## Geometry

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

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? Show Answer

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. Show Answer

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? Show Answer

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? Show Answer

Question 8 - construction/4.triangles.6.lines
Can you construct 4 equilateral triangles with 6 toothpicks? Show Answer

Question 9 - construction/5.lines.with.4.points:
Arrange 10 points so that they form 5 rows of 4 each. Show Answer

Question 10 - construction/square.with.compass:
Construct a square with only a compass and a straight edge. Show Answer

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? Show Answer

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? Show Answer

Question 13 - cycle.polynomial
What are the cycle polynomials for the Platonic solids? Show Answer

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? Show Answer

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). Show Answer

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.? Show Answer

Question 18 - dissections/square.five:
Can you dissect a square into 5 parts of equal area with just a straight edge? Show Answer

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. Show Answer

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? Show Answer

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? Show Answer

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?

```             +---+---+---+
|   |   |   |
+---+-+-+---+
|     |     |
+-----+-----+

```

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? Show Answer

Question 29 - lattice/area:
Prove that the area of a triangle formed by three lattice points is integer/2. Show Answer

Question 30 - lattice/equilateral:
Can an equlateral triangle have vertices at integer lattice points? Show Answer

Question 31 - manhole.cover:
Why is a manhole cover round? Show Answer

Question 32 - Pentomino:
Arrange pentominos in 3x20, 4x15, 5x12, 6x10, 2x3x10, 2x5x6 and 3x4x5 forms. Show Answer

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. Show Answer

Question 34 - points.on.sphere:
What are the odds that n random points on a sphere lie in the same hemisphere? Show Answer

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? Show Answer

Question 36 - rotation:
What is the smallest rotation that returns an object to its original state? Show Answer

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? Show Answer

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? Show Answer

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? Show Answer

Question 42 - tiling/count.1x2:
Count the ways to tile an MxN rectangle with 1x2 dominos. Show Answer

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. Show Answer

Question 44 - tiling/rectangles.with.squares:
Given two sorts of squares, (axa) and (bxb), what rectangles can be tiled? Show Answer

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? Show Answer

Question 49 - trigonometry/euclidean.numbers:
For what numbers x is sin(x) expressible using only integers, +, -, *, / and square root? Show Answer

Question 50 - trigonometry/inequality:
Show that (sin x)^(sin x) < (cos x)^(cos x) when 0 < x < pi/4. Show Answer