Instructions: Engage in the following Desmos activity, either by yourself or in a group / as a class. Read the prompts carefully, and then respond.

Before we continue, it worth noting that it is reasonably apparent that stepping into the pool in one step (\(C_0\)) and stepping into the pool in three steps ((\(C_1\)) can each only happen in one way. To enter the pool in one step, the Robot needs to take a single step toward the pool: T. To enter the the pool in three steps, the Robot needs to take their first Step away from the pool before taking two steps toward the pool: ATT

Therefore, \[ C_0 = C_1 = 1. \]

Paths to the Perilous Pool

1. First, select the number of steps desired to be taken before falling in the pool.
• Remember that we consider 'falling in' to occur when the number of steps the robot is away from the pool is equal to 0.
2. Next, enter the path the robot should take into the input box where "A" coresponds to taking a step away from the pool and "T" corresponds to taking a step toward the pool.
• For example, the path of length 3 would be entered as "ATT."
3. Hit "Try it!" to plot the path.
• If the path entered doesn't end up in the pool, or enters the pool too soon, you will recieve an error message.
4. Repeat until you think you have found all of the valid paths into the pool of the desired number of steps.
5. Once you think you have them all, click the "Check Paths" button at the bottom of the window to verify if you found them all, or if there are more paths still to be found.

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