a. Use the properties of a magic square to fill in the

Chapter 1, Problem 48

(choose chapter or problem)

The sudoku (pronounced: sue-DOE-koo) craze, a number puzzle popular in Japan, hit the United States in 2005. A sudoku (“single number”) puzzle consists of a 9-by-9 grid of 81 boxes subdivided into nine 3-by-3 squares. Some of the square boxes contain numbers. Here is an example:

The objective is to fill in the remaining squares so that every row, every column, and every 3-by-3 square contains each of the digits from 1 through 9 exactly once. (You can work this puzzle in Exercise 66, perhaps consulting one of the dozens of sudoku books in which the numerals 1 through 9 have created a cottage industry for publishers. There’s even a Sudoku for Dummies.)

Trying to slot numbers into small checkerboard grids is not unique to sudoku. In Exercises 47–50, we explore some of the intricate patterns in other arrays of numbers, including magic squares. A magic square is a square array of numbers arranged so that the numbers in all rows, all columns, and the two diagonals have the same sum. Here is an example of a magic square in which the sum of the numbers in each row, each column, and each diagonal is 15:

Exercises 47–48 are based on magic squares. (Be sure you have read the preceding discussion.)

a. Use the properties of a magic square to fill in the missing numbers.

b. Show that if you reverse the digits for each number in the square in part (a), another magic square is generated. (Source for the alpha magic square in Exercise 47 and the mirror magic square in Exercise 48: Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Inc., 2005)

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