Hands On Puzzles Grade 7-8

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The M Puzzle
The M Puzzle
The object is to make of these four pieces a symmetric letter M. The pieces must not overlap each other in the final configuration.
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A Battle Royal
A Battle Royal
by Sam Loyd
The object is to rearrange the pieces in such a way that to form a perfect 8x8 chess board.
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Three Queens
Three Queens
by Martin Gardner
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.
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The Four Queens
The Four Queens
by Martin Gardner
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.
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Four Circles
Four Circles
by Thomas H. O'Beirne
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.
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Circle Circumference and the Number Pi
Circle Circumference and the Number Pi
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?
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Consecutive Rectangle Squares
Consecutive Rectangle Squares

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.

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Circling Numbers II
Circling Numbers II
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.
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Lights On!
Lights On!
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?
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Sword of Knowledge
Sword of Knowledge
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?
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Building Polygons
Building Polygons
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?
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Counting the Triangles
Counting the Triangles
by Henry E. Dudeney
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?
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Counting the Triangles 2
Counting the Triangles 2
How many triangles (counting ones of all possible sizes) can you find in the illustration?
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Hard 14
Hard 14
Rearrange the fourteen pieces so that to form the 8x8 checkerboard shown in the center of the illustration.
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Triangular Numbers
Triangular Numbers
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?
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The 7 Knights Problem
The 7 Knights Problem
by Martin Gardner
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.
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Two Containers
Two Containers
How can you use these two containers, which are 11 and 7 liters, to measure 10 liters of water?
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Four Stacks
Four Stacks
by Martin Gardner
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.
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Red, White, and Blue Balls
Red, White, and Blue Balls
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?
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How Many Squares?
How Many Squares?
by Professor Louis Hoffmann
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.
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Babylon
Babylon
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.
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Babylon II
Babylon II
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.
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Egyptian Triangle
Egyptian Triangle
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?
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The Testa
The Testa
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.
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Puzzling Journey
Puzzling Journey
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.
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The Unicursal Marathon
The Unicursal Marathon
by Serhiy Grbarachuk
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.
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Semicirclechain
Semicirclechain
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.
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Switch the Cars
Switch the Cars
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.
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In Ancient Greece
In Ancient Greece
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.
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Snake
Snake
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.)
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The H Puzzle
The H Puzzle
by Harry Lindgren
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.
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Six Knights
Six Knights
by Henry E. Dudeney
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.
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Strip to Cube
Strip to Cube
by Martin Gardner
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.
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Get 4L
Get 4L
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?
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Pythagorean Hexagons
Pythagorean Hexagons
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?
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Last Updated: September 10, 2009 top
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