Fill in the blanks: 2 5 1 7 5 8

We know this both are magic square. Also, we know the sum of the numbers in each row, column and diagonal is 3 times to the middle number. So, here in both the magic square have the middle number 5. Means the sum of the numbers in each row, column and diagonal is: 5 × 3 = 15 Now we find numbers for 1st magic square In second column two value is given 1 and 5. So we have to find the third value which makes its sum 15 So, 15 - ( 1 + 5 ) = 9 Same way in row 1 now we have 2 and 9. So we have to find the third value which makes its sum 15 So, 15 - ( 2 + 9 ) = 4 Same way we find all the blank spaces in the magic square. 2 9 4 7 5 3 6 1 8 Same way we can find for 2nd magic square. 6 7 2 1 5 9 8 3 4

Show that each square is a magic square by showing that the rows, columns and diagonals all have the same sum. 2 x - 2 y - x + 3 y 2 x - y x + y x x - y y 3 x - 3 y 2 y 3 y - 2 x - x - 5 y 6 x + 2 y 9 x - y x y - 7 x - 4 x - 2 y 3 x + 5 y 4 x - 3 y

Taking squares one by one First, we talk about 1st square. 2 x - 2 y - x + 3 y 2 x - y x + y x x - y y 3 x - 3 y 2 y To show this square is a magic square we have to do the sum of the numbers in each row, column and diagonal and if it is the same then this square is a magic square. First, we sum all the rows Row1: s u m = 2 x - 2 y - x + 3 y + 2 x - y = 3 x Row2: s u m = x + y + x + x - y = 3 x Row3: s u m = y + 3 x - 3 y + 2 y = 3 x Now, we sum all the columns Column1: s u m = 2 x - 2 y + x + y + y = 3 x Column2: s u m = - x + 3 y + x + 3 x - 3 y = 3 x Column3: s u m = 2 x - y + x - y + 2 y = 3 x Now, we sum both the diagonals Diagonal1: s u m = 2 x - 2 y + x - 2 y = 3 x Diagonal2: s u m = 2 x - y + x - y = 3 x Here the sum of the number in each row, column and diagonal is the same. That means it is a ' MAGIC SQUARE '. Now, we talk about 2nd square 3 y - 2 x - x - 5 y 6 x + 2 y 9 x - y x y - 7 x - 4 x - 2 y 3 x + 5 y 4 x - 3 y To show this square is a magic square we have to do the sum of the numbers in each row, column and diagonal and if it is the same then this square is a magic square. First, we sum all the rows Row1: s u m = 3 y - 2 x - x - 5 y + 6 x + 2 y = 3 x Row2: s u m = 9 x - y + x + y - 7 x = 3 x Row3: s u m = - 4 x - 2 y + 3 x + 5 y + 4 x - 3 y = 3 x Now, we sum all the columns Column1: s u m = 3 y - 2 x + 9 x - y - 4 x - 2 y = 3 x Column2: s u m = - x - 5 y + x + 3 x + 5 y = 3 x Column3: s u m = 6 x + 2 y + y - 7 x + 4 x - 3 y = 3 x Now, we sum all the diagonal Diagonal1: s u m = 3 y - 2 x + x + 4 x - 3 y = 3 x Diagonal2: s u m = 6 x + 2 y + x - 4 x - 2 y = 3 x Here the sum of the number in each row, column and diagonal is the same. That means it is a ' MAGIC SQUARE '.

Find the missing terms in the given magic squares. x - 2 x + 2 y 4 x + y

Given magic square: x - 2 x + 2 y 4 x + y We know that in the magic square the sum of the numbers in each row, column and diagonal is 3 times to middle number. Means the sum of the numbers in each row, column, and diagonal is: 3 × x = 3 x . Now, we can easily find the missing terms. Take a standard equation( a x + b y ) at blank space in different row and column at different times. And putting the sum of the numbers in each row, column and diagonal is 3 x . Now, we take row3: a x + b y - 2 x + 2 y + 4 x + y = 3 x ( a - 2 + 4 ) x + ( b + 2 + 1 ) y = 3 x n o w c o m p a i r i n g b o t h s i d e a - 2 + 4 = 3 a n d b + 2 + 1 = 0 a = 1 a n d b = - 3 So, the value of row3, column1 is x - 3 y Now, we take diagonal1 a x + b y + x + 4 x + y = 3 x ( a + 1 + 4 ) x + ( b + 1 ) y = 3 x n o w c o m p a i r i n g b o t h s i d e a + 1 + 4 = 3 a n d b + 1 = 0 a = - 2 a n d b = - 1 So, the value of row1, column1 is - 2 x - y Now, we take diagonal2 a x + b y + x + x - 3 y = 3 x ( a + 1 + 1 ) x + ( b - 3 ) y = 3 x n o w c o m p a i r i n g b o t h s i d e a + 1 + 1 = 3 a n d b - 3 = 0 a = 1 a n d b = 3 So, the value of row1, column3 is x + 3 y Now, we take row1 - 2 x - y + a x + b y + x + 3 y = 3 x ( - 2 + a + 1 ) x + ( - 1 + b + 3 ) y = 3 x n o w c o m p a i r i n g b o t h s i d e - 2 + a + 1 = 3 a n d - 1 + b + 3 = 0 a = 4 a n d b = - 2 So, the value of row1, column2 is 4 x - 2 y Now, we take column1 - 2 x - y + a x + b y + x - 3 y = 3 x ( - 2 + a + 1 ) x + ( - 1 + b - 3 ) y = 3 x n o w c o m p a i r i n g b o t h s i d e - 2 + a + 1 = 3 a n d - 1 + b - 3 = 0 a = 4 a n d b = 4 So, the value of row2,column1 is 4 x + 4 y Now, we take row2 4 x + 4 y + x + a x + b y = 3 x ( 4 + 1 + a ) x + ( 4 + b ) y = 3 x n o w c o m p a i r i n g b o t h s i d e 4 + 1 + a = 3 a n d 4 + b = 0 a = - 2 a n d b = - 4 So, the value of row2, column3 is - 2 x - 4 y So, the complete magic square is: - 2 x - y 4 x - 2 y x + 3 y 4 x + 4 y x - 2 x - 4 y x - 3 y - 2 x + 2 y 4 x + y

Find the missing terms in the following magic squares. 5 y - 2 x + 3 y x

Given magic square: 5 y - 2 x + 3 y x We know that in the magic square the sum of the numbers in each row, column and diagonal is 3 times to middle number. Means the sum of the numbers in each row, column and diagonal is: 3 × x = 3 x . Now we can easily find the missing terms. By assuming a x + b y in different row and column at different times. And putting the sum of the numbers in each row, column and diagonal is 3 x . Now, we take row2 - 2 x + 3 y + x + a x + b y = 3 x ( - 2 + 1 + a ) x + ( 3 + b ) y = 3 x n o w c o m p a i r i n g b o t h s i d e - 2 + 1 + a = 3 a n d 3 + b = 0 a = 4 a n d b = - 3 So, the value of row2, column3 is 4 x - 3 y Now, we take column3 5 y + 4 x - 3 y + a x + b y = 3 x ( 4 + a ) x + ( 5 - 3 + b ) y = 3 x n o w c o m p a i r i n g b o t h s i d e 4 + a = 3 a n d 5 - 3 + b = 0 a = - 1 a n d b = - 2 So, the value of row3,column3 is - x - 2 y Now, we take diagonal1 a x + b y + x - x - 2 y = 3 x ( a + 1 - 1 ) x + ( b - 2 ) y = 3 x n o w c o m p a i r i n g b o t h s i d e a + 1 - 1 = 3 a n d b - 2 = 0 a = 3 a n d b = 2 So, the value of row1,column1 is 3 x + 2 y Now, we take column1 3 x + 2 y - 2 x + 3 y + a x + b y = 3 x ( 3 - 2 + a ) x + ( 2 + 3 + b ) y = 3 x n o w c o m p a i r i n g b i t h s i d e 3 - 2 + a = 3 a n d 2 + 3 + b = 0 a = 2 a n d b = - 5 So, the value of row3,column1 is 2 x - 5 y Now, we take row1 3 x + 2 y + a x + b y + 5 y = 3 x ( 3 + a ) x + ( 2 + b + 5 ) y = 3 x n o w c o m p a i r i n g b o t h s i d e 3 + a = 3 a n d 2 + b + 5 = 0 a = 0 a n d b = - 7 So, the value of row1, column2 is - 7 y Now, we take row3 2 x - 5 y + a x + b y - x - 2 y = 3 x ( 2 + a - 1 ) x + ( - 5 + b - 2 ) y = 3 x n o w c o m p a i r i n g b o t h s i d e 2 + a - 1 = 3 a n d - 5 + b - 2 = 0 a = 2 a n d b = 7 So, the value of row3, column2 is 2 x + 7 y So, the complete magic square is: 3 x + 2 y - 7 y 5 y - 2 x + 3 y x 4 x - 3 y 2 x - 5 y 2 x + 7 y - x - 2 y

David Weber, Talei Kunkel, Alexander Martinez and, Rebecca Shultis Solutions for Chapter: Algebraic Expressions and Equations Puzzles and Tricks, Exercise 8: Practice 2

Author:David Weber, Talei Kunkel, Alexander Martinez & Rebecca Shultis

David Weber Mathematics Solutions for Exercise - David Weber, Talei Kunkel, Alexander Martinez and, Rebecca Shultis Solutions for Chapter: Algebraic Expressions and Equations Puzzles and Tricks, Exercise 8: Practice 2

Attempt the practice questions on Chapter 4: Algebraic Expressions and Equations Puzzles and Tricks, Exercise 8: Practice 2 with hints and solutions to strengthen your understanding. MYP Mathematics A concept-based approach 2 solutions are prepared by Experienced Embibe Experts.

Questions from David Weber, Talei Kunkel, Alexander Martinez and, Rebecca Shultis Solutions for Chapter: Algebraic Expressions and Equations Puzzles and Tricks, Exercise 8: Practice 2 with Hints & Solutions

MEDIUM

MYP:2

IMPORTANT

MYP Mathematics A concept-based approach 2>Chapter 4 - Algebraic Expressions and Equations Puzzles and Tricks>Practice 2>Q 3

How does the sum is relate to the number in the middle of the following 'magic square'?

8	1	6
3	5	7
4	9	2

MEDIUM

MYP:2

IMPORTANT

MYP Mathematics A concept-based approach 2>Chapter 4 - Algebraic Expressions and Equations Puzzles and Tricks>Practice 2>Q 3

How does the sum of all of the numbers is relate to the number in the middle? Does this mean a calendar is a magic square?

8	1	6
3	5	7
4	9	2

HARD

MYP:2

IMPORTANT

MYP Mathematics A concept-based approach 2>Chapter 4 - Algebraic Expressions and Equations Puzzles and Tricks>Practice 2>Q 4

Fill in the blanks:

2
	5
	1

	7
	5
8

HARD

MYP:2

IMPORTANT

MYP Mathematics A concept-based approach 2>Chapter 4 - Algebraic Expressions and Equations Puzzles and Tricks>Practice 2>Q 4

Create a magic square of your own where every row, column and diagonal adds to 15.

HARD

MYP:2

IMPORTANT

MYP Mathematics A concept-based approach 2>Chapter 4 - Algebraic Expressions and Equations Puzzles and Tricks>Practice 2>Q 5

Show that each square is a magic square by showing that the rows, columns and diagonals all have the same sum.

$2 x - 2 y$	$- x + 3 y$	$2 x - y$
$x + y$	$x$	$x - y$
$y$	$3 x - 3 y$	$2 y$

$3 y - 2 x$	$- x - 5 y$	$6 x + 2 y$
$9 x - y$	$x$	$y - 7 x$
$- 4 x - 2 y$	$3 x + 5 y$	$4 x - 3 y$

HARD

MYP:2

IMPORTANT

MYP Mathematics A concept-based approach 2>Chapter 4 - Algebraic Expressions and Equations Puzzles and Tricks>Practice 2>Q 5

Create a magic square that contains numbers by substituting x = l and y = 2 in the following magic squares. Verify that they are magic squares.

$2 x - 2 y$	$- x + 3 y$	$2 x - y$
$x + y$	$x$	$x - y$
$y$	$3 x - 3 y$	$2 y$

$3 y - 2 x$	$- x - 5 y$	$6 x + 2 y$
$9 x - y$	$x$	$y - 7 x$
$- 4 x - 2 y$	$3 x + 5 y$	$4 x - 3 y$

HARD

MYP:2

IMPORTANT

MYP Mathematics A concept-based approach 2>Chapter 4 - Algebraic Expressions and Equations Puzzles and Tricks>Practice 2>Q 6

Find the missing terms in the given magic squares.


	$x$
	$- 2 x + 2 y$	$4 x + y$

HARD

MYP:2

IMPORTANT

MYP Mathematics A concept-based approach 2>Chapter 4 - Algebraic Expressions and Equations Puzzles and Tricks>Practice 2>Q 6

Find the missing terms in the following magic squares.

		$5 y$
$- 2 x + 3 y$	$x$

David Weber, Talei Kunkel, Alexander Martinez and, Rebecca Shultis Solutions for Chapter: Algebraic Expressions and Equations Puzzles and Tricks, Exercise 8: Practice 2

David Weber Mathematics Solutions for Exercise - David Weber, Talei Kunkel, Alexander Martinez and, Rebecca Shultis Solutions for Chapter: Algebraic Expressions and Equations Puzzles and Tricks, Exercise 8: Practice 2

Questions from David Weber, Talei Kunkel, Alexander Martinez and, Rebecca Shultis Solutions for Chapter: Algebraic Expressions and Equations Puzzles and Tricks, Exercise 8: Practice 2 with Hints & Solutions

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