Name: Matchstick Patterns
Uploaded: 2019-07-19
Duration: 5 min 53 s
Description: We can use matchsticks to create several patterns and series. Can you calculate the number of matchsticks required without actually making the model of the p...

Question

A magic square is an array of numbers having the same number of rows and columns and the sum of numbers in each row, column or diagonal being the same. Fill in the blank cells to complete the magic square.

Accepted Answer

We represent the blank places of the square as $a, b, c, d, e, f$ and $g$ as shown below.

Given that the sum of numbers in each row, column, or diagonal is the same.

We observe the sum of the numbers in the fourth column $15 + 21 + 27 + 28 + 9 = 100$

Also, the sum of the numbers in the diagonals are $24 + 12 + 20 + 28 + 16 = 100$ and $22 + 21 + 20 + 19 + 18 = 100$

Considering the sum of the numbers in the first row, we obtain, $24 + a + 8 + 15 + 22 = 100$

$\Rightarrow a + 69 = 100$

$\Rightarrow a = 100 - 69 = 31$

Considering the sum of the numbers in the first column, we obtain, $24 + b + 11 + 17 + 18 = 100$

$\Rightarrow b + 70 = 100$

$\Rightarrow b = 100 - 70 = 30$

Considering the sum of the numbers in the second row, we obtain, $b + 12 + 14 + 21 + c = 100$

$\Rightarrow 30 + 12 + 14 + 21 + c = 100$

$\Rightarrow c + 77 = 100$

$\Rightarrow c = 100 - 77 = 23$

Considering the sum of the numbers in the third row, we obtain, $11 + 13 + 20 + 27 + d = 100$

$\Rightarrow d + 71 = 100$

$\Rightarrow d = 100 - 71 = 29$

Considering the sum of the numbers in the fourth row, we obtain, $17 + 19 + 26 + 28 + e = 100$

$\Rightarrow e + 90 = 100$

$\Rightarrow e = 100 - 90 = 10$

Considering the sum of the numbers in the second column row, we obtain, $a + 12 + 13 + 19 + f = 100$

$\Rightarrow 31 + 12 + 13 + 19 + f = 100$

$\Rightarrow f + 75 = 100$

$\Rightarrow f = 100 - 75 = 25$

Considering the sum of the numbers in the third column row, we obtain, $8 + 14 + 20 + 26 + g = 100$

$\Rightarrow g + 68 = 100$

$\Rightarrow g = 100 - 68 = 32$

With the values of $a, b, c, d, e, f$ and $g$ we fill the blank spaces of the square as follows.

A magic square is an array of numbers having the same number of rows and columns and the sum of numbers in each row, column or diagonal being the same. Fill in the blank cells to complete the magic square.

Important Questions on Whole Numbers

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