Uniform step magic hyper-squares

<p>Given any integers <span class="math inline">\(q\ge 2\)</span> and <span class="math inline">\(p\ge 3\)</span>, a multidimensional array <span class="math display">\[[m(i_1,i_2, \dots, i_q)\colon 1\le i_1\le p, \dots , 1\le i_q\le p],\]</span> with non-repeated entries from the set <span class="math inline">\(\{1,2,\dots, p^q\}\)</span> will be called a <span class="math inline">\(q\)</span> dimensional magic hyper-square of order <span class="math inline">\(p\)</span> if the sum of the numbers in any of its axes or diagonals is <span class="math inline">\(p(p^q+1)/2\)</span>. In this paper, we study odd-order uniform step magic hyper-squares of the form <span class="math display">\[m(i_1, \dots, i_q)=1+\sum\limits_{j=1}^{q} p^{j-1}\left[\left(\sum\limits_{k=1}^qa_{j,k} i_k+d_j\right) \bmod p \right] .\]</span> We derive necessary and sufficient conditions for these coefficients to generate magic hyper-squares and use a specific choice of coefficients to get new formula that generates magic hyper-squares of all odd orders greater than two.</p>