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Methodology “Solving Sudoku using Linear Algebra simultaneous equations” A Sudoku puzzle has two basic rules:
- Each column, each row and each box ( 3 × 3 sub grid) must have the numbers 1 to 9.
- No column, row or box can have two squares with the same number. Someone has replaced each number from 1 to 9 in the Sudoku puzzle above with a letter. Given that the number at the end of a row or at the bottom of a column is the sum of the letters shown in that row or column, can you find which letter corresponds to which number, and then solve the Sudoku? Altogether a set of 16 equations can be formed.
For example, in the first and second columns beginning from the left of the 9 × 9 grid, we can form the following equations: c+g+k=17, f+g+a=19. In the fourth and fifth rows beginning from the top of the 9×9 grid, the following equations can be formed: k+g+m+c=23, g+p=11.
- Write out all of the equations you have. Which are the simplest looking equations? If you have an equation involving only two letters, can you write down the possible values those
Results and Conclusion
Here is one way we could solve this puzzle.
We first look at the most simple equations with only two unknowns. Equation 12 says that p+e=5. This means that both p and e must be less than or equal to 4. Equation (2) says that f+e=10, and since we know that e is less than or equal to 4 , this means that f≥6. Equation (4) says that g+p=11, so we must have g≥7. Equation (5/16) tells us that h+f=14, so h≤8. Now look at equation (7), a+e+k+h=11. The only four different positive whole numbers that can add up to 1 are 1+2+3+5. So we must have that each of a, e, k, h must be one of 1 , 2 , 3 , 5. We know that e≤4, but we now know that e≤3, and so 2≤p≤4 Also, since h≤8 and must add up with a number less than or equal to 9 to make 14 by equation 5, we have that: h=5. So by equation 5, f=9. But now by equation 2 e=1,
Letter Number
- a) c+m+h= Here are the 16 equations:
- b) f+e=
- c) k+g+m+c=
- d) g+p=
- e) h+f=
- f) g+m=
- g) a+e+k+h=
- h) k+c+f+a=
- i) c+g+k=
- j) f+g+a=
- k) m+k+c=
- l) p+e=
- m) g+m+f=
- n) h+m+a=
- o) e+c+h+k=
- p) f+h=
- a
- c
- e
- f
- g
- h
- k
- m
- p
So the Sudoku now looks like: The solution to the Sudoku is:
{1,2,3,4,5,6,7,8,9}. These filled-in cells are called givens. The goal is to fill in the whole grid using the nine digits so that each row, each column, and each block contains each number exactly once. We call this constraint on the rows, columns, and blocks the One Rule. The above-described puzzle is called a Sudoku of rank 3. A Sudoku of rank n is an n^2 ×n^2 square grid, subdivided into n^2 blocks, each of size n×n. The numbers used to fill the grid in are 1, 2, 3, ..., n^2 , and the One Rule still applies. Here is an example of a Sudoku puzzle and its solution
OBJECTIVES OF THE STUDY
This study is design to: To understand what is a Sudoku. Enhance student’s ability in solving mathematical problem especially in Linear Algebra Equations. To give motivation to the students that learning mathematics is fun and enjoyable through Sudoku. To create good impression towards mathematics and appreciate its contribution in other branches of mathematics. To give students some information on how mathematics is very important and interesting.
https://undergroundmathematics.org/thinking_about_algebra/ equation-sudoku https://www.math.cornell.edu/mec/summer2009/Mahmood/Solve.html http://www.math.cornell.edu/mec/summer2009/mahmood/Intro.html
