Chapter 5.3, Problem 2QE

a.

Explanation of Solution

Whether the remaining board with tiles could be covered such that tiles do not overlap or hang off the edge of the board:

• Divide the $2^{\text{n}}\times 2^{\text{n}}$ board into four $2^{\text{n}-1}\times 2^{\text{n}-1}$ board square quadrants

b.

Explanation of Solution

The solution obtained in part (a) can be used to show that $2^{\text{2n}}-1$ divisible by 3 for all positive integers $n$:

• In the above part (a), the one quadrant is uncovered and three quadrants are covered.
• This concept is used to show $2^{2\text{n}}-1$ is divisible by 3

c.

Explanation of Solution

The ways in which the answer obtained in part (a) and (b) are related to Polya’s phases of problem solving:

Polya’s phases:

There are four Polya’s phases as:

• Understands the problem
• Devise a plan
• Implement the plan
• Checking the solution...