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#    Lab 13: Fruitful recursion w/ Turtle graphics

<img src='turtle.gif' alt='Turtle cursor'>
For this part of lab, open the file `fruitfulTurtles.py`. This is where you
will add your code.

## Task 0. Consider a different approach for rows of boxes

We designed recursive Turtle functions that would draw a pattern a given number of times.

For example, when we invoked `row(3, 50)` it would draw **3** squares.

<img src='row3.png' alt='row(3, 50)' style='max-width:200px'>

To make things more challenging, let's change it up a bit. We will tell the function the starting side length of the first square, and a shrink factor and a minimum side length that must be met in order for a square to be drawn. Consider the function `newRow` below:

```py
def newRow(size, shrinkFactor, minimumSize):
    if size < minimumSize:
        pass
    else:
        square(size)
        fd(size)
        newRow(size*shrinkFactor, shrinkFactor, minimumSize)
        bk(size)
```

Here are two sample invocations of `newRow`:

```py
newRow(size, shrinkFactor, minimumSize)
```
### Example invocation 1:
```py
newRow(50, 0.5, 20)
```
produces

<img src='twoboxes.png' alt='newRow(50, 0.5,  20)' style='max-width:200px'>
The first box has size 50, the second 25, and the third would be 12.5, but 12.5 < 20, so it stops.

### Example invocation 2:
```py
newRow(50, 0.8, 20)
```
produces

<img src='fiveboxes.png' alt='newRow(50, 0.8,  20)' style='max-width:300px'>

The box sides are 50, 40, 32, 25.6, and 20.48. The next one would be 16.38, but that is < 20, so it is not drawn.


## Task 1. Write fruitfulRow
 Using `newRow` (code above) as your starting point, write `fruitfulRow`  so that the fruitful function returns the number of boxes that are drawn. Remember that with fruitful recursion, a value **always** should be returned.

How to count the number of boxes drawn?
+  How many boxes are drawn in the base case? That's the value to return in the base case.
+  How many boxes are drawn in the recursive case? Use **wishful thinking** and pretend that
your recursive case works correctly and returns the number of boxes drawn in your recursive call.

### Example invocations of `fruitfulRow`:
```py
In []: fruitfulRow(50, 0.5, 20)
Out[]: 2
In []: fruitfulRow(50, 0.8, 20)
Out[]: 5
In []: fruitfulRow(100, 0.75, 18)
Out[]: 6
In []: fruitfulRow(100, 0.9, 10)
Out[]: 22

```



## Task 2. fruitfulRowImproved
In this next task, we want to create an improved version of the row function that instead of just returning a single value (`squareCount`) it **returns a tuple** indicating how many squares were drawn *and* the total distance travelled by the Turtle/cursor.

<br>
__Capturing tuples returned from a function__

In order to complete this task, study the following example that demonstrates a function that returns a tuple...

```py
def pad(x, y, padding):
    return x + padding, y + padding
```

Given the above code, make predictions of what the following print statements would produce:

```py
x, y = pad(300, 200, 10)
print(x) # ???
print(y) # ???
```

Run your code in Canopy to check your predictions. Note how the function `pad` returns a tuple. Because of this, we can capture its results in a tuple: `x, y = pad(300, 200, 10)`.


<br> Once you understand the above code, copy and complete the
following `fruitfulRowTuple` function in your `lab13/fruitfulTurtles.py`
file:

```py
def fruitfulRowTuple(size, shrinkFactor, minimumSize):
    if size < minimumSize:
        return ???
    else:
        square(size)
        fd(size)
        ??? = fruitfulRowTuple(size*shrinkFactor, shrinkFactor, minimumSize)
        ???
        bk(size)
        return ???
```


__Important observation about the above function:__ Note how the base case also needed to return a tuple, `0,0`. It's essential that the value type returned by the base case is consistent with what is returned in the recursive case.

Come up with some test cases for your `fruitfulRowTuple`. Try using a shrink factor of 0.5 for easier initial tests and calculations.



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