Sections: Pure, Games, Geometry, Physics, Large.

If you make any progress on any of these puzzles, or if you'd like to contribute a new one, then please do tell me about it.

Puzzles in this section:

- Bridging a river with unit rods
- Fly on 2 x 1 x 1 brick --- spoiler
- Lawnmower --- spoiler
- Offball: object whose projections are disks --- spoiler
- Paper-folding
- Platonic solids inside and outside a unit sphere --- spoiler
- Portion of planets from which others are invisible --- spoiler
- Sofas and ladders round corners --- spoiler
- Sphere packings --- spoiler
- Sunrise in the west --- spoiler

Q: ??? Consider a river of width 1 with a right-angle bend. Given two
rods of length 1 we can bridge the gap at the bend: put one rod
diagonally across the outside of the bend, then we have a gap of length
sqrt(2) - sqrt(1/2) = 0.707... to bridge with the second rod. Now
consider wider rivers, but allow ourselves an arbitrary supply of
rods. Is there a bound on the width of river that we can bridge?

[Andrew Usher, 30 Apr 1996]

Q: A (pointlike) fly is sitting at a corner of a 2 x 1 x 1
brick. What's the furthest point from it, measuring along the
surface?

[rec.puzzles, 15 Apr 1996]

Q: ??? On a centrally-symmetric body, must it be the case that every, or indeed any, longest geodesic's endpoints are antipodes?

Spoiler --- Top of page

Q: ??? An automatic lawnmower works by moving in a straight line until
it hits an edge of the lawn, turning through a fixed angle, and starting
again (in fact it should turn through the angle repeatedly until it's
heading back onto the lawn). Say it succeeds if it eventually passes
within epsilon of every point of the lawn. Does there exist a lawn (and
a starting point) for which it fails? For rational angles the answer is
yes - pick a suitable polygon - but what about irrational angles? What
if we insist the lawn is simply connected? Convex?

[TB, 12 Jul 1996]

Spoiler --- Top of page

Q: ??? An "offball" is an object such that its projections in three
orthogonal directions are unit disks. What's the max/min volume it can
have, assuming it's convex?

[rec.puzzles, 16 Jul 1996]

Spoiler --- Top of page

Q: ??? Given a nice subset of the plane, is there a way to fold it (origami-style) that increases the circumference? What's the best definition of "nice" here?

What if we allow a general piecewise-isometric continuous map? Is
this the same thing (ie are there such maps which can't be produced by a
sequence of physical folds)?

[rec.puzzles, 15 Apr 1996]

Q: What are the volumes of the platonic solids inscribed
in/circumscribed about a unit sphere?

[TB, 11 Apr 1996]

Spoiler --- Top of page

Q: Consider a system of `n` identical stationary
spherical planets. For each planet, find all points such that none of
the others is visible, and paint them red. How much paint do you
need?

Equivalently: shrink-wrap a finite set of identical spheres (ie take their convex hull). What is the area of contact between the spheres and the wrapping?

Spoiler --- Top of page

Two corridors of widths `a` and `b` intersect at
right angles.

Q: What is the longest ladder (line segment) you can take round the corner?

Q: What is the largest (wrt area) rectangular sofa you can take round the corner?

Q: ??? What about a non-rectangular sofa? What if it must be convex?

Spoiler --- Top of page

Q: What are the densities of various nice packings of spheres (cubic, triangular close-packed, diamond molecule, etc)?

Spoiler --- Top of page

Q: Where and when on Earth can the Sun rise in the west?

Spoiler --- Top of page

Sections: Pure, Games, Geometry, Physics, Large.

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and was last modified on Sat 23 Jan 2021.

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