# Workshop 1

Welcome to the first workshop of the Sudoku Experience from the Empirical Modelling Research Group at the University of Warwick.

## Preparation

Most -- perhaps all -- of you will already be familiar with Sudoku puzzles. They require inserting the digits 1 to 9 into a 9-by-9 grid in such a way that all 9 digits appear in each row, column and each 3-by-3 subregion (marked in bold on the grid). For this Workshop you will use the Sudoku Environment - you can see the link caption in the panel on the left - but wait until the end of this paragraph before you go there! If you are new to Sudoku you might like to look at the Wikipedia article listed in the Technical page, but come back here as soon as you think you have got the idea. Notice that there is no arithmetic involved, the symbols 1 to 9 are just convenient - any distinct symbols would do. But the puzzle solving - partly because it's logical - has a lot to do with mathematics. As the Oxford mathematician Andrew Hodges has observed: "Sudoku may not require long multiplication or division, but it is a very good puzzle that replicates the pattern of thinking required to solve quite complex logical problems in maths." There is something attractive to many people about abstract puzzles that admit solution by pure reasoning. They are the kind of puzzles that have been of special interest in computer science because they are the puzzles that computers can be readily programmed to solve. Here we are equally interested in how humans solve the puzzle, so have a go now. There is probably not enough time to finish the puzzle but spend 10 minutes putting in as many numbers as you can (but don't guess!). Have a look now at the Sudoku Environment, click on Play Colour Sudoku, read the How to play and work on Puzzle1 (ignore the Colour version for the moment, if you can!). After about 10 minutes come back here.

How many numbers did you put in? Perhaps you are an expert and finished it in 10 minutes?! That's very good, but it really doesn't matter if you only did a few entries. The whole point in this workshop is to think about how you (or your brain or mind - together with eyes and fingers probably!) went about the process. We'd like you to think about Sudoku puzzles from a broader perspective than just their solution. You will in passing be introduced to one way in which you can program a computer to solve Sudoku puzzles. But you will also be seeing how computers can help to support the thinking you do when solving Sudoku puzzles. To understand the distinction between "solving Sudoku puzzles" and "the thinking done when solving Sudoku puzzles", it helps to bear in mind that there is also something quite unattractive to some people about abstract puzzles that admit solution by pure reasoning! For such people, solving a cryptic crossword puzzle is an altogether more rewarding activity, as it engages the human mind, memory and imagination in wholly unpredictable ways. Of course, Sudoku is not just about reasoning - you have to be good at observing quickly and accurately too. So although Puzzle1 is relatively very easy try to reflect upon how you scanned the puzzle for the next entry. Did you go for where the numbers seemed most 'dense'? Did you systematically work through where the 1's go, then the 2's etc?

Beware! You will have gathered from the story so far that it is crucial for this Workshop that you reflect on your own personal activity (observing and reasoning) as you do a puzzle. So if you solve the puzzles just using one of the solver programs on the web you will have no evidence to reflect upon! In Workshops 2 and 3 we shall be thinking about computer support for human solving as well as (the quite different idea of) computer solutions.

The aim of the following tasks is to begin our investigation into the process of solving a puzzle.  Any standard Sudoku puzzle requires filling in about 50 or so squares. Every square correctly filled in is a step on the journey to the solution. But for each entry there is a separate journey (of observation and reasoning) to find out the correct digit to enter. It is this latter, invisible, 'journey' that we want to focus upon here. Sometimes it is a very simple step - especially near the end of a puzzle - if there is only one square left in a column it should be obvious what goes there! But sometimes it may be very complicated indeed to find any correct entry. As you probably saw, we have included five puzzles in the Environment and we refer in the tasks to some of them. They all have a unique solution and none of them need any complicated logic or guessing to complete. You will meet some of them again in the other Workshops. But note there is no need to complete any of them (unless you want to, of course!).

## What to do now

It might be useful to have some paper ready (or a word processor) so that you can make notes and prepare your feedback to each task. Instead of completely solving a puzzle we shall typically ask you to spend 10 minutes on it in order to get the experience of some approach or technique. You will need to work back and forth between this page and the Sudoku Environment page. Imagine the columns of the grid are labelled A- I, the rows labelled 1-9 and the regions numbered 1-9 from top left to bottom right in 'reading order'. So for example in Puzzle1 the middle square of region3 (H2) is given as 8 and the top left square of region4 (A4) is given as 3. We shall sometimes refer to a 'sector' meaning a row or column or 3x3 region.

At the end of these tasks you will be asked to report (in Feedback1) on what you have learned or discovered. It is important to be able to express your thoughts about what you have learned. Note we have called it 'Feedback' because for most of these questions we don't know any 'right answers', but we are very interested in your impressions and thoughts on the experience of puzzle-solving. You may find it convenient to write in your feedback directly onto the form on Feedback1. Notice the questions there refer to these Tasks but don't always ask exactly the same questions as appear in the Tasks.

(a) Did you notice that we've described Sudoku in two rather different ways? The second sentence in the 'Preparation' section above used the phrase '... all 9 digits appear in each row ...'  while in the first sentence under 'How to play' (Sudoku Environment) we used the phrase '.. each digit occurs exactly once in each row ...'. Are you convinced these two descriptions are equivalent? If so, can you explain why?

(b) During the time you spent on Puzzle1 how did you find new entries? For this you need to think about your own thinking! Can you describe the rules you used? Did you use any particular overall strategy?

(a) In the Sudoku Environment there is a box called 'Enable plausibles'. Click there to activate it. It is explained in the last paragraph on that page under the section 'How to play'. Spend 10 minutes on another puzzle of your choice making use of the plausibles.  Save your progress as described on the Technical page.

(b) Do you feel using the 'Enable plausibles' button is somehow cheating? Does it take away something that is actually quite an enjoyable part of the human solving, or does it just automate a routine scanning process that the human (you!) was glad to be without?

### Task 3: the invisible journey

(a) If you have done a lot of Sudokus you probably found that Puzzle1 a bit boring. Look now at Puzzle5. Can you get started using the same methods you used for Puzzle1? Can you find any method of getting started? Of course, you can always use the 'plausibles' now. Spend 5 - 10 minutes on this if you need to. If you still have not found a way of starting try the following hint. [Hint: Don't read this until you have tried Puzzle5 for at least a few minutes!! One approach is to notice first that there are four empty squares in region4 where an 8 cannot go, leaving 2 possibles. Similarly for 7 in region6. To eliminate one of the possible positions for the 8 in region4 think about what digits must go in the top row of region6. Do something similar to place the 7 in region6.]  If you used the hint you will have seen a good example of how two pieces of 'knowledge' about the grid can be combined to give a correct entry. Find at least three other (independent) bits of useful knowledge about Puzzle5.

(b) [The rest of this task is just reading these two paragraphs!] Gaining a lot of such knowledge will help a solver finish a puzzle quickly once they have got started.  Acquiring relevant knowledge about a puzzle (from observation, memory and reasoning) and putting it together to justify insertion of a digit is what we mean here by an 'invisible journey'. Of course it is not invisible to the person who takes the journey first. But it is personal and so invisible to everyone else. Similar journeys occur in many subjects: in mathematics they are called 'proofs', in science they produce 'theories' (often based on mathematics, experiments and observations) and so on. A good deal of what you learn at school (and university) has to do with effective ways of making such journeys of personal discovery visible to others. These 'journeys' are presumably connected with how well we understand our knowledge (rather than just storing it like a parrot, or a database, can do).

So this has been an important task. It has drawn attention to the difference between the simple and routine methods that worked with Puzzle1 and the more creative insight that seemed necessary for Puzzle5. The former is typical of what computer programming is good for, the latter is typical of what humans are good at.  The idea behind the Empirical Modelling you will meet in the next Workshops is to develop a framework and tools which conveniently includes both kinds of activity.

(a) Now try the colour version by ticking on the 'Enable colour' box. This fills the grid with colours that represent the actual number of an occupied cell, or a mixture of the plausible numbers for an empty cell. Read what it says in the 'How to play' section about the colour version. Spend 10 minutes again with the same puzzle that you chose for Task 2. (You could start again or make use of the new entries you saved.) Look for any ways in which the colour might help you towards the solution.

(b) Try Puzzle3 and put the cursor over the middle cell of the bottom row (E9). It is white and has 5 plausible entries. But it is easy to see that it can only be one digit, which? Stay in Puzzle3 and find an empty square with the same colour as an occupied square. Fill it in accordingly and notice how the surrounding empty squares changed. Now you can instantly spot two 'mutual pairs' which would probably (?) not have been so easy to spot just from the 'Enable plausibles' tool. Look at Puzzle2 and cell G5 for another 'misleading' white. In a later workshop you will see how we could overcome this problem.

(c) Is it any easier to solve puzzles using the colour version? Why?  Can you imagine puzzles for which using colour is not very useful?

(a) You may have already noticed that you can modify the colours used to represent a particular digit. For example, try setting the digit 1 to white (red, green and blue at maximum), and see if this can be of benefit when solving. What's a disadvantage of setting a digit to white? Does this 'setting to white' of a digit correspond with any simple rules you have found useful for puzzle-solving? Which?

(b) Can you describe some strategies for solving using colour or for changing the colours? Can you give examples of where such use of colour could be useful with some of the puzzles given?

(c) You have seen some limitations of the use of colour. What other limitations can you see in this (Flash) implementation of Sudoku? In the next Workshop you will be introduced to an Empirical Modelling environment and a more flexible model of Sudoku. We shall see then whether the limitations you have found can be overcome in that model.