__Geometry__

An archive of questions and answers that may be of interest to puzzle enthusiasts.

__Question 1 - K3,3:__

Can three houses be connected to three utilities without the pipes crossing?

_______ _______ _______ | oil | |water| | gas | |_____| |_____| |_____| _______ _______ _______ |HOUSE| |HOUSE| |HOUSE| | one | | two | |three|

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__Question 2 - bear:__

If a hunter goes out his front door, goes 50 miles south, then goes 50
miles west, shoots a bear, goes 50 miles north and ends up in front of
his house. What color was the bear?
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__Question 3 - bisector:__Prove if two angle bisectors of a triangle are equal, then the triangle is isosceles (more specifically, the sides opposite to the two angles being bisected are equal). Show Answer

__Question 4 - calendar:__

Build a calendar from two sets of cubes. On the first set, spell the
months with a letter on each face of three cubes. Use lowercase
three-letter English abbreviations for the names of all twelve months
(e.g., "jan", "feb", "mar"). On the second set, number the days with a
digit on each face of two cubes (e.g., "01", "02", etc.).
Show Answer

__Question 5 - circles.and.triangles:__

Find the radius of the inscribed and circumscribed circles for a triangle.
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__Question 6 - coloring/cheese.cube:__

A cube of cheese is divided into 27 subcubes. A mouse starts at one
corner and eats each subcube, one at a time. Can it finish in the middle?
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__Question 7 - coloring/triominoes:__

There is a chess board (of course with 64 squares). You are given 21
"triominoes" of size 3-by-1 (the size of an individual square on a
chess board is 1-by-1). Which square on the chess board can you cut out
so that the 21 triominoes exactly cover the remaining 63 squares? Or is
it impossible?
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__Question 8 - construction/4.triangles.6.lines__

Can you construct 4 equilateral triangles with 6 toothpicks?
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__Question 9 - construction/5.lines.with.4.points:__

Arrange 10 points so that they form 5 rows of 4 each.
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__Question 10 - construction/square.with.compass:__

Construct a square with only a compass and a straight edge.
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__Question 11 - corner:__

A hallway of width A turns through 90 degrees into a hallway of width
B. A ladder is to be passed around the corner. If the movement is
within the horizontal plane, what is the maximum length of the ladder?
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__Question 12 - cover.earth:__

A thin membrane covers the surface of the (spherical) earth. One
square meter is added to the area of this membrane to form a larger
sphere. How much is added to the radius and volume of this membrane?
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__Question 13 - cycle.polynomial__

What are the cycle polynomials for the Platonic solids?
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__Question 14 - dissections/disk__

Can a disk be cut into similar pieces without point symmetry about the
midpoint? Can it be done with a finite number of pieces?
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__Question 15 - dissections/hexagon:__

Divide the hexagon into:

1) 3 identical rhombuses.

2) 6 identical kites(?).

3) 4 identical trapezoids (trapeziums in Britain).

4) 8 identical shapes (any shape).

5) 12 identical shapes (any shape).
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__Question 16 - dissections/largest.circle__What is the largest circle that can be assembled from two semicircles cut from a rectangle with edges a and b? Show Answer

__Question 17 - dissections/square.70:__

Since 1^2 + 2^2 + 3^2 + ... + 24^2 = 70^2, can a 70x70 square be dissected into
24 squares of size 1x1, 2x2, 3x3, etc.?
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__Question 18 - dissections/square.five:__

Can you dissect a square into 5 parts of equal area with just a straight edge?
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__Question 19 - dissections/tesseract__

If you suspend a cube by one corner and slice it in half with a
horizontal plane through its centre of gravity, the section face is a
hexagon. Now suspend a tesseract (a four dimensional hypercube) by one
corner and slice it in half with a hyper-horizontal hyperplane through
its centre of hypergravity. What is the shape of the section
hyper-face?
Show Answer

__duck.and.fox:__

A duck is swimming about in a circular pond. A ravenous fox (who cannot
swim) is roaming the edges of the pond, waiting for the duck to come close.
The fox can run faster than the duck can swim. In order to escape,
the duck must swim to the edge of the pond before flying away. Assume that
the duck can't fly until it has reached the edge of the pond.

How much faster must the fox run that the duck swims in order to be always able to catch the duck? Show Answer

__Question 20 - earth.band:__

How much will a band around the equator rise above the surface if it is
made one meter longer? Assume the equator is a circle.
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__Question 21 - fence:__

A farmer wishes to enclose the maximum possible area with 100 meters of fence.
The pasture is bordered by a straight cliff, which may be used as part of the
fence. What is the maximum area that can be enclosed?
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__Question 22 - ham.sandwich:__

Consider a ham sandwich, consisting of two pieces of bread and one of
ham. Suppose the sandwich was dropped into a machine and spindled,
torn and mutilated. Is it still possible to divide the ham sandwich
with a straight knife cut such that both the ham and each slice of
bread are divided in two parts of equal volume?
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__Question 23 - hike__

You are hiking in a half-planar woods, exactly 1 mile from the edge,
when you suddenly trip and lose your sense of direction. What's the
shortest path that's guaranteed to take you out of the woods? Assume
that you can navigate perfectly relative to your current location and
(unknown) heading.
Show Answer

__Question 24 - hole.in.sphere:__

Old Boniface he took his cheer,

Then he bored a hole through a solid sphere,

Clear through the center, straight and strong,

And the hole was just six inches long.

Now tell me, when the end was gained,

What volume in the sphere remained?

Sounds like I haven't told enough,

But I have, and the answer isn't tough!
Show Answer

__Question 25 - hypercube:__

How many vertices, edges, faces, etc. does a hypercube have?
Show Answer

__Question 26 - kissing.number:__How many n-dimensional unit spheres can be packed around one unit sphere? Show Answer

__Question 27 - konigsberg:__

Can you draw a line through each edge on the diagram below without crossing
any edge twice and without lifting your pencil from the paper?

+---+---+---+ | | | | +---+-+-+---+ | | | +-----+-----+Show Answer

__Question 28 - ladders:__

Two ladders form a rough X in an alley. The ladders are 11 and 13 meters
long and they cross 4 meters off the ground. How wide is the alley?
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__Question 29 - lattice/area:__

Prove that the area of a triangle formed by three lattice points is integer/2.
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__Question 30 - lattice/equilateral:__

Can an equlateral triangle have vertices at integer lattice points?
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__Question 31 - manhole.cover:__

Why is a manhole cover round?
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__Question 32 - Pentomino:__

Arrange pentominos in 3x20, 4x15, 5x12, 6x10, 2x3x10, 2x5x6 and 3x4x5 forms.
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__Question 33 - points.in.sphere:__

What is the expected distance between two random points inside a sphere?
Assume the points are uniformly and independently distributed.
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__Question 34 - points.on.sphere:__

What are the odds that n random points on a sphere lie in the same hemisphere?
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__Question 35 - revolutions:__

A circle with radius 1 rolls without slipping once around a circle with radius
3. How many revolutions does the smaller circle make?
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__Question 36 - rotation:__

What is the smallest rotation that returns an object to its original state?
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__Question 37 - shephard.piano:__

What's the maximum area shape that will fit around a right-angle corner?
Show Answer

__Question 38 - Smuggler:__

Somewhere on the high sees smuggler S is attempting, without much
luck, to outspeed coast guard G, whose boat can go faster than S's. G
is one mile east of S when a heavy fog descends. It's so heavy that
nobody can see or hear anything further than a few feet. Immediately
after the fog descends, S changes course and attempts to escape at
constant speed under a new, fixed course. Meanwhile, G has lost track
of S. But G happens to know S's speed, that it is constant, and that S
is sticking to some fixed heading, unknown to G.

How does G catch S?

G may change course and speed at will. He knows his own speed and course at all times. There is no wind, G does not have radio or radar, there is enough space for maneuvering, etc. Show Answer

__Question 39 - spiral:__

How far must one travel to reach the North Pole if one starts from the
equator and always heads northwest?
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__Question 40 - table.in.corner:__

Put a round table into a (perpendicular) corner so that the table top
touches both walls and the feet are firmly on the ground. If there is
a point on the perimeter of the table, in the quarter circle between
the two points of contact, which is 10 cm from one wall and 5 cm from
the other, what's the diameter of the table?
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__Question 41 - tetrahedron:__

Suppose you have a sphere of radius R and you have four planes that are
all tangent to the sphere such that they form an arbitrary tetrahedron
(it can be irregular). What is the ratio of the surface area of the
tetrahedron to its volume?
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__Question 42 - tiling/count.1x2:__

Count the ways to tile an MxN rectangle with 1x2 dominos.
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__Question 43 - tiling/rational.sides:__

A rectangular region R is divided into rectangular areas. Show that if
each of the rectangles in the region has at least one side with
rational length then the same can be said of R.
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__Question 44 - tiling/rectangles.with.squares:__

Given two sorts of squares, (axa) and (bxb), what rectangles can be tiled?
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__Question 45 - tiling/scaling:__

A given rectangle can be entirely covered (i.e. concealed) by an
appropriate arrangement of 25 disks of unit radius.

Can the same rectangle be covered by 100 disks of 1/2 unit radius? Show Answer

__Question 46 - tiling/seven.cubes:__

Consider 7 cubes of equal size arranged as follows. Place 5 cubes so
that they form a Swiss cross or a + (plus) (4 cubes on the sides and
1 in the middle). Now place one cube on top of the middle cube and the
seventh below the middle cube, to effectively form a 3-dimensional
Swiss cross.

Can a number of such blocks (of 7 cubes each) be arranged so that they are able to completely fill up a big cube (say 10 times the size of the small cubes)? It is all right if these blocks project out of the big cube, but there should be no holes or gaps. Show Answer

__Question 47 - topology/fixed.point:__

A man hikes up a mountain, and starts hiking at 2:00 in the afternoon
on a Friday. He does not hike at the same speed (a constant rate), and
stops every once in a while to look at the view. He reaches the top in
4 hours. After spending the night at the top, he leaves the next day
on the same trail at 2:00 in the afternoon. Coming down, he doesn't
hike at a constant rate, and stops every once in a while to look at the
view. It takes him 3 hours to get down the mountain.

Q: What is the probability that there exists a point along the trail that the hiker was at on the same time Friday as Saturday?

You can assume that the hiker never backtracked. Show Answer

__Question 48 - touching.blocks:__

Can six 1x2x4 blocks be arranged so that each block touches n others, for all n?
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__Question 49 - trigonometry/euclidean.numbers:__

For what numbers x is sin(x) expressible using only integers, +, -, *, / and
square root?
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__Question 50 - trigonometry/inequality:__

Show that (sin x)^(sin x) < (cos x)^(cos x) when 0 < x < pi/4.
Show Answer