## Read an Excerpt

#### 100 Geometric Games

**By Pierre Berloquin, Denis Dugas**

**Dover Publications, Inc.**

**Copyright © 1976 Pierre Berloquin**

All rights reserved.

ISBN: 978-0-486-80567-2

All rights reserved.

ISBN: 978-0-486-80567-2

CHAPTER 1

**PROBLEMS**

**Game 1**

Twelve points are connected above by five straight lines, without raising the pencil.

You can do better: connect the same twelve points:

• without raising your pencil

• in five straight lines

• ending on the first point, thereby making a closed circuit

• without going through any point twice (but the lines can cross each other).

How?

**Game 2**

Only two squares are shown out of many squares whose vertexes lie on four of the twenty points in the figure.

How many points do you have to erase so that *no square* can be formed on any four of the remaining points?

**Game 3**

Is the rope a simple loop?

or is it knotted once?

or is it knotted several times?

**Game 4**

One of two identical coins remains motionless while the other coin rotates around it, touching it without slipping.

When the second coin has completed a turn around the first coin, how many turns has it made around its own axis?

(Solve the problem without using actual coins.)

**Game 5**

Go through the maze.

**Game 6**

Can you place four chess queens on the board so that none of them threatens another?

(A queen can move any number of squares horizontally, vertically, or diagonally.)

**Game 7**

How can you go through the garden:

• going along each walk once and only once

• without crossing your path

• finishing at your starting point?

**Game 8**

The sixteen matches form five squares. Can you change the position of three matches so that only four squares are formed by the sixteen matches?

**Game 9**

How many triangles are there in the diagram? Can you count them methodically enough not to miss any?

**Game 10**

Among the six drawings five are identical, but rotated by multiples of sixty degrees.

The sixth drawing is different.

Which one is it?

Why?

**Game 11**

Traverse the maze before you proceed on your journey.

**Game 12**

How many triangles are there in the diagram?

**Game 13**

A snail has undertaken to climb a pile of ten bricks. It can climb four bricks in an hour. But then, since the effort has been extremely tiring, it must sleep an hour, during which it slips down three bricks.

How long will the snail take to reach the top of the pile?

**Game 14**

Sixteen points are connected above by six straight lines, without raising the pencil and without going through any point twice.

You can do better: connect the same sixteen points:

• without raising your pencil

• in six straight lines

• without going through the same point twice (but the lines can cross each other)

• ending on the first point, thereby making a closed circuit.

How?

**Game 15**

The side of the small square is one meter and the side of the large square one and a half meters.

One vertex of the large square is at the center of the small square.

The side of the large square cuts two sides of the small square into one-third parts and two-thirds parts.

What is the area where the squares overlap?

**Game 16**

How many knots are on this rope?

**Game 17**

Place five chess queens on the board so that none of them threatens another. There are two independent solutions.

**Game 18**

Which two matches should you remove so that only two squares are left?

**Game 19**

Where should you start and where should you finish to go through the garden:

• going along each walk once and only once

• without crossing your path?

**Game 20**

Among the six drawings five are identical, but rotated by multiples of sixty degrees.

The sixth drawing is different.

Which one is it?

Why?

**Game 21**

Can you reach the center of the maze?

**Game 22**

How many regular hexagons are there in the diagram?

**Game 23**

The figure above can be cut to make two identical parts.

Can you cut the figure below to make two identical parts?

**Game 24**

An air squadron has about fifty planes. Its flight pattern is an equilateral triangle; every plane except the first is halfway between two planes ahead of it. Several planes are shot down in combat. When the squadron returns, the planes form four equilateral triangles. The lost planes could have formed another equilateral triangle.

If all these triangles are different in size how many planes were there to begin with?

**Game 25**

From three points you can form three rows of two points each.

Can you arrange ten points to form five rows of four points each?

**Game 26**

If you enter this garden through its door, how can you go through it:

• going along each walk once and only once

• without crossing your path?

**Game 27**

There are two loops in the rope. Are they independent?

Or are they interlocked?

**Game 28**

Which six matches should you remove, without changing the position of the others, so that only three squares are left?

**Game 29**

There is already one chess queen on the board. Place five more so that none of the six queens threatens another.

**Game 30**

Among the six drawings five are identical but are rotated by multiples of sixty degrees.

The sixth drawing is different.

Which one is it?

Why?

**Game 31**

Enter the maze, and exit as shown.

**Game 32**

How many triangles are there in the diagram?

**Game 33**

Cut the figure to make two identical parts.

**Game 34**

Place eight chess queens on the white squares of the board so that none of them threatens another.

One queen is already on the board.

The queens cannot be placed on black squares, but they can move through them.

**Game 35**

Timothy wants to saw a cube of wood into twenty-seven equal cubes.

During the work, if several pieces are already sawed, he can arrange them as he pleases, then saw through all of them with one cut.

Working this way, how many operations are needed?

**Game 36**

From twelve points you can form six rows of four points each.

Can you rearrange the points to keep six rows of four with only two rows parallel?

**Game 37**

There are three loops in the rope. How many are independent? How many are interlocked?

**Game 38**

Where should you place an odd number of matches inside the square so that four lots of equal area are fenced off?

**Game 39**

Can you go through the forest:

• going along every road once and only once

• without crossing your path

• finishing at your starting point?

**Game 40**

Among the six drawings five are identical but are rotated by multiples of sixty degrees.

The sixth drawing is different.

Which one is it?

Why?

**Game 41**

How can you go from one eye to the other of this owl-maze?

**Game 42**

How many rectangles are there in the diagram? (Note: A square is a rectangle.)

**Game 43**

Cut the figure to make two identical parts.

**Game 44**

Can you arrange thirteen points to form twelve rows of three points each?

**Game 45**

The bicycle is stationary on the ground; its tires do not slide. A man kneels by the bicycle and pulls the bottom pedal backward (arrow).

Will the bicycle go forward or backward?

**Game 46**

Ten coins are in a row, five heads on the right and five tails on the left.

In as few moves as possible, we want to alternate heads and tails.

The only move permitted takes two consecutive coins and places them in a two-coin-wide space in the same order they were picked up in. If there is no space between coins, the two coins can be placed at one of the ends of the row.

For example, here are the first and second moves of one attempt.

**Game 47**

How many loops are there? How many are free? How many are interlocked?

**Game 48**

Can you change the position of four matches so that exactly three equilateral triangles are formed? (Don't remove any matches.)

**Game 49**

You will discover that it is impossible to go through the garden:

• going along each walk once and only once

• without crossing your path.

Only one short walk has to be added to make it possible. Where?

**Game 50**

Among the six drawings five are identical but are rotated by multiples of sixty degrees.

The sixth drawing is different.

Which one is it?

Why?

**Game 51**

Here is a maze of a new kind. It has three dimensions and its roads must be followed logically, even when they disappear momentarily from sight under other roads.

Can you get from one side of the maze to the other?

**Game 52**

How many triangles are there in the diagram?

**Game 53**

How many rectangles are there in the diagram?

**Game 54**

Cut the figure to make two identical parts.

**Game 55**

From sixteen points you can form fifteen rows of four points each.

Can you rearrange the points so they form ten rows of four, and no two rows are parallel?

**Game 56**

Nine glasses are in a row, all right side up. We want them all upside down.

A permissible move reverses any six glasses, putting each one upside down if it is right side up, or right side up if it is upside down.

For example, we start with

A first move reverses the six glasses on the right.

A second move reverses the six glasses on the left.

A third move reverses glasses 2 to 7.

And so on ...

Can you get all the glasses upside down? How many moves does it take?

**Game 57**

How many loops are there? How many are free? How many are interlocked?

**Game 58**

A chess knight threatens eight squares at most.

How many knights do you have to place on the 8 × 8 chessboard so that each square is:

• occupied by a knight

• or threatened by at least one knight?

**Game 59**

What is the smallest number of matches you can remove so that no square of any size is left?

**Game 60**

Among the six drawings five are identical, but rotated by multiples of sixty degrees.

The sixth drawing is different.

Which one is it?

Why?

**Game 61**

Do you know how to cross this maze? It is in three dimensions (roads may disappear momentarily from sight under other roads) and you must obey this rule: on entering a traffic circle, exit by either the first road on your right or the first road on your left.

**Game 62**

How many quadrilaterals are there in the diagram? A quadrilateral is *any* four-sided figure. (Beware—there are more than ten.)

**Game 63**

How many quadrilaterals are there in the diagram?

**Game 64**

How many loops are there? How many are free, interlocked, knotted?

**Game 65**

Cut the figure to make two identical parts.

**Game 66**

A monkey weighing fifty kilograms is climbing a rope. The rope goes over a pulley and is fastened on the other side to a fifty-kilogram weight. The pulley rotates without friction around a fixed axis.

The monkey is doing enough work to climb forty centimeters per second if the rope was fixed.

Does the monkey go up or down? How fast?

**Game 67**

How many straight lines are needed to separate each star from all the others? Draw them.

**Game 68**

Two knights are already on the chessboard. How many knights do you have to add so that each square is occupied or threatened by a knight?

**Game 69**

Can you change the position of four matches in this spiral so that exactly three squares are formed? (Use all the matches.)

**Game 70**

In this diagram, eight equal line segments (four horizontal and four vertical) form 14 squares:

1 × 1 9 squares

2 × 2 4 squares

3 × 3 1 squares

Can you rearrange the eight line segments in one diagram so you have:

2 squares

24 isosceles triangles?

The squares are of different sizes. There are 4 big triangles, 8 of intermediate size, and 12 small ones.

**Game 71**

It is possible to cross this maze made of pipes. How?

**Game 72**

How many triangles are there in the diagram?

**Game 73**

Cut the figure to make two identical parts.

**Game 74**

From twenty-one points you can form eleven rows of five points each.

Can you arrange the points to form twelve rows of five?

**Game 75**

How many straight lines are needed to separate each star from all the others? Draw them.

**Game 76**

How many loops are there? How many are free, interlocked, knotted?

**Game 77**

How many bishops do you have to place on a chessboard so that each square is:

• occupied by a bishop

• or threatened by at least one bishop?

(A bishop can move any number of squares diagonally.)

**Game 78**

The two enclosures are made of twenty matches. Using all the matches, can you form two new separated enclosures so that one area is three times the other?

**Game 79**

A chess knight is in a corner of the board, ready to tour it in a series of moves, occupying each square once and only once and finishing where it started.

Actually, the tour is impossible. Why?

**Game 80**

Among the six drawings five are identical, but rotated by multiples of sixty degrees.

The sixth drawing is different.

Which one is it?

Why?

**Game 81**

Can you go through the maze of this electronic circuit?

**Game 82**

How many hexagons, regular or not, are there in the diagram? Crossed hexagons (that is, hexagons with sides that continue through an intersection) aren't allowed—which still leaves more than three hundred hexagons ...

**Game 83**

How many loops are there? How many are free, interlocked, knotted?

**Game 84**

Timothy, Urban, and Vincent are running the hundred-meter dash.

Timothy and Urban will reach the tape together if Timothy is given a head start of twenty meters. Urban and Vincent will reach the tape together if Urban is given a head start of twenty-five meters.

Timothy and Vincent want to reach the tape together. Who gets a handicap, and how much? (Assume each man always runs at the same speed.)

**Game 85**

Can you arrange twenty-two points to form twenty-one rows of four points each?

**Game 86**

Fill each square with one of the five symbols so that the same symbol does not appear twice:

• in any horizontal row

• in any vertical column

• in either of the two main diagonals.

**Game 87**

How many bishops do you have to place on the chessboard so that each square is threatened by at least one bishop. (If it is occupied, it must be threatened by at least one other bishop.)

**Game 88**

Using twelve matches, can you form a quadrilateral with the same area as this rectangle?

**Game 89**

How many loops are there? How many are free, interlocked, knotted?

**Game 90**

Among the six drawings five are identical, but rotated by multiples of sixty degrees.

The sixth drawing is different.

Which one is it?

Why?

**Game 91**

Beware! Experimental maze, in three dimensions (paths may disappear momentarily from sight under other paths). You may have to scale vertical walls to do it, but it is possible to reach the uppermost terrace. How?

**Game 92**

How many loops are there? How many are free, interlocked, knotted?

**Game 93**

Can you cut the vase in three pieces and assemble them to form a square? Note that the vase is entirely composed of curves.

**Game 94**

The board has forty-nine squares. One square is marked with a star. Can you cover the forty-eight remaining squares with twenty-four two-square dominoes?

**Game 95**

On the chessboard place fifty-one pieces:

• eight queens

• eight rooks

• fourteen bishops

• twenty-one knights

so that no queen threatens a queen, no rook threatens a rook, no bishop threatens a bishop, and no knight threatens a knight.

Of course, two pieces cannot occupy the same square; but in this game, pieces can move through squares occupied by other kinds of pieces.

(The moves of all the pieces except the rook have already been defined. A rook can move any number of squares horizontally or vertically.)

**Game 96**

How many quadrilaterals are there in the diagram?

**Game 97**

How many loops are there? How many are free, interlocked, knotted?

**Game 98**

A rectangular tiled floor in Timothy's house has 93 square tiles on the short dimension and 231 on the other.

Timothy draws a diagonal from one corner to the opposite corner. How many tiles does it cross?

**Game 99**

Among the six drawings five are identical, but rotated by multiples of sixty degrees.

The sixth drawing is different.

Which one is it?

Why?

**Game 100**

This maze contains some one-way streets. You only have to cross the bridge between the two halves of the maze once, but if you find yourself returning on it, don't give up: keep going. (Note: You can go on roads under the bridge.)

*(Continues...)*

Excerpted from100 Geometric GamesbyPierre Berloquin, Denis Dugas. Copyright © 1976 Pierre Berloquin. Excerpted by permission of Dover Publications, Inc..

All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

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