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Games and Maths Puzzles

Games

Why Play?

Is there any need to actually play the games if the object is merely to provide a setting for doing some mathematics? After all the game could be explained very quickly and the follow-up work started immediately. However, it is best that the games are played properly for three reasons. First of all there is the intrinsic mathematics which always present. Second, there is the high level of interest and motivation which games-playing generates. Third, and perhaps most important, is the deeper understanding of the situation to be worked on which can be gained only by playing through several games. Far too often students are expected to try and analyse a situation of which they usually have little or no previous experience. Here at least there is a chance to remedy that defect. To provide everyone with experience of a common activity, a system needs to be put in place which affords playing opportunities to all.  Click here to download the full document

Maths games are so easy to find nowadays that it is tempting to think their availability is reason enough to include them in our children's mathematical experiences. But is it enough that children might "have fun" playing them? Can we justify their use in the classroom? In writing this article, I will try to address these questions using some examples of mathematical games from NRICH - Click here

The NRICH website has a series of 4 articles looking at the use of games in the classroom
     *  Why use games
     *  Types of games
     *  Creating your own games
     *  From Strategy Games to Investigations


Blog on Maths Games by @mathhombre

This is a page to organize the math games he's created or modified significantly with some notes about content and a collection of the best math games he's seen - Click here

Build an Army @mrbartonmaths

Mr Barton has uploaded a bundle of free activities I have created to the TES website, called "Build an Army: Strength in Numbers". And yes, the title came before the activity.

“Build an Army” is a fun (well, hopefully anyway!), strategy game that can be used to consolidate understanding of key mathematical concepts. After students have played the game and described their strategy, there are opportunities for differentiation via various lines of inquiry and probing questions for the students to investigate. Full instructions are provided in the “General Rules” PowerPoint. 

These are all freely available on TES, and can be accessed via the links found here

What can you do with a Pack of Cards?

 

 

Teachers can go crazy thinking of different ways to practice the same facts to help students learn without getting bored. What a card game does is to give the students something to hold, touch, and move around while they see the facts on the cards and say them as well. Manipulating the cards in a variety of games, whether it is matching, making decisions on which answer is higher, or creating groups of similar attributes, is a highly effective multisensory tool.

The students will be happy to play games and the games will help their memories absorb the facts;

Number Levels 1-4 Number Set 1 Levels 1-4 Number Set 2 Simple Number Facts Four Operations Multiples
Indices Fractions and Decimals Percentages Place Value  
Algebra Sequences Co-ordinates      
Data Averages        
Mixed Card War Games Magic Tricks      

In addition to using a standard Pack of Cards you might prefer to play Mathematical Top Trumps - Click here

Simulation Games

The following two games are simple simulations for running a business and are based upon an understanding of simple probabilities

The Great Elf Game                 Alternative Link -    Rules   Record Sheet with Formulas (Fill in red cells)      Record Sheet pdf                             Virtual Dice
The Lobster Game                  To project               Pupil Record Sheet                                                               Advanced Pupil Record Sheet              

Strategy Games

Several games of strategy that can be used for a competition

Pencil and Pen Games Creating Squares 5 by 5 Game Ultimate Tic Tac Toe Breakthrough

Online Version
Backgammon Variations Simple Board Games mathpickle Galileo.org Bridge Problems & Other Classics

Board Here

One of many types hosted on www.cleavebooks.co.uk/trol/indexgam.htm

Other Historical Problems

I have downloaded several simple to play strategy games, some requiring just pencil and paper and others use counters.  Click here to download this word document. 

You might like the challenge of designing a set of rules (like a computer programme does) to play Noughts and Crosses successfully against a human opponent - Click here

This list is an indication of the games available at the Maths Arcade. We tried to get a number of games that would be suitable for a group of students to play – particularly important at the start. Some, such as Giant Blokus, have been particularly successful as students like to watch even if they are not playing.

Abalone

2 player strategy game. The objective is to push six of the opponent’s fourteen marbles off the edge of the hexagonal board following a set of simple rules. A good game but you don’t want too many sets. Many students like this.

Blokus 3D/Blockus Giant/Blockus Trigon

2-4 player strategy game. Involves placing polyomino-based tiles onto a board to capture available space. Giant version is great for several students to play and watch.

Callisto

2-4 player game. Looks bright and childish but students that have played it have got a lot out of it. Easy to pick up but tricky to come up with the best strategy.

Chaos

2 player strategy game. Involves stacking counters. Excellent strategy game but hard to learn. Takes longer than other games once both players understand the rules.

Gambit

2 player strategy game. Involves sliding rows of coloured tiles. Enjoyable but not one of the most popular games.

Gobblet

2 player strategy game. Placing or moving already placed pieces, including larger pieces covering smaller ones, to make a row of four on a 4x4 grid. A good, easy to master, strategy game.

Gygès

2 player strategy game. The object of the game is to move a piece to your opponent’s last row. The catch is no one owns the pieces.

Hex

2 player game said to be invented by mathematician John Nash. It can be bought from Nestor Games (see below). Hex boards can also be made relatively easily or can be played on paper. See boardgamegeek for paper grid.

Ingenious

1-4 player strategy game. Placing tiles on a board with a clever scoring system. Good for strategy. Travel version is easier to play but full-size version can be watched by more students.

Joggle

2 player strategy game. Dice-based placement of marbles on a board. First to build a 6 marble rectangle wins.

MiQube

1-4 players. It can be played by one player reassembling the cube or as a board game encouraging strategic thinking. It encourages special thinking. A good review on boardgamegeek: http://boardgamegeek.com/thread/261372/review-miqube

Neuron

1-4 players. Similar to Tantrix but less complex rules. Aim is to make the longest line of your colour.

Novem

2 player tactical game. Lots of game theory involved in this. One person controls rows and the other columns and you try and capture tiles with the highest value. Strategy involves trying to work out what your opponent is going to do.

Pentago

2 player strategy game. Placing coloured marbles on a 6x6 board, the quadrants of which can be rotated, to form five in a row. Rules are easy to follow but the strategies take longer to assimilate. Excellent short game.

Pylos

2 player strategy game. Placing marbles to form a pyramid according to simple rules. Whoever places the top marble wins. Best with players of equal ability.

Q-bitz

2-4 player game. Three rounds based on speed, chance and memory. Arrange mini-cubes to make patterns. Good for encouraging symmetric visualisation.

Quads magnetic

2 player game. The tiles have different combinations of colours or lines. Players take turns placing a tile so that the colours or lines on each edge match that of adjacent tiles. The player who stops his or her opponent from being able to place a tile wins.

Quarto

2-4 player strategy games. Object is to get a row of 4 pieces with the same attribute. The catch is that your opponent chooses your piece for you.

Quirky

2-3 player strategy game. Uses coloured tokens to build equilateral triangles. Not got the hang of this game yet! Instructions are not well-explained.

Quixo

2-4 player strategy game. Adding and shifting tokens to form five in a row. Excellent strategy game that requires 100% concentration.

Quorridor

2-4 player strategy game. Each player aims to move a pawn to the other side of the board but can place walls to obstruct their opponent. Good for programmers.

Rubik’s cube/Hollow cube/Sudoku cube

The classic puzzle and variants. There are many books and websites explaining how to explain Group Theory using a Rubik’s Cube.

Rubik’s 360

Puzzle. Involves changing the position of six coloured balls in a central sphere to six coloured compartments in an outer sphere, by manoeuvring them through a middle sphere that only has two holes.

Rubik’s magic

Puzzle. Folding connected tiles to form a pattern.

Rumis

2-4 player strategy game. Involves placing blocks onto a board as part of a 3D structure to capture available space. Not played much.

Rush Hour

1 or more player strategy game. The objective is to move a red car out of a six-by-six grid by moving the other vehicles out of its way. There are many variations on this including online versions and apps for iPads etc. Similar to Red Donkey.

Saikoro 2 player game with lots of dice. Easy to earn and demonstrated on ‘Board Games with Scott’. Involves finding a way of removing dice to trap your opponent. http://www.boardgameswithscott.com/?p=84
Solomon’s stones 2 player strategy game. A variant on Nim’s game. Excellent for encouraging thinking and strategy and possibly not too difficult for the more able to program.
Sprocket 2-4 player strategy game. Using rotor pieces to create gears and lugs. An excellent strategy game once you have understood all the rules – not easy!
Square up Like Sam Lloyd’s 15-puzzle. We have developed a program in Excel that produces a random 5x5 grid as the provided 4x4 is too easy. This now means that we can have 6 people play together with the program.
Stratum 2-4 player strategy game. Trying to cover the opponents’ pieces by placing tiles. An excellent game for discussing what the best strategy is.
Sudoku cards 2-5 player strategy game. Involves placing numbered cards according to sudoku rules.       
Tantrix 1-4 player strategy game. Hexagonal tile-based placement. Good but hard to teach complexities of rules and time consuming.
Totemland 2-4 player. Excellent strategy game. You build totem poles but you want to have as many of your pieces on top at the end. The rules of placing pieces make this harder than it seems. Excellent game.
Tower of Hanoi The classic puzzle

The classic games Backgammon, Chess, Draughts, Go, Reversi (Othello), as well as playing cards are available at the Arcade but are not described in the list above.

There are several websites that I have found useful for sourcing games and finding out about them although most of the above can be purchased via Amazon.
I recommend registering with BoardGameGeek.com for their reviews and files containing English rules – necessary as often the translations provided are tricky to comprehend!
BoardGameswithScott.com is another interesting resource with video instructions for some games.
I have recently come across NestorGames.com which seems to have a good supply of unusual strategy games.

Mixture of Games 

This site presents a collection of games and puzzles with mathematical contents and a few practice exercises disguised as games - Click here

Thousands of Sudoku Puzzles to print and solve - Click here

http://www.popmaths.com/games/
16 Games such as Connect Four, Tower of Hanoi and Battleships played online

http://hoodamath.com/ 
A mix of fun and maths super stacker is excellent

http://www.math.com/students/puzzles/puzzleapps.htm
Collection of online logic puzzles

http://www.madras.fife.sch.uk/maths/homelearning/games/index.htm
Range of categorised games: Board, Geometry, Logic, Mazes, Graphical and Arithmetical.

www.nrich.maths.org.uk/public/games.php  
This is an excellent list of games from NRICH.  The games are all played on paper and the rules can be copied from this site.

http://students.itec.sfsu.edu/itec815/loosli/studentpage.html
This is a page set up by a teacher which encourages students to review some games on the web
 

Games to Download

www.nrich.maths.org.uk/public/games.php  
This is an excellent list of games from NRICH.  The games are all played on paper and the rules can be copied from this site.

http://atschool.eduweb.co.uk/ufa10/games.htm
Some mathematical games with a multicultural flavour.

http://students.itec.sfsu.edu/itec815/loosli/studentpage.html
This is a page set up by a teacher which encourages students to review some games on the web

www.cs.uidaho.edu/~casey931/conway/games.html
Site for mathematical games, i.e. games involving no chance, complete information and the outcome of which can be entirely predicated given optimum strategy on the part of both players. A resource suitable perhaps for the more able but at any age. Some interesting variations on familiar games.

http://www.bbc.co.uk/education/mathsfile/index.shtml
This BBC site has 12 games aimed at KS3 with 3 levels of difficulty (Level 3-6)

Games to Buy

ATM
These games are mostly suitable for KS3 and can be purchased from ATM by phoning 
01332 346599 or locate at www.atm.org.uk.

The Fourbidden Card Game promotes the good use of mathematical language.  It is based on the concept that you have to describe a mathematical word without mentioning four other words which are listed on the card.  Eg. Identify the word perimeter without mentioning: distance, circumference, round or edge. Code ACT 012

Fourbidden Too is 52 different cards of the type described above. Code ACT 013

Two Number Jigsaws - Order of Operations are two sets of 25 cards that form a square by matching up calculations that give the same result. Eg. 3 + 7 - 4 x 2 and (3 + 4) x 2 - 12  Code ACT 025

Two Algebraic Jigsaws is an algebra equivalent of the item above. Code ACT 014

 

Tarquin Team Games

These are special mathematical puzzles and problems which produce real cooperation between the members of a team. The mathematical content is that of the normal curriculum and whether you call them games, puzzles or problems they undoubtedly offer a very positive experience at a variety of different levels. Each player only gets some of the information and so all must play a part in arriving at a solution. Sixteen tried and tested team games are provided in photocopiable form and once it is realised how well this format works, it will not be difficult to construct more for yourself.

Mathematical Team Games - by Tarquin - 1 899618 56 2 - Vivien Lucas

 

Puzzles

Bridges is an accessible daily puzzle from BrainBashers.com - Click here          Alternatively KrazyDad has lots of these and other puzzles - Click here

Great site with lots of different puzzles - Click here

Thousands of Sudoku Puzzles to print and solve - Click here

Puzzles.com: A great website covering a wide range of puzzles, for all abilities - Click here

Online logic problems, logic puzzles, cryptograms and other puzzles too - Click here

A Wide range of Logic Puzzles - Click here  

A website dedicated to the puzzling world of mathematics - Click here

Lots of Maths Challenges designed to encourage discussion and risk taking - Click here

Fantastic site with games requiring logical thinking and problem solving - Click here

 

River Crossings

Crossing a Rickety Bridge

Four people need to cross a river at night. A rickety bridge over the river can hold only up to two people at a time. There's one flashlight that must be used when crossing. After it's used, it must be brought all the way back to the people remaining on the near side.

Each of the four people takes a different amount of time to get across. If two people cross together, they travel at the slower person's rate. Annie takes 10 minutes to get across, Barry takes 5 minutes, Charlie takes 2 minutes, and Speedy takes 1 minute.

What's the shortest time that it would take all four people to end up on the far side of the river?

Dangerous Crossing.

Four creatures A, B, C and D come to a river at night. The bridge is very thin and narrow, and can only hold any two of them at a time. Besides, it is dark and they need to keep their torch on while on the bridge.

It takes A one minute to cross the bridge, B - 2, C - 5, and D - 8 minutes.

Can they all cross to the other side if the batteries in the torch last only 15 minutes? 

A 16th century version

There are three beautiful brides and their young, handsome, and intensely jealous husbands. The small boat that is to take them across the river holds no more than two people. To avoid compromising situations, the crossings must be arranged so that no woman is left with a man unless her husband is also present. How many trips does it take to ferry all six across the river without an angry outburst?

Online Version

http://pbskids.org/cyberchase/games/modeling/modeling.html

Treasure to Ship


There are 2 pirates and 4 treasure chests on an island. The pirates have 1 small boat to take the treasure to their ship. The boat can take 2 pirates or 1 pirate and 1 chest of treasure.

How many trips do the pirates have to take to get all the treasure and both pirates onto the ship?

Note: The ship needs 2 pirates to sail it. Don't worry about one pirate sailing off with all the treasure!

Magic Squares

The story of Lo Shu is basically one of a huge flood in ancient China, whereby sacrifices to the river god -- to calm his anger -- seem ineffective.  Each time a turtle came out of the river and walked around the sacrifice, as if to suggest that the river god had not accepted the sacrifice.  Until a child noticed a curious figure on the turtle shell - in effect, the 3x3 magic square shown above.  On this basis, the people realized the correct amount of sacrifice to make, and thus appeased the river god,

Franklin Magic Square & Durer's Square

Numberphile video looking at Magic Squares and the Magic Hexagon

Chessboards

Knights Tour, One of the earliest problem involving chess pieces is due to Guarini di Forli who in 1512 asked how two white and two black knights could be interchanged if they are placed at the corners of a 3 × 3 board (using normal knight's moves) - Knights Tour

Grains on Chessboard - Arabic mathematician Ibn Kallikan who, in 1256 - YouTube Clip

 

Konigsberg Bridges

Our story begins in the 18th century, in the quaint town of Königsberg, Prussia on the banks of the Pregel River.  In 1254, Teutonic knights founded the city of Königsberg.  In the Middle Ages, Königsberg became a very important city and trading center with its location strategically positioned on the river.  The healthy economy allowed the people of the city to build seven bridges across the river, most of which connected to the island of Kneiphof; their locations can be seen in the accompanying picture.  

As the river flowed around Kneiphof, literally meaning pub yard, and another island, it divided the city into four distinct regions linked by seven bridges According to lore, the citizens of Königsberg used to spend Sunday afternoons walking around their beautiful city.  While walking, the people of the city decided to create a game for themselves, their goal being to devise a way in which they could walk around the city, crossing each of the seven bridges only once. 

Tower of Hanoi
The Tower of Hanoi or Towers of Hanoi , also called the Tower of Brahma or Towers of Brahma, is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.

The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:

Conway's Game of Life
The Game of Life is not your typical computer game. It is a 'cellular automaton', and was invented by Cambridge mathematician John Conway.

This game became widely known in 1970. It consists of a collection of cells which, based on a few mathematical rules, can live, die or multiply. Depending on the initial conditions, the cells form various patterns throughout the course of the game - Rules, online animation and free program download - Click here

Frogs

·       There are two families of frogs – purple and blue.

·       Each family contains 3 frogs.

·       The purple frogs live on the left of the pond, the blue frogs on the right.

·       The purple frogs want to get to the right side of the pond as they think the blue frogs get the juiciest flies.

·       The blue frogs, on the other hand, think the purple frogs get fatter flies and want to get to the left of the pond.

·       There are 7 lily pads which the frogs must use to cross the pond.

·       Frogs can only jump to EMPTY lily pads.

·       Frogs can only jump over ONE other frog at a time.

·       Frogs don’t know how to jump backwards!

Work out how the families swap sides.  What is the smallest number of jumps they have to make to get there?

Task Version 1         Task Version 2                Online Version            Alternative Task

Mix of Investigations

Tower of Hanoi     Pentominoes     Polyhedra     Handshakes     Paving Slabs     Frogs     Tetrahedra     Painted Cube

Penguin Sliders     Pick's Theorem     Stacking Boxes     Interactive Number Square     Mix of Investigations

Squares Investigation

Introductory PowerPoint         Handout

15 Puzzles Version 1       Version 2
Jug Measuring Problems Online Problems

Interesting Applet and Representation

 
Sudoku Online Games     
OkiDoku

Many people enjoy working on grid puzzles as small, quick challenges of their mathematical and logical skills. Here is one you may not have seen, the OkiDoku. How does it work? Looking at the grid above, try to find four different numbers and put them in these 16 squares in a way that will satisfy the following two conditions:

1. Each of these four numbers must appear exactly once in each row and in each column.

2. The blocks with thick borders are called cages. Each cage shows a target number and a mathematical operation. The operation applied to the numbers in the cage should produce the target number - Click here

Bedouins Breaking Bread

 

Two Bedouins were travelling across the desert to a distant village. In the middle of the day, they sat down to eat the loaves of bread that they had brought with them for lunch. One of them had five loaves and the other had three.

Just as they were ready to eat, a stranger comes along and asks if he might share their meal.

He said he had plenty of money but no food. The two agreed to divide their loaves equally among the three of them.

After the meal was finished, the stranger laid down eight coins of equal value for what he had eaten and he went away. The traveller who had five loaves took up five coins and left three for the other guy. But the other guy disputed it, saying, "We shared the bread, we should each get four coins." Since they could not agree, they called in a magistrate.

The magistrate listened to the story and then figured out who should get what.

The question is, who's right? Or, is neither of them right?

Map Colouring Since the time that mapmakers began making maps that show distinct regions (such as countries or states), it has been known among those in that trade, that if you plan well enough, you will never need more than four colours to colour the maps that you make.

The basic rule for colouring a map is that no two regions that share a boundary can be the same colour. (The map would look ambiguous from a distance.) It is okay for two regions that only meet at a single point to be coloured the same colour, however. If you look at a some maps or an atlas, you can verify that this is how all familiar maps are coloured - Click here