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 ?

Calculate the magic constant

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

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

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

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

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

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

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

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

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

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

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

$a_1=16$