Can you place three chess queens on a 6x6 board
so that all vacant cells are attacked? A vacant cell is considered to be
attacked when it is in the same row, column or diagonal with at least one
of the queens.

Place four chess queens on a 7x7 chessboard so
that all vacant cells are under attack, and no queen attacks another. A
vacant cell (or a queen) is under attack when it is in the same row,
column or diagonal with at least one of the queens.

Draw this pattern of four crossing circles with pencil in one continuous
line so that you don't take the pencil point off the paper. You aren't
allowed to go over any part of the line twice, or cross it.

Roll a circle one full turn along a line. The line is equal to the
circumference of the circle. Then imagine rolling more circles of various
sizes along the line always equal to the circumference of the circle. What
can you tell about the relationship between the circumference and the
diameter of a given circle? Is it the same for all circles?

This interesting question crops up in
recreational mathematics literature: Using each of the first
consecutive integers (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) once each to
create the dimensions of five rectangles, how many combinations of
those five rectangles can be assembled into a square? In each of these
four cases, place the colored rectangles at right on the grid at left.

Here is a harder variation of
Circling Numbers. Write four 1’s, three 2’s and three 3’s in the empty
circles so that no three numbers that are beside each other have a sum
that is divisible by three.

There are seven lights in a circle – as shown in the upper illustration.
They are all turned off and the problem is to turn all of them on. Each
light has a switch. The trouble is that the electrician didn’t use proper
switches. When you press a switch, if a light is on it turns it off, and
if a light is off, it turns it on. The electrician also made a mistake in
writing the switches. If you press the switch for one light, it not only
changes that light, but also the lights on either side. So, if you pressed
the switch for light B in the left diagram of the bottom Example
illustration, lights A and B would turn on and light C would turn off as
shown in the right diagram of that illustration. Can you figure out a way
to turn all the lights on?

The dragon of ignorance has three heads and three tails. You can slay it
with the sword of knowledge, by chopping of all of its heads and all of
its tails. With one stroke of the sword, you can chop off either one head,
two heads, one tail, or two tails. But the dragon is hard to slay! If you
chop off one head, a new one grows in its place. If you chop off one tail,
two new tails replace it. If you chop off two tails, one new head grows.
If you chop off two heads nothing grows. Show how to slay the dragon of
ignorance. How many chops do you need?

You have many squares and equilateral triangles all with the same side
length. In this problem we want to build polygons by putting the squares
and triangles as shown in the upper illustration. The illustration shows
how we can build polygons with 3, 4, 5, or 6 sides. But, you are not
allowed to build polygons like the ones in the bottom illustration. If you
could put an elastic band around those polygons, there would be empty
spaces. Can you build polygons with 7, 8, 9, 10, 11 and 12 sides like the
ones in the upper illustration?

Draw the figure shown in the illustration; it's a pentagon with each its
vertex connected with every other. The question is how many different
triangles are hidden in this figure?

Triangular numbers can be found by stacking a group of objects in
equilateral fashion – two objects are placed under one, three objects are
placed under the two that are under the one and so on – as shown in the
illustration. The fourth triangular number – 10 – was called the tetraktys
by Pythagoras and his followers. They considered it sacred and revered it.
What is so special about the triangular pattern? Can you work out how many
objects there are in the eighteenth triangular number?

You have 7 chess knights. Place one of them on any empty cell of a 3x3
board and then move it to another empty cell using knight's move (some
examples of such moves are shown in the bottom diagrams, a and b). Then in
exactly the same way place and move another knight. Repeat your
"place-move" steps till you have all 7 knights placed on the board.

Place eight coins in a row as shown in the illustration. The object is to
make from all the coins four stacks of two coins each and it should be
done in four moves only. Every move consists of jumping of a coin over any
two coins (no matter lying flat or in a stack) in one direction, and
stopping on the top of the next coin.

There are three pairs of balls - red, white, and blue. In each pair one
ball is a little bit heavier than another one. All the heavy balls weigh
the same, and all the light balls weigh the same. Also you have a balance
scale. Now, in just two weighings you have to determine the light and the
heavy balls in each pair. How can it be done?

Puzzle 1. Count how many perfect squares of all possible sizes are
hidden in the cross of dots on the left. A square is counting if any four
dots are placed exactly in its respective corners. Puzzle 2. It is more difficult than previous one. You have to
remove exactly 6 dots so that any four dots from those remaining would not
lie in the corners of a square. So you'll get the "no-squares" position
for which there are no four dots that form a perfect square.

This puzzle is a variation of the classic Tower of Hanoi. You can play it
on several different levels of difficulty and with variant sets of rules.
The puzzle begins with a stack of disks in the left hand column as shown
in the insets below. Your object in each puzzle is to transfer the disks
to the right-hand column, keeping the same numerical order. The basic rule
is do not place a disk on another disk of smaller value. Otherwise,
shuttle the disks one at a time among the three columns until you have the
proper arrangement in the right-hand column. Puzzles 1, 2, 3 and 4 (see
first diagram below, left) – find the minimum number of moves to transfer
2, 3, 4 and 5 disks respectively to the right-hand column. Puzzle 5
(second diagram below) – find the minimum number of moves to transfer the
four disks, observing an additional rule that a disk can not be placed on
another disk of the same color. That means that disk 1 cannot be placed on
disk 4.

This puzzle is a variation of the classic Tower of Hanoi. You can play it
on several different levels of difficulty and with variant sets of rules.
The puzzle begins with a stack of disks in the left hand column as shown
in the insets below. Your object in each puzzle is to transfer the disks
to the right-hand column, keeping the same numerical order. The basic rule
is do not place a disk on another disk of smaller value. Otherwise,
shuttle the disks one at a time among the three columns until you have the
proper arrangement in the right-hand column. Puzzles 1, 2, 3 and 4 (see
first diagram below, left) – find the minimum number of moves to transfer
2, 3, 4 and 5 disks respectively to the right-hand column. Puzzle 5
(second diagram below) – find the minimum number of moves to transfer the
four disks, observing an additional rule that a disk can not be placed on
another disk of the same color. That means that disk 1 cannot be placed on
disk 4.

Surveyors in ancient Egypt had a simple tool for making near-perfect right
triangles: a loop of rope divided by knots into twelve equal sections.
When they stretched the rope to make a triangle whose sides were in the
ratio 3:4:5, they knew the largest angle was a right angle. Can you fit
the five pieces at bottom into the two smaller squares above the right
triangle? Then can you fit the same five pieces into the larger square
below the right triangle? If you can do both, what have you done?

The objective of this puzzle is to make from all the nine pieces shown in
the illustration a 5x5 square with exactly one of each color in every row
and column.

The goal is to draw a path from A to B so that it goes through each empty
square of the board only once and has no self-crossings. Your path must go
horizontally and vertically (never diagonally), and it has to avoid the
four squares with the mushrooms in them. There is an additional condition:
the second square of the path must be exactly under the A square as shown
in the illustration.

The object of this puzzle is to figure out which of the 12 patterns in the
illustration can't be drawn with pencil in one continuous line so that you
don't take the pencil point off the paper. You are not allowed to go over
any part of the line twice, or cross it.

Can you attach the eight semicircles to the pegs on the line so that no
two semicircles cross? Although semicircles may hang from either side of
the line, no two are allowed to share a peg.

Two sidings join the main track of a railroad. They meet together and lead
on to a dead end. All this is shown in the center of the illustration
above. The dead end is long enough to hold a car or an engine at a time.
Currently a blue car is located on the left siding and a green car - on
the right siding. An engine is located on the main track on the midway
between the two cars. The objective of the challenge is to exchange the
positions of the cars and then return the engine to its initial position
as is shown in the lower right corner. And it has to be done with the
smallest number of couplings and uncouplings.

Can you draw the ancient symbol shown in the illustration with one
continuous line, making the minimum possible number of turns? You're
allowed to go over the same lines more than once.

Nine disks are arranged as shown, with the eye of the snake on the left.
The object of this puzzle is to transfer the eye to the other end in the
fewest possible number of moves. (In this puzzle a move counts as an
instance in which you place a disk in one of the three spaces in the side
of the snake.)

The object is to make from all the six pieces a symmetric
capital H. You're allowed to rotate the pieces as you wish and even turn
them over, but they must not overlap each other in the final letter.

Place three black and three white chess knights on a 3x4 board as shown in
the upper illustration. The object is to exchange black and white knights
in the fewest possible number of their moves (some examples of such moves
are shown in the bottom diagrams, a and b). Moves should be made by black
and white knights in turn. And after every move none of the knights should
be under attack of any of the knights of the opposite color. Only one
knight can be on a square at the same time.

The minimal strip that can be folded into a one-unit cube is one unit wide
and seven units long (1x7) as shown in the illustration. The object is to
show how it can be done.

There are a 3L container and a 5L container available as shown in the
illustration. The object is to measure exactly 4L of water with the help
of these two containers and some immense supply of water (say, river or
lake). How this can be done?

A set of regular hexagons with sides 3, 4 and 5 is extended on the sides
of a right triangle. This seems to suggest that the Pythagorean theorem
can be extended beyond squares and is valid for hexagons as well. Is that
really the case?