In the Tower of Hanoi (huh-NOY) puzzle, a set of discs sits on 1 peg, and there are 2 other empty pegs.

A move in the Tower of Hanoi puzzle involves taking a disc and moving it to another peg. There are two rules:

• Move only 1 disc at a time.
• Never put a larger disc on top of a smaller one.

You complete the puzzle by building the complete tower on any peg other than the starting peg.

1. Using 3 discs, complete the puzzle. What is the smallest number of moves you can find?
2. Using 4 discs, complete the puzzle. What is the smallest number of moves you can find?
3. Jada says she used the solution for 3 discs to help her solve the puzzle for 4 discs. Describe how this might happen.

Some checkers are lined up. Blue checkers are on one side, red are on the other, and there is 1 empty space between them. A move in this checker game pushes any checker forward 1 space or jumps over any 1 checker of the other color. Jumping the same color is not allowed, moving backward is not allowed, and 2 checkers cannot occupy the same space.

A puzzle that consists of a row of 7 connected squares. From left to right, the squares contain the following, blue checker, blue checker, blue checker, empty, red checker, red checker, red checker.

The puzzle is completed when the colors are completely switched: ending up with blue on the right, red on the left, and 1 empty space between them.

1. Using 1 checker on each side, complete the puzzle. What is the smallest number of moves needed?
2. Using 3 checkers on each side, complete the puzzle. What is the smallest number of moves needed?
3. Estimate the number of moves needed if there are 4 checkers on each side, then test your guess.
4. Noah says he used the solution for 3 checkers on each side to help him solve the puzzle for 4 checkers. Describe how this might happen.