1 eg. Given the digits in black, we know that digit 1 must be at Row 2 Column 9. X stands for any digit other than 1.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "1 eg. Given the digits in black, we know that digit 1 must be at Row 2 Column 9. X stands for any digit other than 1."

Transcription

1 Sudoku Solving: Dot method and more. David A Paterson (aka mollwollfumble) 2 Apr 204 This is a quick summary giving examples of how I solve Sudokus of all difficulties. It contains useful mental and written methods, some of which haven t appeared on the web before. Low hanging fruit Fill in any digits that jump out at you. Don t spend too long on this, if you spend more than two seconds looking for the next digit then you re wasting time eg. Given the digits in black, the digit must be at Row 4 Column 4. Scanning Scanning is where you go through each digit in turn, eg. to 9, and look along rows and columns for new places to put each digit. We all do it. X eg. Given the digits in black, we know that digit must be at Row 2 Column 9. X stands for any digit other than.

2 While scanning, you will come across distractions in the form of digits that can be filled in out of sequence. Don t allow yourself to be distracted, come back to these at a later time. Keep cycling through each digit in turn until you re sure that you can do no more. While scanning, I use three tricks that find more digits than simple scanning alone without appreciably slowing the scanning process but. I call these Arrows, Left-Middle-Right, and Casual Dots. Arrows and Left-Middle-Right are mental techniques whereas Casual Dots is a written aid these are described below. After you ve passed through the scanning phase of solution into more advanced techniques, come back to scanning every now and then to search for more digits. Arrows Mentally draw lines on the Sudoku grid along rows and columns that are eliminated by each digit in turn. I find it handy to add a mental arrowhead to these lines so I know in which direction they re going. X X X X X X X eg. Given the digits in black, following rows and columns allows you to place a new digit in Row 2 Column 9. Left-Middle-Right While scanning, I keep up a mental commentary of Left-Middle-Right and Top-Middle- Bottom. You can say this out loud if you like. For each digit scanned and for each group of three rows or columns keep watch on which of the three is a possible place. See example.

3 Top Mid X X X X X Bot eg. Along the bottom three rows, reading from left to right, say the possible positions for digit as top-bottom, top-middle-bottom, top-bottom. The centre-bottom box is the only one with a middle so digit must be in the middle row of the centre-bottom box. A challenge in solving Sudokus is knowing when NOT to search, when all digits findable using that search method have already been found. The method of Left-Middle-Right gives an immediate knowledge of when scanning for any particular digit is complete. X eg. A commonly encountered case where the method of Left-Middle-Right says that no more digit s can be found by scanning. Reading down the three left columns gives middle-right, middle-right, left so there is no unique middle or right. Reading across the top three rows gives middle-bottom, top, middle-bottom so there is no unique middle or bottom. The presence or absence of a digit marked X in the top left box has no influence on the result. Dot Method You will already know about using pencil marks in a cell. The dot method is a more compact way of showing the same information. Imagine the digits,2,3,4,5,6,7,8,9 placed in a cell with,2,3 across the top, 4,5,6 across the middle, and 7,8,9 across the bottom. Instead of writing the digit in pencil, place a dot where the digit would be. Also, rather than pencilling

4 in digits that are possible, place a dot wherever that digit is eliminated. In this way, there is never any need to rub out dots and I recommend putting the dots in with a pen, not a pencil, unless you re doing an extremely difficult Sudoku using trial and error. Just touch the pen to the paper once for a dot, don t waste time drawing a filled circle. When you finally discover the unique digit for a cell, write that digit over the top of the dots, it will be easily visible eg. The nine positions within a cell and the nine dots at those positions eg. Pencilled digits in a cell and the equivalent information using the dot method. Saying that, 3, 4, 8 & 9 are not eliminated is the same as saying that 2, 5, 6 & 7 are eliminated. 8 eg. A digit written over dots is easily visible. I ve been told that some people never make mistakes. I m not one of them. If you add a dot in the wrong place by accident, strike it out by putting a single diagonal line through it. Later if you find that that digit is indeed eliminated, put a second diagonal line through it to form a cross. eg. Correcting accidental errors. a) Digit is eliminated. b) Oops, I made a mistake and digit is still possible. c) Digit is indeed eliminated. Advantages of the dot method over pencilled digits are: you never need to rub out or scrub out a digit, the more compact form allows you to do Sudokus with smaller printed cells, the final digits are perfectly visible when written over the top of dots, and certain more advanced techniques such as groups of four are more easily visible using the dot method than when using pencilled digits. I ll come back to the last of these later.

5 The Dot Method for Puzzles other than Sudoku. The dot method is also useful for a whole range of puzzles other than Sudokus. The dots can replace letters or symbols equally as well as replacing digits. For puzzles that have other than nine symbols I use the following dot placements eg. Dot placements when there are three, four, five, six or sixteen possibilities. It is rare to have a puzzle format where it becomes easier to mark where a digit may be rather than where it can t be. Such puzzles are those where we are given information on the jump from one digit to another, see example below. I mark a possible location using a small circle rather than a dot. And then, when the digit is finally eliminated, place a dot in the centre of the circle. Don t use this technique for Sudoku. 2 eg. On this non-sudoku puzzle, the puzzle rules say that number is a knight s jump away from the number 2. Possible positions for the number are marked by circles. When the position is eliminated place a dot in the circle. Casual Dots. Pencilled digits can be casual, placed wherever you feel like it, or complete, placed in every cell. Casual dots can be placed while scanning without slowing down the process much. For Sudokus you know in advance will be difficult, place dots every time a digit is eliminated when there is no matching digit in the same row, column or box.

6 X X X eg. Two examples of the placement of casual dots. Casual dots go in places where digits are eliminated despite there being no matching digit in the same row, column or box. More often than not, placing casual dots doesn t help you find any extra digits, but occasionally it does, the following is an example. In addition, placing casual dots saves a lot of time if you need to progress to complete dots eg. An example where casual dots may help. Three dots appear in the cell in Row Column 3. When combined with the other digits present in the same row, column and box, there is only one candidate left, the digit 7. Quick Checks. Some quick checks help to find digits that have been accidentally missed. While still in the scanning phase of solving a Sudoku, count the number of times the digit appears in the puzzle. If it s eight times, find the ninth. Once all nine occurrences of a digit are found, write that digit outside the puzzle, as a quick way to avoid later searching for digits that have all already been found.

7 Another type of quick check is illustrated in the following example eg. I know that I know the digit at Row 6 Column 7, because the three other digits in Column 7 are fixed by the digits in the bottom right box, but what is it? A quick check comparing the digits in the box to the digits in the column finds the digit 6. Line End Digits. My sister-in-law likes solving black-belt Sudokus. She introduced me to the method of Line End Digits. At the end of each row or column that you are interested in, outside the puzzle, write all possible unplaced digits in that row or column, or part of row or column. This is most useful when the printed Sudoku is too small to pencil in digits within the cells eg. In Column 5, the top 3 digits form a group of three digits listed at the top of the column. The remaining possibilities are listed at the bottom of the column. The only overlap between the digits and 3456 is the digit 6, so the digit at Row 6 Column 5 is 6.

8 n-blank. Go through the digits to 9 in turn for each row or column or box with n blank cells, mentally marking each possible digit by bending a finger of the left hand starting with the little finger. Then for each blank cell within that row, column or box mentally figure out which of these is eliminated, looking for cells where there is only one candidate as well as for groups of two and three related cells. See example below. When n is, 2 and 3 this is a trivial mental exercise. When n is 4 it is still fairly easy using the fingers of the left hand. When n is 5 and 6 it is still possible. After n = 3 or 4, I tend to turn to the written method of complete dots given below eg. 3-blank and 4-blank. For Row mark off the only possibilities, 2, 5 using fingers of the left hand. Row Column cannot be 2 or 5 so must be. For Column mark off the only possibilities 3, 7, 9 using fingers of the left hand. Row 6 Column cannot be 7 or 9 so must be 3. For the left middle box mark the only possibilities 2, 6, 7, 9 using fingers of the left hand. Row 6 Column 3 cannot be 2, 7 or 9 so must be 6. Complete Dots. This is the use of the Dot Method where every blank cell is filled with dots. It helps a lot if casual dots have been previously used. Visibility is reduced if pencilled digits have previously been used. Line end digits are an unwanted distraction. Be systematic in adding dots. I fill in the dots box by box, by first eliminating (by placing a dot) every digit that casual dots would have picked up, then every digit in the box, then every digit in the row and in the column. The method of complete dots carries exactly the same amount of information as the method of pencilling digits in each cell.

9 eg. The top Sudoku is the original puzzle. The second is after scanning with casual dots and after 3-blank. The third is with complete dots. On Row 6, Column 3 must be digit 6, so Row 6 Column 2 is digit 2 and Row 6 Columns 7 & 8 can t be digits 6 or 2. On Row 7, Columns 3 and 7 form a pair using up digits & 3, so Row 7 Column 9 must be 2 and Row 7 Column 6 is 9. On Row 8, Columns 6 and 8 form a pair using up digits 4 & 5, so Row 8 Columns 4 & 7 can t have digit 5. Working from there, the Sudoku is easily solved without trial and error.

10 With the method of complete dots, groups of 2, 3 and even 4 connected cells can be seen immediately. More advanced techniques such as X-wing can also be seen. eg. An X-wing. Row 4 Column 2 cannot be digit 3, because then that would give the Sudoku multiple solutions. Line Trial and Error. About midway through my Sudoku career I came upon a Sudoku that defeated me utterly. Six months later I learnt that someone had solved it with minimal difficulty using the method of Line Trial and Error. I haven t been completely defeated by a Sudoku since. Select a row or column and try in turn all the different ways to complete it. I usually choose a row or column that can be completed in six to twelve different ways. I use pen for all dots and digits before trying trial and error and use pencil only for trial and error

11 eg. At top is the original Sudoku puzzle. Below gives complete dots. There are four possibilities for Column 3 reading from top to bottom these are 75293, 75392, 75932, Try each in turn. It will be found that is the correct solution. Sudoku difficulty vs method It is frequently said that you can t tell how difficult a Sudoku puzzle will be in advance. This isn t quite true. If there are more than 28 digits in the puzzle then it s likely to be very easy. Further, Sudokus with digits arranged along diagonal lines tend to be more difficult than those with digits arranged along vertical or horizontal lines. If you have a fair idea how difficult a Sudoku will be, you can choose an appropriate solving method in advance. For example, with a Sudoku you know in advance will be extremely difficult, don t waste time with pencilled digits, line end digits or n-blank for n > 3. Go straight from scanning with casual dots to the method of complete dots and then use row trial and error. Solution methods for Sudoku puzzles from trivially easy to extremely difficult in turn, in order of increasing difficulty: Low hanging fruit Scanning Scanning with arrows Scanning with left-middle-right Casual dots Casual pencilled digits Line end digits n-blank Complete pencilled digits Complete dots (skip pencilled digits) Line trial and error with complete dots Never balk at the prospect of copying the Sudoku onto a new grid. It s seldom needed but it often helps.

How to Draw a Tessellation

How to Draw a Tessellation How to Draw a Tessellation PART ONE By David B Sullivan www.dbsullivan.com Introduction A tessellation is when a specific shape can be repeated within a plane (or flat surface) and interlock together without

More information

OA3-10 Patterns in Addition Tables

OA3-10 Patterns in Addition Tables OA3-10 Patterns in Addition Tables Pages 60 63 Standards: 3.OA.D.9 Goals: Students will identify and describe various patterns in addition tables. Prior Knowledge Required: Can add two numbers within 20

More information

Unit 6 Number and Operations in Base Ten: Decimals

Unit 6 Number and Operations in Base Ten: Decimals Unit 6 Number and Operations in Base Ten: Decimals Introduction Students will extend the place value system to decimals. They will apply their understanding of models for decimals and decimal notation,

More information

Sudoku puzzles and how to solve them

Sudoku puzzles and how to solve them Sudoku puzzles and how to solve them Andries E. Brouwer 2006-05-31 1 Sudoku Figure 1: Two puzzles the second one is difficult A Sudoku puzzle (of classical type ) consists of a 9-by-9 matrix partitioned

More information

Battleship. Big bands

Battleship. Big bands Ball Fill in the grid so that every row, column (six smaller cells and three bigger circles or stars), outlined figures (eight smaller cells and a bigger circle), nine bigger circles and nine bigger stars

More information

Grade 7 & 8 Math Circles February 2-3, 2016 Logic Puzzles

Grade 7 & 8 Math Circles February 2-3, 2016 Logic Puzzles Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7 & 8 Math Circles February 2-3, 2016 Logic Puzzles Introduction Math is not always numbers, equations

More information

THIS MEANS THAT EACH NUMBER SHOULD ONLY APPEAR ONCE IN EACH ROW, COLUMN AND BOX! TOP TIPS FOR SOLVING SU DOKU

THIS MEANS THAT EACH NUMBER SHOULD ONLY APPEAR ONCE IN EACH ROW, COLUMN AND BOX! TOP TIPS FOR SOLVING SU DOKU THE RULES Each row must contain each number from to 9 Each column must contain each number from to 9 Each box (small by grid shown by thicker lines) must contain each number from to 9 THIS MEANS THAT EACH

More information

1st UKPA Sudoku Championship. Saturday 6th Sunday 7th November 2010

1st UKPA Sudoku Championship. Saturday 6th Sunday 7th November 2010 1st UKPA Sudoku Championship I NS TRU C TIO N B O O K L E T Saturday 6th Sunday 7th November 2010 Competition Rules & General Information REGISTRATION To participate in the championship, you will need

More information

Games ~ 1. Triangle Sum. Poly Pick. In & Out. Skittles. Cover Up. Home Run. Cross Nim. To and Fro. Thirty one. Put Down. One or Two.

Games ~ 1. Triangle Sum. Poly Pick. In & Out. Skittles. Cover Up. Home Run. Cross Nim. To and Fro. Thirty one. Put Down. One or Two. Games ~ 1 List of Contents Triangle Sum Poly Pick In & Out Skittles Cover Up Home Run Cross Nim To and Fro Thirty one Put Down One or Two Triangle Sum Players take turns moving a single marker or counter

More information

Organizing Information by Using Tables

Organizing Information by Using Tables In this chapter Find out what tables are and why you should use them Learn how to create tables and put information in them Discover how easy it is to format columns and rows Learn how to customize table

More information

Advanced Techniques for the Walkingbass

Advanced Techniques for the Walkingbass Advanced Techniques for the Walkingbass I have seen guys with 5 string basses who can t get half the sounds that you are getting out of just three. -Buddy Fo of the Invitations If you have read the Beginners

More information

This chapter introduces the sudoku-solving ground rules,

This chapter introduces the sudoku-solving ground rules, hapter Simplifying Sudoku This chapter introduces the sudoku-solving ground rules, giving you all the tools you need to approach each puzzle with the best strategy. It also shows you how to take your sudoku-solving

More information

Solving a Puzzle Ring

Solving a Puzzle Ring Solving a Puzzle Ring Copyright 2008 by Gary Wooding These instructions are for solving a 4-ring puzzle ring something like this one. Your ring may not be identical to this one, but the important point

More information

Chapter 2. Making Shapes

Chapter 2. Making Shapes Chapter 2. Making Shapes Let's play turtle! You can use your Pencil Turtle, you can use yourself, or you can use some of your friends. In fact, why not try all three? Rabbit Trail 4. Body Geometry Can

More information

Unit 6 Geometry: Constructing Triangles and Scale Drawings

Unit 6 Geometry: Constructing Triangles and Scale Drawings Unit 6 Geometry: Constructing Triangles and Scale Drawings Introduction In this unit, students will construct triangles from three measures of sides and/or angles, and will decide whether given conditions

More information

Shapes Puzzle 1. Shapes Puzzle 2. Shapes Puzzle 3

Shapes Puzzle 1. Shapes Puzzle 2. Shapes Puzzle 3 Shapes Puzzle The twelve pentominoes are shown on the left. On the right, they have been placed together in pairs. Can you show which two pentominoes have been used to make each shape? (Each pentomino

More information

TIMBERVIEW HIGH PRESENTS: CRAFTING CORNELL NOTES FOCUSED NOTE-TAKING THE CORNELL WAY

TIMBERVIEW HIGH PRESENTS: CRAFTING CORNELL NOTES FOCUSED NOTE-TAKING THE CORNELL WAY TIMBERVIEW HIGH PRESENTS: CRAFTING CORNELL NOTES FOCUSED NOTE-TAKING THE CORNELL WAY OUR GOAL OUR GOAL IS TO CREATE A COLLEGE- GOING CULTURE AND PREPARE EVERY STUDENT TO COMPLETE COLLEGE AND ENTER INTO

More information

Grade 3 Module 3 Lessons 1 21

Grade 3 Module 3 Lessons 1 21 Eureka Math 2015 2016 Grade 3 Module 3 Lessons 1 21 Eureka Math, Published by the non-profit Great Minds. Copyright 2015 Great Minds. No part of this work may be reproduced, distributed, modified, sold,

More information

Kenken For Teachers. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles June 27, 2010. Abstract

Kenken For Teachers. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles June 27, 2010. Abstract Kenken For Teachers Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles June 7, 00 Abstract Kenken is a puzzle whose solution requires a combination of logic and simple arithmetic skills.

More information

Cognitive Abilities Test Practice Activities. Teacher Guide. Form 7. Quantitative Tests. Level. Cog

Cognitive Abilities Test Practice Activities. Teacher Guide. Form 7. Quantitative Tests. Level. Cog Cognitive Abilities Test Practice Activities Teacher Guide Form 7 Quantitative Tests Level 9 Cog Test 4: Number Analogies, Level 9 Part 1: Overview of Number Analogies An analogy draws parallels between

More information

10 th POLISH SUDOKU CHAMPIONSHIP INSTRUCTION BOOKLET. February 22, 2015 IMPORTANT INFORMATION:

10 th POLISH SUDOKU CHAMPIONSHIP INSTRUCTION BOOKLET. February 22, 2015 IMPORTANT INFORMATION: 10 th POLISH SUDOKU CHAMPIONSHIP February 22, 2015 INSTRUCTION BOOKLET IMPORTANT INFORMATION: 1. Answer form can be sent many times, but only the last version will be considered. 2. In case of a tie, the

More information

Grade 4 Mathematics Geometry: Lesson 4

Grade 4 Mathematics Geometry: Lesson 4 Grade 4 Mathematics Geometry: Lesson 4 Read aloud to the students the material that is printed in boldface type inside the boxes. Information in regular type inside the boxes and all information outside

More information

Unit 1 Number Sense. All other lessons in this unit are essential for both the Ontario and WNCP curricula.

Unit 1 Number Sense. All other lessons in this unit are essential for both the Ontario and WNCP curricula. Unit Number Sense In this unit, students will study factors and multiples, LCMs and GCFs. Students will compare, add, subtract, multiply, and divide fractions and decimals. Meeting your curriculum Many

More information

Multiplication and Division

Multiplication and Division E Student Book 6 7 = 4 Name Series E Contents Topic Multiplication facts (pp. 7) 5 and 0 times tables and 4 times tables 8 times table and 6 times tables Date completed Topic Using known facts (pp. 8 )

More information

Setting up a basic database in Access 2003

Setting up a basic database in Access 2003 Setting up a basic database in Access 2003 1. Open Access 2. Choose either File new or Blank database 3. Save it to a folder called customer mailing list. Click create 4. Double click on create table in

More information

Drawing an Umbrella using Bezier Curves in CS3 Fireworks

Drawing an Umbrella using Bezier Curves in CS3 Fireworks Bezier Curves are the mathematical means of making nice smooth and balanced curves in most computer graphics programs. They can be extremely annoying and difficult to use for complete beginners. The things

More information

If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

More information

Scanning and Patching Certificates in Corel Photo-Paint 10 or later.

Scanning and Patching Certificates in Corel Photo-Paint 10 or later. Scanning and Patching Certificates in Corel Photo-Paint 10 or later. Don t Give Up After the First Try! It will take a long time to patch together your first image. But the more you do it, the better you

More information

Tasks to Move Students On

Tasks to Move Students On Maths for Learning Inclusion Tasks to Move Students On Part 1 Maths for Learning Inclusion (M4LI) Tasks to Move Students On Numbers 1 10 Structuring Number Maths for Learning Inclusion (M4LI) Tasks to

More information

GRAPH PAPER PROGRAMMING 1

GRAPH PAPER PROGRAMMING 1 GRAPH PAPER PROGRAMMING 1 VOCABULARY Algorithm - A list of steps that you can follow to finish a task Programming Language - a formal constructed language designed to communicate instructions to a machine,

More information

How to Play Chess. There are six different chess pieces. (King, Queen, Rook, Bishop, Knight, Pawn)

How to Play Chess. There are six different chess pieces. (King, Queen, Rook, Bishop, Knight, Pawn) How to Play Chess The chess board is placed between the two players as shown in the diagram, White sitting on one side (the bottom of the diagram) and Black on the other side (the top of the diagram).

More information

100 40 5 4 = 149. how many left? what s the difference? 5 60 80 4 = 149. take away. Maths in School. minus. Subtraction in School.

100 40 5 4 = 149. how many left? what s the difference? 5 60 80 4 = 149. take away. Maths in School. minus. Subtraction in School. 100 40 5 4 = 149 0 635 735 775 780 784 1000 8 5 2 1 5 3 how many left? what s the difference? 5 60 80 4 = 149 take away 0 635 640 700 780 784 1000 Maths in School minus Subtraction in School by Kate Robinson

More information

Grade 4 Mathematics Patterns, Relations, and Functions: Lesson 1

Grade 4 Mathematics Patterns, Relations, and Functions: Lesson 1 Grade 4 Mathematics Patterns, Relations, and Functions: Lesson 1 Read aloud to the students the material that is printed in boldface type inside the boxes. Information in regular type inside the boxes

More information

Qixi Festival - Chinese Valentine's Day

Qixi Festival - Chinese Valentine's Day LogicMastersIndia August 2010 Sudoku Test Qixi Festival - Chinese Valentine's Day Date:14 th (00:00 GMT) ~ 16 th (23:59 GMT) August 2010 Time : 120 minutes In late summer, the stars Altair and Vega are

More information

Hints and tips on making your application

Hints and tips on making your application BPS 2016: Hints and tips on making your application Version 5 This document provides hints and tips about how to make your BPS 2016 application. Visit our YouTube channel - Rural Payments Agency - BPS

More information

USER S MANUAL. Thank you for buying a Wide Body Boomerang Bead Loom. We re sure you re going to love working with this professional quality loom.

USER S MANUAL. Thank you for buying a Wide Body Boomerang Bead Loom. We re sure you re going to love working with this professional quality loom. USER S MANUAL Thank you for buying a Wide Body Boomerang Bead Loom We re sure you re going to love working with this professional quality loom. It s quite easy to set up and use. You ll be started on your

More information

Elementary Algebra. Section 0.4 Factors

Elementary Algebra. Section 0.4 Factors Section 0.4 Contents: Definitions: Multiplication Primes and Composites Rules of Composite Prime Factorization Answers Focus Exercises THE MULTIPLICATION TABLE x 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5

More information

CHAPTER 1: PATTERNS AND ALGEBRAIC THINKING

CHAPTER 1: PATTERNS AND ALGEBRAIC THINKING CHAPTER 1: PATTERNS AND ALGEBRAIC THINKING Algebra skills are essential for your future students. Why? Here are just a few reasons: Mathematics, and especially algebra, is the language of science and modern

More information

General Excel Instructions. Excel Instructions

General Excel Instructions. Excel Instructions Excel Instructions This is a general document designed to assist you with understanding the Excel functions (formulas) and how they work. The contents are: Entering Excel Formulas... 2 How Formulas Work...

More information

Colour by Numbers Image Representation

Colour by Numbers Image Representation Activity 2 Colour by Numbers Image Representation Summary Computers store drawings, photographs and other pictures using only numbers. The following activity demonstrates how they can do this. Curriculum

More information

Sudoku Puzzles and How to Solve Them

Sudoku Puzzles and How to Solve Them 1 258 NAW 5/7 nr. 4 december 2006 Sudoku Puzzles and How to Solve Them Andries E. Brouwer Andries E. Brouwer Technische Universiteit Eindhoven Postbus 513, 5600 MB Eindhoven aeb@win.tue.nl Recreational

More information

MATHEMATICS Y3 Using and applying mathematics 3810 Solve mathematical puzzles and investigate. Equipment MathSphere www.mathsphere.co.

MATHEMATICS Y3 Using and applying mathematics 3810 Solve mathematical puzzles and investigate. Equipment MathSphere www.mathsphere.co. MATHEMATICS Y3 Using and applying mathematics 3810 Solve mathematical puzzles and investigate. Equipment Paper, pencil, ruler Dice, number cards, buttons/counters, boxes etc MathSphere 3810 Solve mathematical

More information

T e 8 h W r o l r d S d u o d k o u C a h m m i p o i n o s n h s i h p I s n t s r t u r c u t c i t o i n B o o k o l k e l t

T e 8 h W r o l r d S d u o d k o u C a h m m i p o i n o s n h s i h p I s n t s r t u r c u t c i t o i n B o o k o l k e l t The 8th World Sudoku Championship Instruction Booklet Instruction Booklet for the 8th World Sudoku Championship This booklet explains the competition procedures, scoring, and ranking in the league table,

More information

Barsch Learning Style Inventory Jeffery Barsch, EdD.

Barsch Learning Style Inventory Jeffery Barsch, EdD. Barsch Learning Style Inventory Jeffery Barsch, EdD. Name Date To gain a better understanding of yourself as a learner, it is useful to identify the way you prefer to learn. Learning is easier when study

More information

BATTLESHIPS SOLVING GUIDE

BATTLESHIPS SOLVING GUIDE Figure. Figure. Figure. BAEHI VIG GUIDE 6 Figure. Figure. Figure 6. BAEHI VIG GUIDE Battleships puzzles are a solitaire version of the classic paper-and-pencil game of the same name. he object of each

More information

Creating Simple Tables and Charts using Microsoft Excel 2013

Creating Simple Tables and Charts using Microsoft Excel 2013 2015 Bow Valley College 1 Microsoft Excel Vocabulary Creating Simple Tables and Charts using Microsoft Excel 2013 Column: A grouping of information or data organized from top to bottom. In Excel columns

More information

quick start guide A Quick Start Guide inflow Support GET STARTED WITH INFLOW

quick start guide A Quick Start Guide inflow Support GET STARTED WITH INFLOW GET STARTED WITH INFLOW quick start guide Welcome to the inflow Community! This quick-start guide includes all the important stuff to get you tracking your inventory before you know it! Just follow along

More information

How to Make the Most of Excel Spreadsheets

How to Make the Most of Excel Spreadsheets How to Make the Most of Excel Spreadsheets Analyzing data is often easier when it s in an Excel spreadsheet rather than a PDF for example, you can filter to view just a particular grade, sort to view which

More information

Microsoft Office Excel 2007 Training. Created & Hosted by: Hagop (Jack) Hadjinian I.A., Information Technology

Microsoft Office Excel 2007 Training. Created & Hosted by: Hagop (Jack) Hadjinian I.A., Information Technology Microsoft Office Excel 2007 Training Created & Hosted by: Hagop (Jack) Hadjinian I.A., Information Technology Course Contents Lesson 1: Meet the workbook Lesson 2: Enter data Lesson 3: Edit data and revise

More information

Using Quizzes To Enhance Your Safety Program

Using Quizzes To Enhance Your Safety Program Using Quizzes To Enhance Your Safety Program Quizzes can improve and enhance your safety training program. This guide will help you make the most of the Quizzes by helping you make Quiz delivery easy and

More information

Graph Paper Programming

Graph Paper Programming 4 LESSON NAME: Graph Paper Programming Lesson time: 45 60 Minutes : Prep time: 10 Minutes Main Goal: Help students understand how coding works. OVERVIEW: By programming one another to draw pictures, students

More information

Bon Voyage! Puzzle Cruise / 18-19th June 2011 UK Puzzle Association Instructions Booklet

Bon Voyage! Puzzle Cruise / 18-19th June 2011 UK Puzzle Association Instructions Booklet / 18-19th June 2011 UK Puzzle Association http://www.ukpuzzles.org/contests.php?contestid=8 Anchors Aweigh! Welcome aboard the perplexing! The national puzzle test for both the United Kingdom and the United

More information

State Spaces Graph Search Searching. Graph Search. CPSC 322 Search 2. Textbook 3.4. Graph Search CPSC 322 Search 2, Slide 1

State Spaces Graph Search Searching. Graph Search. CPSC 322 Search 2. Textbook 3.4. Graph Search CPSC 322 Search 2, Slide 1 Graph Search CPSC 322 Search 2 Textbook 3.4 Graph Search CPSC 322 Search 2, Slide 1 Lecture Overview 1 State Spaces 2 Graph Search 3 Searching Graph Search CPSC 322 Search 2, Slide 2 State Spaces Idea:

More information

An Australian Microsoft Partners in Learning (PiL) Project

An Australian Microsoft Partners in Learning (PiL) Project An Australian Microsoft Partners in Learning (PiL) Project 1 Learning objects - Log on to the website: http://www.curriculumsupport.education.nsw.gov.au/countmein/ - Select children Select children - This

More information

Copyright 2006 James E. Riley. All rights reserved. SUPER WORDOKU is a trademark of Good Year Books.

Copyright 2006 James E. Riley. All rights reserved. SUPER WORDOKU is a trademark of Good Year Books. Are available for most basic curriculum subjects plus many enrichment areas. For more Good Year Books, contact your local bookseller or educational dealer. For a complete catalog with information about

More information

5CHAMPIONSHIP. Puzzle Examples SUDOKU. th WORLD. from PHILADELPHIA. Lead Sponsor

5CHAMPIONSHIP. Puzzle Examples SUDOKU. th WORLD. from  PHILADELPHIA. Lead Sponsor th WORLD SUDOKU CHAMPIONSHIP PHILADELPHIA A P R M A Y 0 0 0 Puzzle Examples from http://www.worldpuzzle.org/wiki/ Lead Sponsor Classic Sudoku Place the digits through into the empty cells in the grid (a

More information

Would You Like To Earn $1000 s With The Click Of A Button?

Would You Like To Earn $1000 s With The Click Of A Button? Would You Like To Earn $1000 s With The Click Of A Button? (Follow these easy step by step instructions and you will) This Version of the ebook is for all countries other than the USA. If you need the

More information

Hacking. Hacking Challenges

Hacking. Hacking Challenges Hacking Hacking is the process of creating and modifying programs in the program memory of your flotation device and using those programs to accomplish various tasks. Hacking Challenges The basic unit

More information

Grade 1 Module 1 Lessons 1 39

Grade 1 Module 1 Lessons 1 39 Eureka Math 2015 2016 Grade 1 Module 1 Lessons 1 39 Eureka Math, Published by the non-profit Great Minds. Copyright 2015 Great Minds. No part of this work may be reproduced, distributed, modified, sold,

More information

10 Body Language Blunders. 1. A Closed Body. 2. Clenched Fists. SalesHQ

10 Body Language Blunders. 1. A Closed Body. 2. Clenched Fists. SalesHQ 10 Body Language Blunders SalesHQ 1. A Closed Body Whether sitting or standing, your body should be loose and open. Do not cross your arms. Do not cross your legs. While you may think it conveys a sense

More information

Support Booklet for. Student Name: Coach Name(s): Start Date: What does the student want to achieve from this programme?

Support Booklet for. Student Name: Coach Name(s): Start Date: What does the student want to achieve from this programme? Support Booklet for Student Name: Name(s): Start : What does the student want to achieve from this programme? Target End : Sarah Emson Page 1 Why and Who? Using Turn Tables Using Turn Tables Individual

More information

Whiteboard Music Lessons

Whiteboard Music Lessons Whiteboard Music Lessons Module Two Middle Primary/Elementary Lesson Plans Whiteboard Music Lessons - Module 2. Lesson Plans Page 1 Contents Introduction page 3 Lesson Plan Guides: Lesson 1 page 8 Lesson

More information

Session 6 Number Theory

Session 6 Number Theory Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple

More information

The Kakuro Kraze. The Rules of Kakuro. Kakuro puzzles are currently sweeping the world!

The Kakuro Kraze. The Rules of Kakuro. Kakuro puzzles are currently sweeping the world! The Kakuro Kraze Kakuro puzzles are currently sweeping the world! In Japan, the puzzles are well known and second in popularity only to Sudoku puzzles, which spawned a puzzle publishing craze in Europe

More information

Instructions for applying data validation(s) to data fields in Microsoft Excel

Instructions for applying data validation(s) to data fields in Microsoft Excel 1 of 10 Instructions for applying data validation(s) to data fields in Microsoft Excel According to Microsoft Excel, a data validation is used to control the type of data or the values that users enter

More information

Trial 9 No Pill Placebo Drug Trial 4. Trial 6.

Trial 9 No Pill Placebo Drug Trial 4. Trial 6. An essential part of science is communication of research results. In addition to written descriptions and interpretations, the data are presented in a figure that shows, in a visual format, the effect

More information

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 3 Greater Than, Less Than, Equal To

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 3 Greater Than, Less Than, Equal To Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Please watch Section 3 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm

More information

Table of Contents. 10 Drawing Portraits: Fundamentals. Introduction and how this book is organized 13

Table of Contents. 10 Drawing Portraits: Fundamentals. Introduction and how this book is organized 13 10 Drawing Portraits: Fundamentals Table of Contents Introduction and how this book is organized 13 Section One: 15 Drawing fundamentals for the new artist. 15 Some introductory words for the newbie artist

More information

NUMB3RS Activity: Sudoku Puzzles. Episode: All s Fair

NUMB3RS Activity: Sudoku Puzzles. Episode: All s Fair Teacher Page 1 NUMBRS Activity: Sudoku Puzzles Topic: Sudoku Puzzles Grade Level: 8-10 Objective: Explore sudoku puzzles and Latin squares Time: 60-90 minutes Introduction In All s Fair, Alan, Larry, and

More information

Unit 6. Five daily lessons. Year 3 Autumn term. Unit Objectives Year 3. Resources needed to teach this unit: Shape and space Reasoning about shapes

Unit 6. Five daily lessons. Year 3 Autumn term. Unit Objectives Year 3. Resources needed to teach this unit: Shape and space Reasoning about shapes Unit 6 Shape and space Reasoning about shapes Five daily lessons Year 3 Autumn term Classify and describe 3-D and 2-D shapes, referring to reflective symmetry, faces, sides/edges, vertices, angles. (pages

More information

The Five Basic Skills of Drawing

The Five Basic Skills of Drawing The Five Basic Skills of Drawing by Dan Johnson A free guide for improving your drawing skills, originally published on Right Brain Rockstar http://rightbrainrockstar.com/the-five-basic-skills-of-drawing/

More information

5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1

5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1 MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing

More information

GUITAR THEORY REVOLUTION. Part 1: How To Learn All The Notes On The Guitar Fretboard

GUITAR THEORY REVOLUTION. Part 1: How To Learn All The Notes On The Guitar Fretboard GUITAR THEORY REVOLUTION Part 1: How To Learn All The Notes On The Guitar Fretboard Contents Introduction Lesson 1: Numbering The Guitar Strings Lesson 2: The Notes Lesson 3: The Universal Pattern For

More information

UNPLUGGED Graph Paper Programming. Basic lesson time includes activity only. Introductory and Wrap-Up suggestions can be used

UNPLUGGED Graph Paper Programming. Basic lesson time includes activity only. Introductory and Wrap-Up suggestions can be used U UNPLUGGED Graph Paper Programming Lesson time: 20 Minutes to delve deeper when time allows. Basic lesson time includes activity only. Introductory and Wrap-Up suggestions can be used LESSON OVERVIEW

More information

SHELL INDUSTRIAL APTITUDE BATTERY PREPARATION GUIDE

SHELL INDUSTRIAL APTITUDE BATTERY PREPARATION GUIDE SHELL INDUSTRIAL APTITUDE BATTERY PREPARATION GUIDE 2011 Valtera Corporation. All rights reserved. TABLE OF CONTENTS OPERATIONS AND MAINTENANCE JOB REQUIREMENTS... 1 TEST PREPARATION... 2 USE OF INDUSTRIAL

More information

TI-86 Graphing Calculator Keystroke Guide

TI-86 Graphing Calculator Keystroke Guide TI-86 Graphing Calculator Keystroke Guide In your textbook you will notice that on some pages a key-shaped icon appears next to a brief description of a feature on your graphing calculator. In this guide

More information

Creating Clueless Puzzles Gerard Butters Frederick Henle James Henle Colleen McGaughey 1

Creating Clueless Puzzles Gerard Butters Frederick Henle James Henle Colleen McGaughey 1 Creating Clueless Puzzles Gerard Butters Frederick Henle James Henle Colleen McGaughey This is an invitation and a report. The report is of our efforts to create puzzles of a special kind. The invitation

More information

POLAR COORDINATES: WHAT THEY ARE AND HOW TO USE THEM

POLAR COORDINATES: WHAT THEY ARE AND HOW TO USE THEM POLAR COORDINATES: WHAT THEY ARE AND HOW TO USE THEM HEMANT D. TAGARE. Introduction. This note is about polar coordinates. I want to explain what they are and how to use them. Many different coordinate

More information

Thinking Skills. Lesson Plan. Introduction

Thinking Skills. Lesson Plan. Introduction xxx Lesson 18 Thinking Skills Overview: This lesson provides basic foundational information about two types of thinking skills: critical and creative. Students have used critical and creative skills each

More information

Fraction Tools. Martin Kindt & Truus Dekker. ------------ 3n 4 -----

Fraction Tools. Martin Kindt & Truus Dekker. ------------ 3n 4 ----- Fraction Tools - + - 0 - n + n Martin Kindt & Truus Dekker 0 Section A Comparing Fractions - Parts and pieces (). Of each figure, color part. Be as precise as possible.. Of each figure, color part. Each

More information

Coordinates. List of Contents. Coords (1st Quadrant) Practice ~ 1. Coords (1st Quadrant) Practice ~ 2. Coords (4 Quadrants) Practice ~ 3

Coordinates. List of Contents. Coords (1st Quadrant) Practice ~ 1. Coords (1st Quadrant) Practice ~ 2. Coords (4 Quadrants) Practice ~ 3 oordinates List of ontents oords (st Quadrant) Practice ~ oords (st Quadrant) Practice ~ oords ( Quadrants) Practice ~ Tests (st Quadrant) to Tests ( Quadrants) to ards for playing oord ingo oordinate

More information

1974 Rubik. Rubik and Rubik's are trademarks of Seven Towns ltd., used under license. All rights reserved. Solution Hints booklet

1974 Rubik. Rubik and Rubik's are trademarks of Seven Towns ltd., used under license. All rights reserved. Solution Hints booklet # # R 1974 Rubik. Rubik and Rubik's are trademarks of Seven Towns ltd., used under license. All rights reserved. Solution Hints booklet The Professor s Cube Solution Hints Booklet The Greatest Challenge

More information

Inspiration Quick Start Tutorial

Inspiration Quick Start Tutorial Inspiration Quick Start Tutorial This tutorial is a great starting point for learning how to use Inspiration. Please plan on about 45 minutes from start to finish. If you re new to Inspiration and like

More information

USING PLAYING CARDS IN THE CLASSROOM. What s the Pattern Goal: To create and describe a pattern. Closest to 24!

USING PLAYING CARDS IN THE CLASSROOM. What s the Pattern Goal: To create and describe a pattern. Closest to 24! USING PLAYING CARDS IN THE CLASSROOM You can purchase a deck of oversized playing cards at: www.oame.on.ca PRODUCT DESCRIPTION: Cards are 4 ½ by 7. Casino quality paperboard stock, plastic coated. The

More information

RECREATIONAL MATHEMATICS IN THE PARK

RECREATIONAL MATHEMATICS IN THE PARK PART 2 EXAMPLES OF MATH TRAILS We give four examples of trails in particular situations: A park playground, a city, a zoo, and a shopping mall. The playground and the shopping mall are each a single actual

More information

Hooray for the Hundreds Chart!!

Hooray for the Hundreds Chart!! Hooray for the Hundreds Chart!! The hundreds chart consists of a grid of numbers from 1 to 100, with each row containing a group of 10 numbers. As a result, children using this chart can count across rows

More information

Rational Number Project

Rational Number Project Rational Number Project Fraction Operations and Initial Decimal Ideas Lesson 10: Overview Students develop an understanding of thousandths and begin to look at equivalence among tenths, hundredths, and

More information

MATH 111. 3. Numbers & Operations

MATH 111. 3. Numbers & Operations MATH 111 3. Numbers & Operations Chapter 3 - Numbers & Operations 2013 Michelle Manes, University of Hawaii Department of Mathematics These materials are intended for use with the University of Hawaii

More information

PQRST 11 PUZZLE COMPETITION

PQRST 11 PUZZLE COMPETITION PQRST PUZZLE COMPETITION PUZZLE 0 (0+0 points penalty for wrong answers) + points Wild Operations You re writing mathematical operations, using letters instead of digits to represent numbers. You take

More information

Lesson #13 Congruence, Symmetry and Transformations: Translations, Reflections, and Rotations

Lesson #13 Congruence, Symmetry and Transformations: Translations, Reflections, and Rotations Math Buddies -Grade 4 13-1 Lesson #13 Congruence, Symmetry and Transformations: Translations, Reflections, and Rotations Goal: Identify congruent and noncongruent figures Recognize the congruence of plane

More information

MAKING MATH MORE FUN BRINGS YOU FUN MATH GAME PRINTABLES FOR HOME OR SCHOOL

MAKING MATH MORE FUN BRINGS YOU FUN MATH GAME PRINTABLES FOR HOME OR SCHOOL MAKING MATH MORE FUN BRINGS YOU FUN MATH GAME PRINTABLES FOR HOME OR SCHOOL THESE FUN MATH GAME PRINTABLES are brought to you with compliments from Making Math More Fun at and Math Board Games at Copyright

More information

Table of Contents. Graphing with Excel 1

Table of Contents. Graphing with Excel 1 Table of Contents Graphing with Excel 1 1. Graphing Data 1.1. Starting the Chart Wizard 1.2. Selecting the Data 1.3. Selecting the Chart Options 1.3.1. Titles Tab 1.3.2. Axes Tab 1.3.3. Gridlines Tab 1.3.4.

More information

Microsoft PowerPoint Exercises 4

Microsoft PowerPoint Exercises 4 Microsoft PowerPoint Exercises 4 In these exercises, you will be working with your Music Presentation file used in part 1 and 2. Open that file if you haven t already done so. Exercise 1. Slide Sorter

More information

First Grade Exploring Two-Digit Numbers

First Grade Exploring Two-Digit Numbers First Grade Exploring Two-Digit Numbers http://focusonmath.files.wordpress.com/2011/02/screen-shot-2011-02-17-at-3-10-19-pm.png North Carolina Department of Public Instruction www.ncdpi.wikispaces.net

More information

Math Skill Development Games

Math Skill Development Games Part One Math Skill Development Games COPYRIGHTED MATERIAL Chapter One Paper - and - Pencil Games Paper - and - pencil games are popular among children and require little preparation. The games in this

More information

Improvisation Tools: Pentatonic by Gunharth Randolf

Improvisation Tools: Pentatonic by Gunharth Randolf Improvisation Tools: Pentatonic by Gunharth Randolf This is the first of a series of articles to familiarize you with the basic tools used in improvisation. In order to get the most out of this article

More information

GRADE 3 FLORIDA. Multiplication WORKSHEETS

GRADE 3 FLORIDA. Multiplication WORKSHEETS GRADE 3 FLORIDA Multiplication WORKSHEETS Multiplication explore You will need: 4 counters What to do: Chef Charlie has cupcakes on some trays in the oven. There are the same number of cupcakes on each

More information

How to use Microsoft Excel 2007

How to use Microsoft Excel 2007 Microsoft Office Excel is a powerful tool used to create and format spreadsheets. Spreadsheets allow information to be organized in rows and tables and analyzed with automatic mathematics. Spreadsheets

More information

1 Mathematics Curriculum

1 Mathematics Curriculum New York State Common Core 1 Mathematics Curriculum G R A D E GRADE 1 MODULE 1 Topic F Development of Addition Fluency Within 10 1.OA.3, 1.OA.6 Focus Standard: 1.OA.3 Apply properties of operations as

More information

Multiplication and Division Fact Families

Multiplication and Division Fact Families Multiplication and Division Fact Families Objectives To review fact families and the Multiplication/Division Facts Table; and to guide children as they practice multiplication and division facts. www.everydaymathonline.com

More information