"Where the mind is without fear and the head is held high; Where knowledge is free; Where the world has not been broken up into fragments by narrow domestic walls; Where words come from the depth of truth; Where tireless striving stretches its arms toward perfection; Where the clear stream of reason has not lost its way into the dreary desert sand of dead habit; Into that heaven of freedom it should be our intent. ( my adaption from Rabindranath Tagores famous lines)

Saturday, July 16, 2005

Sudoku_Beginners 4x4 with color symbols

Probably this would be more appealing than the digits 1 to 4 used in the last teaching exercise.

Tuesday, July 12, 2005

Sudoku _ beginners logic for 4 x 4 grid

Sudoku - for beginners

These images need no explanation.The concept of Sudoku is shown under various grid sizes. I have done this to encourage beginners to attempt the main 9 x 9 Sudoku. No maths or calculations are involved, just logic. Colours or symbols can be substituted for the numbers. It is just like a Crossword Puzzle

Sudoku - Tutorials, Contests and Cheating

SUDOKU : Tutorials, Contests and Cheating.
The main site of the pioneer , Wayne Gould, trading as Pappocom is http://www.sudoku.com/ .Clicking on the link on the left column “how to solve” leads you to a very good tutorial.
Another good tutorial is found here along with examples : http://www.puzzle.jp/letsplay/sudokuruleflash-e.html

Every month one or more newspapers/magazines presently hold contests. One of the sites that hold this is http://www.sudoku.com/contest.htm where every month there is a contest and the solution and winner is declared the following month. You may want to try it !

If you want an unfair (Cheat !) advantage, or need to know the answer to the latest one that's driving you nutty, or you just want to know what the fuss is all about, why not check out Guy's Sudoku Solver at http://www.sudoku-solver.org.uk/ ?
This will solve your Sudoku problem in seconds while you watch!
You can also use it to create your own puzzles and solutions to give to friends. Either way you'll be onto a winner. Unfortunately it does not make use of the techniques stated herein but solves them inelegantly using Computer Brute Force.

Another site, where, given a puzzle you can solve it there is : http://www.sudokusolver.co.uk/

Sudoku - Solving techniques

SUDOKUSolving techniques
Fundamentally there exist 2 logical methods of solution:
First: In elimination, progress is made by successively eliminating candidate numbers from one or more cells to leave just one choice. After each answer has been achieved, another scan may be performed - usually checking to see the effect of the latest number. There are a number of elimination tactics. One of the most common is "unmatched candidate deletion". Cells with identical sets of candidate numbers are said to be matched if the quantity of candidate numbers in each is equal to the number of cells containing them. For example, cells are said to be matched within a particular row, column, or region if two cells contain the same pair of candidate numbers (p,q) and no others, or if three cells contain the same triple of candidate numbers (p,q,r) and no others. These are essentially coincident contingencies. These numbers (p,q,r) appearing as candidates elsewhere in the same row, column, or region in unmatched cells can be deleted.
Second: In the what-if approach, a cell with only two candidate numbers is selected and a guess is made. The steps above are repeated unless a duplication is found, in which case the alternative candidate is the solution. In logical terms this is known as reductio ad absurdum : For each candidate for a cell, the question is posed: will entering a particular number prevent completion of the other placements of that number? If the answer if yes, then that candidate can be eliminated. The what-if approach requires a pencil and eraser. This approach may be frowned on by logical purists as too much Trial and Error but it can arrive at solutions fairly rapidly.

Ideally one needs to find a combination of techniques which avoids some of the drawbacks of the above elements. The counting of regions, rows, and columns can feel boring. Writing candidate numbers into empty cells can be time-consuming. The what-if approach can be confusing unless you are well organised. The SOLUTION is to find a technique which minimises counting, marking up, and rubbing out.
To get this drudgery out of the way try this link http://www.angusj.com/sudoku/ which is “Simple Sudoku” and it does the “markings” for you. In this site also see the "a step by step guide for solving Sudoku" on the very first page. It is excellent.

For persons not familiar with reductio ad absurdum, it is a mode of argumentation that that seeks to establish a contention by deriving an absurdity from its denial, thus arguing that a thesis must be accepted because its rejection would be untenable. It is a style of reasoning that has been employed throughout the history of mathematics and philosophy from classical antiquity onwards. Say, I have a Su Doku puzzle partially filled in and I see there are two squares within a box, row or column where some symbol could go. If I show that putting it in one box leads to a chain of logical deductions from which a contradiction follows, then the alternative must be correct. Many mathematical theorems can be proven using proof by use of reductio ad absurdum, for example the fact that the square root of 2 is not the ratio of two integers.( My observation: Even though the proof of the irrationality of the square root of two using this method is accepted by mathematicians ,I still believe it is not the strictest proof ; it is indirect proof.)

Monday, July 11, 2005

Sudoku - History

Sudoku - Its History
The game was created around 1783 by a Swiss mathematical genius named Leonard Euler. He called such puzzles as Latin Squares. These were also grids of equal dimensions ( starting with 4 x 4) in which every number or symbol occurs once in each row and column. It differs from the Sudoku grid only in the subdivision of blocks conditionality. In the 1980’s the game turned up in the United States and was renamed the Number Place Game. However a Japanese puzzle magazine found an example of this in an American magazine, modified the game into a slightly more difficult verssion and christened it as Sudoku which means “number single”. It was introduced to the Japanese readers in 1984 and the game became a craze. However the present domination of the game is due to a retired New Zealand lawyer and puzzle fan Wayne Gould, aged 59, who spent six years developing a computer program to mass produce the puzzles. He promoted this final version to the Times and Daily Mail in November 2004. The immense popularity of Soduku internationally is being dubbed in the world media variously as the “Rubik’s cube of the 20th Century” or the “fastest growing puzzle in the world today’. As I mentioned earlier there is no mathematics involved and a person need not be even literate. It is amenable to symbols or colors.

Sudoku -What is it ?

SODUKU - What is it
For anyone who doesn't know, it's a puzzle now found in most newspapers, books and online. A simple-looking grid of nine rows by nine, split into nine boxes, each containing nine squares, it looks like just another numbers game. Almost every newspaper carries this game now and already it is on cellphones and TV and recently introduced as a board game. It's the latest craze and it is addictive ! Addicts are as obsessed as 1980s teenagers fixated on the Rubik's cube. The internet is awash with chat about Su Doku and programmers are tapping away to find the best system for creating and solving the puzzles. Before I go further into this I quote from the 2 great mathematicians:

"Puzzles and Paradoxes have been popular since antiquity, and, in amusing themselves with these playthings men have sharpened their wits and whetted their ingenuity. But it was not for amusement alone that Kepler, Pascal, Fermat, Leibnitz, Euler, Lagrange, Cayley and many others devoted so much time to puzzles. Researches in Recreational Mathematics were guided by the same principles and required the same faculties as the researches leading to the most profound discoveries in mathematics or science. (Edward Kasner & James R. Newman)"

Again, Wait on. It is nothing to do with Mathematics. Even illiterates can solve them. Sudoku is not a mathematical or arithmetical puzzle. It works just as well if the numbers are substituted with letters or some other symbols, but numbers work best.

SUDOKU _ How to solve.
It is very simple. Fill in the blanks in the 9 x 9 Grid so that every row, column and every 3 x 3 box contains the digit 1 thru 9. There is no math involved. Nothing has to add up to anything else. You solve the puzzle through logic and reasoning. The puzzle rewards Applied Logic as it encourages you to apply the principles of deduction and induction. It is simple and yet fiendish and once played, utterly addictive. If you just try guessing you wont get very far, I assure you !