1. Announcements

2. Magic Number Square Tentaizu Approach
Backtracking
Magic Number Square
Tentaizu Approach

3. Magic Number Squares
A 3x3 magic number square has 9 numbers {1,2,… 9} that must be arranged in such a way that every row, column, and diagonal has the same sum.
For a 3x3 magic number square the sum is 3*(3*3+1)/2 = 15. 15 15

4. Magic Number Square
It is possible to have different sized magic number squares of size nxn,
where numbers {1,2,… n*n} must be arranged in such a way that every row and column has the same sum.
For a nxn magic number square the sum is n*(n*n+1)/2

5. Magic Number Square
Say we want to create a class MagicSquare that takes in an int n and returns all valid Magic Number Squares of size nxn.
Example, n = 3 returns 8 magic squares:

6. Magic Number Square How would we start?
The plan like we did with Eight Queens is just to try a number in the first position.
Then we can recursively solve the board at the next position given that the number has been placed.
Then we systematically “throw out” boards that do not meet our constraints.
What do we need to do that?
First we will want an empty board that we will fill in one square at a time.
Then we will want to mark which numbers in the set (1, … , n*n) have already been used.
We need to calculate the sum that each row, column, and diagonal must add up to.

7. initialize all values to 0, it’s empty to start!
public class MagicSquare { private int[][] square; private boolean[] possible; private int totalSqs; private int sum; private int numsquares; // Creates an empty MagicSquare object. public MagicSquare(int n) { // Fill in an empty square; 0 represents unfilled. square = new int[n][n]; for (int i=0; i<n; i++) for (int j=0; j<n; j++) square[i][j] = 0; // Start with all numbers being possible. totalSqs = n*n; possible = new boolean[totalSqs]; for (int i=0; i<totalSqs; i++) possible[i] = true; sum = n*(n*n+1)/2; numsquares = 0; }
square[][] will hold the numbers we place.
possible[] will hold the numbers that have still not bee used yet.
sum will store what value the rows/columns/diagonals must sum to.
square[][] is size nxn
initialize all values to 0, it’s empty to start!
There are n*n total numbers,
so possible[] is size n*n. At first
all of the numbers can be used
so the entire array is set to true.

8. Try all possible values for this spot
// Recursive method which fills in the square at (row,col). public void fill(int row, int col) { // Screen away incorrect squares. if (!checkRows() || !checkCols() || !checkDiags()) return; // Filled everything, so print !! if (row == square.length) { System.out.println(this); numsquares++; }
// Try each possible number for this square.
for (int i=0; i<totalSqs; i++) {
if (possible[i]) {
square[row][col] = i+1;
possible[i] = false;
// Go to the next square.
int newcol = col+1;
int newrow = row;
if (newcol == square.length) {
newrow++;
newcol = 0; }
// Recursively fill.
fill(newrow, newcol);
// Undo this square.
square[row][col] = 0;
possible[i] = true;
Try all possible values for this spot
Find the new row and col, so we can recursively fill the next spot.
Base Cases:
Either we broke the constraints, so we throw that board out. OR
We’re DONE!!
Recursively call fill() on our new spot.
Undo this square for future recursive calls,
so that we can try ALL valid boards

9. // Recursive method which fills in the square at (row,col)
// Recursive method which fills in the square at (row,col). public void fill(int row, int col) { // Screen away incorrect squares. if (!checkRows() || !checkCols() || !checkDiags()) return; // Filled everything, so print !! if (row == square.length) { System.out.println(this); numsquares++; }
// Try each possible number for this square.
for (int i=0; i<totalSqs; i++) {
if (possible[i]) {
square[row][col] = i+1;
possible[i] = false;
// Go to the next square.
int newcol = col+1;
int newrow = row;
if (newcol == square.length) {
newrow++;
newcol = 0; }
// Recursively fill.
fill(newrow, newcol);
// Undo this square.
square[row][col] = 0;
possible[i] = true;

10. Determine whether each row is filled and the row’s sum.
// Returns true iff all the rows are okay. public boolean checkRows() { // Try each row. for (int i=0; i<square.length; i++) { int test = 0; boolean unFilled = false; // Add up the values in this row. for (int j=0; j<square[i].length; j++) { test += square[i][j]; if (square[i][j] == 0) unFilled = true; } // If the row is filled and adds up to the wrong number, // this is an invalid square. if (!unFilled && test != sum) return false; // Never found proof of an invalid row. return true;
Determine whether each row is filled and
the row’s sum.
If the row is filled and adds
up to the wrong number
then our row constraint is violated,
and we return false.
At this point we have checked all rows,
and none of them violated our constraint.

11. Magic Number Square Decision Tree
Shown on the board…

12. Tentaizu Approach
The game is played on a 7x7 board. Exactly 10 of the 49 squares are each hiding a star. Your task is to determine which squares are hiding the stars.
Other squares in the board provide clues: A number in a square indicates how many stars lie next to the square
No square with a number in it contains a star, but a star may appear in a square with no adjacent numbers.

13. Tentaizu Approach Break down the problem into smaller steps
Read in the board
Determine the squares with numbers
Determine the squares that are blank
Place the stars in a backtracking method based on the possible locations
Check that the current or finished board meets the requirements of the game
Ten stars are present
Each numbered square has the correct number of adjacent stars

14. Optional Homework Problem
Get together with someone in the class and brainstorm how you will solve the Tentaizu assignment.
Turn in a paper describing your strategy.
What matrices and/or arrays will you need?
What constraints will you check against?
How will you know when these constraints are violated?
How will you know when the board is complete?
Show a decision tree diagramming how your Tentaizu algorithm will work. (Like how we did with the Magic Number Square)

15. References
Slides adapted from Arup Guha’s Computer Science II Lecture notes: ctures/
Additional material from the textbook:
Data Structures and Algorithm Analysis in Java (Second Edition) by Mark Allen Weiss
Additional images:
xkcd.com