Question

A magic square is an arrangement of n2 numbers into n rows and n columns using distinct numbers from 1 up to n2 such that the sum in any row, any column or any of the two diagonals is fixed. Consider the 4 by 4 magic square below:

a1 a5 2 13a2 10 11 a7a3 6 a4 124 15 a6 1
What is the value of a1 ?

Answer 1

Calculate the magic constant

$$sum=n[(n^2+1)/2]$$

$$sum=4[(4^2+1)/2]=34$$

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c} a_1 & a_5 & 2 & 13 \\ \hline a_2 & 10 & 11 & a_7 \\ \hline a_3 & 6 & a_4 & 12 \\ \hline 4 & 15 & a_6 & 1 \\ \end{array}$$

$$a_5+10+6+15=34=>a_5=3$$

$$a_1+3+2+13=34=>a_1=16$$

$$16+10+a_4+1=34=>a_4=7$$

$$a_3+6+7+12=34=>a_3=9$$

$$16+a_2+9+4=34=>a_2=5$$

$$5+10+11+a_7=34=>a_7=8$$

$$4+15+a_6+1=34=>a_6=14$$

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c} 16 & 3 & 2 & 13 \\ \hline 5 & 10 & 11 & 8 \\ \hline 9 & 6 & 7 & 12 \\ \hline 4 & 15 & 14 & 1 \\ \end{array}$$

$a_1=16$