91Ó°ÊÓ

A magic square of order \(n\) is a square arrangement of the positive integers 1 through \(n^{2}\) such that the sum of the integers along each row, column, and diagonal is a constant \(k\), called the magic constant. Figure 4.30 shows two magic squares, one of order 3 and the other of order \(4 .\) Prove that the magic constant of a magic square of order \(n\) is \(n\left(n^{2}+1\right) / 2\).

A magie square of order \(n\) is a square arrangement of the positive integers 1 through \(n^{2}\) such that the sum of the integers along each row, column, and diagonal is a constant \(k\) , called the magie constant. Figure 4.30 shows two magic squares, one of order 3 and the other of order \(4 .\) Prove that the magic constant of a magic square of order \(n\) is \(n\left(n^{2}+1\right) / 2 .\)

Find the value of the base \(b\) in each case. $$ 54_{b}=64 $$

Find the value of the base \(b\) in each case. $$144_{b}=49$$

Use the insertion sort algorithm in Algorithm 4.12 to answer Exercises Algorithm insertion sort \((x, n)\) (* This algorithm sorts a list \(x\) of n elements into ascending order by inserting a new element in the proper place at the end of each pass. \(^{\star}\) ) 0\. Begin \(\left(^{\star} \text { al gori thm }^{\star}\) ) \right. 1\. for \(i=2\) to \(n\) do 2\. begin (* for *) 3 \(\operatorname{temp} \leftarrow x_{i}\) \(\left(* \text { temp is a temporary variable }^{\star}\right)\) \(4 . \quad j \leftarrow i-1\) 5\. while \(j \geq 1\) do

Find the number of trailing zeros in the decimal value of each. $$378 !$$

Find the number of trailing zeros in the decimal value of each. $$500 !$$

Let \(S_{n}\) denote the sum of the elements in the \(n\) th set of the sequence of sets of squares \(\\{1\\},\\{4,9\\},\\{16,25,36\\}, \ldots .\) Find a formula for \(S_{n}\).(J. M. Howell, 1989 )