# Geometry and topology

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

Bridging a river with unit rods

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]

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### Fly on 2 x 1 x 1 brick

Fly on 2 x 1 x 1 brick

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?

### Lawnmower

Lawnmower

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]

### Offball: object whose projections are disks

Offball: object whose projections are disks

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]

### Paper-folding

Paper-folding

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]

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### Platonic solids inside and outside a unit sphere

Platonic solids inside and outside a unit sphere

Q: What are the volumes of the platonic solids inscribed in/circumscribed about a unit sphere?
[TB, 11 Apr 1996]

### Portion of planets from which others are invisible

Portion of planets from which others are invisible

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?

### Sofas and ladders round corners

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?

### Sphere packings

Sphere packings

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

### Sunrise in the west

Sunrise in the west

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

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