On a pond there are several lily pads lined up in a row. The center lily pad is empty, but on either side the lily pads are all occupied by a frog. The frogs on the left are all facing right, and the frogs on the right are all facing left.

The groups of frogs wish to trade places. . . the right-facing frogs all on the far right, the left-facing all on the far left, and the center lily pad empty. But the frogs can only move according to these rules:

1) The frogs only move forward. (The right-facing frogs only move right, and vice versa.)

2) A frog can move onto an empty lily pad in front of it if that pad is empty.

3) A frog can jump over one frog and land two lily pads away if that pad is empty. (Frogs can only jump over one other frog.)

Can you find how many moves it takes for these frogs to trade places?

1

To find the number of moves you can simulate with the colored disks or electronically at

https://primefactorisation.com/frogpuzzle

1. Complete the table. This will take much trial and error, but there is a pattern in how the frogs move. You may also realize a pattern in the total number of moves which will help you to fill out the table.

Frogs on each side	Number of moves
1
2
3
4
5
6

Come to think of it, how many moves would it take for zero frogs on each side?

2. Provide a drawing, or series of drawings, to show the moves for 3 frogs on each side.

Before you move to Part 3, participate in the Investigation Discussion Form

3. The table on the previous page represents a sequencean where the inputn is the number of frogs on each side and the outputan is the number of jumps.

Plot the ordered pairs forn= 0, 1, 2, 3, 4, 5, and 6. Scale appropriatelysoall points fit on the grid.

4. The sequence is in fact a quadratic sequence of the form

an=an2+bn+c

a) Can you find the values ofa,b, andc? Use the table and/or the points on the graph.

b) Use the equation you found to predict the number of moves when there are 20 frogs on each side.

c) How about 1000 frogs on each side?

Here is what I need you to submit.

Type a report that includes the following:

1. An introduction that gives an overview of this Investigation.

2. Answers to all the questions for Parts 1 - 4.

• Include any sketches you were asked to make
• Include any tables that you were asked to make.
• Make graphs usingDesmos(www.desmos.com).
• can you please explain and show work and help me with this, I would be very appreciative.