√3

On each edge of the tetrahedron, build a cylindrical sector.
The angle of the sector = 360 – 90 – 90 – dihedral angle of the tetrahedron
= 360 – 90 – 90 – arccos (1/3) = approx. 109.47°
Length of each cylindrical sector = 2, radius = 1
Total volume of the six sectors

= 6 π (1^2) 2 (109.47 / 360) ≈ 11.46

Finally, there are pieces of the four original spheres that are not covered by any of the above. The four pieces together make one complete sphere, V = (4/3)π.

Total volume is the sum
tetrahedron + 4 prisms + 6 cylindrical sectors + sphere
= approx. 23.52

Fox 254

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This is just a model for a very basic structure in Organic Chemistry. Real molecules are more dynamic than this. Atoms vibrate even at very low temperatures.

Fox 245

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Ignore the curvatures, SpongeBob gets wobbly when he's overjoyed.
By the way, SpongeBob and Patrick are owned by Viacom International, Inc.
Freedom to the Bikini-Bottom!

Fox 243 - Solution

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Bleaug claims the following:

Almost a pure geometric proof this time (only requires Pythagorean theorem).

- TIJ is the section of the pyramid by a plane orthogonal to base and parallel to BC and AD

- by construction, TI is orthogonal to AB, and TJ to CD

- hence: AT^2-BT^2 = AI^2+IT^2-BI^2-IT^2 = AI^2-BI^2, similarly DT^2-CT^2 = DJ^2-CJ^2

- since AIB is a translation of DJC: AT^2-BT^2 = DT^2-CT^2

- from which we get DT^2 = AT^2-BT^2+CT^2 = 9-2+1 = 8

Fox 244

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Look into thy heart, and see,

it desireth to fly in Heavens,

whilst there are knots pulling thee

to cruelly darketh corners!
--- Dervish Fox

Fox 243

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Answer not confirmed again.

Fox 240

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Now this asks little more than geometry.

Note that the random process described in the question is discrete.

Ladybugs "appear" in next vertices at the same time.

Fox 54

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Fox 53

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Fox 57

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Fox 94

Observe how the rod moves inside the prism.
What can be the maximum length?