Chapter 12.5, Problem 50E

a.

To determine

Find the $8\times 8and7\times 7$ squares in chess board.

Answer to Problem 50E

  $One8\times 8,Four7\times 7$

Explanation of Solution

Given information:

A standard chess board contains $64$ small black or white squares. These can make other larger squares of different sizes.

  

Calculation:

Consider the given chessboard,

There is one $8\times 8$ square the chess board itself.

There are four $7\times 7$ square.

Hence, the $8\times 8and7\times 7$ squares in chess board are $One8\times 8,Four7\times 7$ .

b.

To determine

Find the all $8$ sizes squares.

Answer to Problem 50E

  $\begin{array}{l}S\left(8\right)=1\\ S\left(7\right)=4\\ S\left(6\right)=9\\ S\left(5\right)=16\\ S\left(4\right)=25\\ S\left(3\right)=36\\ S\left(2\right)=49\\ S\left(1\right)=64\end{array}$

Explanation of Solution

Given information:

A standard chess board contains $64$ small black or white squares. These can make other larger squares of different sizes.

  

Calculation:

Consider the given chessboard,

Let $S\left(n\right)$ be the number of $n\times n$ squares.

  $\begin{array}{l}S\left(8\right)=1\\ S\left(7\right)=2\cdot 2=4\\ S\left(6\right)=3\cdot 3=9\\ S\left(5\right)=4\cdot 4=16\\ S\left(4\right)=5\cdot 5=25\\ S\left(3\right)=6\cdot 6=36\\ S\left(2\right)=7\cdot 7=49\\ S\left(1\right)=8\cdot 8=64\end{array}$

c.

To determine

Find the total number of squares on $8\times 8$ chessboard.

Answer to Problem 50E

  $204$

Explanation of Solution

Given information:

A standard chess board contains $64$ small black or white squares. These can make other larger squares of different sizes.

  

Calculation:

Consider the given chessboard,

The total numbers of squares are,

  $\begin{array}{l}{\displaystyle \sum _{i=1}^n{i}^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}}\\ {\displaystyle \sum _{i=1}^8{i}^2=\frac{8\left(9\right)\left(17\right)}{6}}=204\end{array}$