Lab 7: Recursion!¶

Objectives

In this lab you will gain experience with recursion and recursive thinking. Recursion is a powerful design technique. As is the case with any new technique, it takes a bit of practice to master recursion. In this lab, we will concentrate on a variety of recursion problems. The goal of this lab is to practice writing recursive programs and to train our brains to think recursively.

Thinking Recursively¶

When writing a recursive function, always start with one or more base cases. These are problems that are so small that we can solve them directly. Then, write the recursive cases. There are two parts to each recursive case:

1. taking an action that contributes to the solution (makes progress), and,

2. calling the recursive function on a smaller subproblem (which gets you closer to one of the base cases).

Getting Started¶

Clone the lab resources from the gitlab repository, as usual:

```git clone https://evolene.cs.williams.edu/cs134-labs/23xyz3/lab07.git
```

where your CS username replaces `23xyz3`. You will find five Python files, `sumDigits.py`, `bedtime.py`, `quilt.py`, `shrub.py`, and `fancy.py`, each corresponding to a task described below.

This week, you will solve five problems using recursion: a warm-up recursion (Task 0) (to be completed before lab), a text-based recursion with lists of strings (Task 1), and three graphical recursion tasks (Tasks 2, 3, and 4). Finally, there is an optional extra-credit task which is a more challenging graphical recursion. We encourage you to read through all the task descriptions before writing any code.

Task 0 [Warm-up]: Recursion with Numbers¶

Please complete this task before coming to your scheduled lab session. In this task, you will write the `sumDigits` function in the file `sumDigits.py`. The `sumDigits` function takes a non-negative integer `num` as input and computes and returns the sum of the digits of `num`.

Your function should be recursive and should not use any loops. You are also not allowed to change the type of `num` from an integer to a string. Here are examples of how your function should behave in interactive Python:

```>>> print(sumDigits(0))
0
>>> print(sumDigits(90))
9
>>> print(sumDigits(889832))
38
>>> print(sumDigits(1234567890))
45
```

As you start thinking about the recursive problem structure, consider some specific examples. For example, there are some numbers where you know the answer immediately: `sumDigits(1)` is 1, `sumDigits(8)` is 8, and so on. How does this map to a base case? Now consider a number like 178. You can think of `sumDigits(178)` as `sumDigits(17) + 8`; and `sumDigits(17)` as `sumDigits(1) + 7`; and finally, we know that `sumDigits(1)` is simply 1. To generalize from this example, given an integer `num` that does not meet your base case criteria, your recursive case should separate it into two parts using arithmetic operators:

1. everything except its last digit, and

2. its last digit.

Hint: Think about how you can use the arithmetic operators to extract digits.

Bedtime stories follow a common pattern across cultures where a simple phrase repeats multiple times in a nested fashion. For example,

```The mother of the parrot told a story about a cow...
The mother of the cow told a story about a flamingo...
The mother of the flamingo told a story about a heron...
and then the flamingo fell asleep.
and then the cow fell asleep.
and then the parrot fell asleep.
```

In this task, you will write a function called `bedtimeStory` which, given a list of strings `characters`, should return a list of strings representing the lines of a bedtime story. The above story shows the lines generated using the characters `['parrot', 'cow', 'flamingo', 'heron']`. We’ve printed it nicely to help show the recursive structure, but your `bedtimeStory` function actually returns those lines as a list of strings without indentation:

```['The mother of the parrot told a story about a cow...',
'The mother of the cow told a story about a flamingo...',
'The mother of the flamingo told a story about a heron...',
'and then the flamingo fell asleep.',
'and then the cow fell asleep.',
'and then the parrot fell asleep.']
```

Your implementation should be recursive and cannot use any loops. What is the recursive structure? What sort of `character` lists produce the simplest story? (Hint: the simplest story is `[]`, which should be returned if fewer than 2 characters are provided.) How do you construct the story in the recursive case?

You should use the provided helper functions `firstSentence(object, subject)` and `lastSentence(object)` in your implementation. The following examples illustrate their behavior:

• `firstSentence('parrot', 'cow')` returns `'The mother of the parrot told a story about a cow...'`.

• `lastSentence('parrot')` returns `'and then the parrot fell asleep.'`.

Be careful about the return types of the various functions: `firstSentence` and `lastSentence` return strings, while `bedtimeStory` returns a list of strings (sentences of the story). We have also provided the helper function `formatPrint` which prints out this list of “story strings” in a nicely readable format.

Running the file `bedtime.py` as a Python script takes, as command line arguments, the set of characters you’d like in the story. In your Terminal, command line arguments appear after the word python3 (such as moose, bear, and reindeer in the example below). In Python, we can easily store these values in a list of strings for processing. We have provided the code that handles this for you in the `if __name__ == "__main__"` block. You do not need to modify it.

For instance,

```python3 bedtime.py moose bear reindeer
```

will produce the following story:

```The mother of the moose told a story about a bear...
The mother of the bear told a story about a reindeer...
and then the bear fell asleep.
and then the moose fell asleep.
```

Note that the last character in the list (`reindeer`) in the above example, is just the `object` of a story for the `bear`, and not a `subject` of its own story.

You can test out the `bedtimeStory` function directly in interactive Python; here is is one test you may want to try:

```>>> from bedtime import bedtimeStory
>>> bedtimeStory(['parrot', 'cow', 'flamingo', 'heron'])
['The mother of the parrot told a story about a cow...',
'The mother of the cow told a story about a flamingo...',
'The mother of the flamingo told a story about a heron...',
'and then the flamingo fell asleep.',
'and then the cow fell asleep.',
'and then the parrot fell asleep.']
```

You must write one new doctest for this function. When you are writing your new doctest, note that newlines between elements of lists can cause doctests to fail. Please make the tests as short as you can, but you can go over 80 characters if needed (as in the test provided with the starter code). Hint: You may wish to consider tests involving character lists with only 0 or 1 character name in them.

In this task, you will implement a recursive function to draw concentric squares with our favorite turtle:

In addition to drawing the squares, your function should return the total number of squares drawn (as an int). The starter code for this task is in `squares.py`. The primary function you must complete is `drawNestedSquares(size, gap, color, otherColor)` in `squares.py`. The input parameters are described below:

• `size` is the side length of the whole pattern (that is, the largest square); the side length of each successive square is decreased by twice the `gap`.

• `gap` is the inset for the nested squares. That is, a band of width `gap` is left showing after the nested squares are drawn.

• `color` is the color of the largest square you will draw.

• `otherColor` is the color of the first square to draw inside the outer one. (Note that `color` and `otherColor` alternate roles on the nested squares.)

Here are a few examples of calling `drawNestedSquares(400, gap, PURPLE, WHITE)` with different `gap`s, with the value after the function call `->` indicating the value returned by that invocation. :

`drawNestedSquares(400, 40, PURPLE, WHITE)` ` -> 5`

`drawNestedSquares(400, 20, PURPLE, WHITE)` ` -> 10`

`drawNestedSquares(400, 10, PURPLE, WHITE)` ` -> 20`

You may assume the turtle is facing east and positioned at `(-size/2, -size/2)` when `drawNestedSquares(size, gap, color, otherColor)` is called. As usual, we should first consider the base case. The most simple drawing we may need to create is one with no squares, which will happen when `size < gap`. In that case, we simply return 0. In all other cases, we’ll need to draw a square with side length `size` and then recursively draw nested squares. The next nested square having side length `size - 2 * gap` and be centered inside the outer-most square. You’ll need to think about where that square is positioned relative to the outer square and how to ensure the squares have the right colors. Make sure you keep track of the total number of squares drawn as you go, too.

To help you write your recursive function, we provide one helper function `drawSquare(size, color)` that draws a square of side length `size` filled with color `color`. The turtle should be in the lower left corner of the square to draw and facing east when this is called. In addition to `drawSquare`, the only turtle commands you can use in your function are `forward`, `backward`, `left`, and `right`. You should not use any loops. Also, your turtle should return to where it started after drawing your nested squares.

We have set up `squares.py` to take `size` and `gap` as command line arguments, which means you should supply those numbers when run your program as a script, eg:

```python3 squares.py 400 40
```

That should make it easy to experiment with different values for those parameters. We’ve stuck to gaps that evenly divide the size, but that doesn’t need to be the case – experiment a bit!

You can also change the pattern that gets drawn with small variations to the code:

In this task, you will implement a recursive pattern to graphically build a purple and gold quilt:

The starter code for this task is in `quilt.py`, which is structured in the same way as `squares.py`. This time, the primary function you should complete is `drawQuilt(quiltSize, patchSize, patchColor, otherColor)` in `quilt.py`. Once again, you should return the total number of squares drawn (as an int). The input parameters are described below:

• `quiltSize` is the side length of the whole quilt pattern.

• `patchSize` is the side length of the largest square that can be drawn as a solid square, without further dividing it into four smaller squares.

• `patchColor` is the color to use if you draw a solid patch as part of the pattern.

• `otherColor` is the patch color to use on any recursive calls. (As in the nested squares, `patchColor` and `otherColor` alternate roles on any nested calls to `drawQuilt`…)

To get started on the design of your recursive function `drawQuilt(quiltSize, patchSize, patchColor, otherColor)`, let’s explore how `quiltSize` and `patchSize` are related. Note that the `patchSize` will remain constant across recursive calls. Most importantly, to draw a quilt where the `quiltSize <= patchSize`, we just draw a single square of the `patchColor`, and return 1, since all quilts will have at least one square drawn. This is the base case of our quilt design:

`drawQuilt(512, 512, PURPLE, GOLD)` ` -> 1`

`drawQuilt(512, 512, GOLD, PURPLE)` ` -> 1`

If `quiltSize > patchSize`, then we must divide the quilt into four smaller squares. The smaller squares appearing on the anti-diagonal (top-right and bottom-left squares) are solid squares of `patchColor`. The smaller squares appearing on the main diagonal (top-left and bottom-right squares) are recursive quilts with `patchColor` and `otherColor` swapped to make the pattern more interesting. Thus each recursive case should draw something in each of the four quadrants, yielding two calls to `drawSquare()` and two recursive calls to `drawQuilt()`. In the following two examples, the recursive calls are handled by the base case, leaving four smaller solid squares as solid patches drawn with the appropriate color:

`drawQuilt(512, 256, PURPLE, GOLD)` ` -> 4`

`drawQuilt(512, 256, GOLD, PURPLE)` ` -> 4`

If the recursive quilt squares are still too big to be a single patch, then further recursive calls are made to break them up into four even smaller parts, yielding the following:

`drawQuilt(512, 128, PURPLE, GOLD)` ` -> 10`

`drawQuilt(512, 128, GOLD, PURPLE)` ` -> 10`

We can continue with smaller `patchSize`s to get even more intricate patterns:

`drawQuilt(512, 64, PURPLE, GOLD)` ` -> 22`

`drawQuilt(512, 32, PURPLE, GOLD)` ` -> 46`

`drawQuilt(512, 16, PURPLE, GOLD)` ` -> 94`

In the examples above, the value after the `->` indicates the value returned by that invocation. You may assume the turtle is facing east and positioned at `(-quiltSize/2, -quiltSize/2)` when `drawQuilt(quiltSize, patchSize, patchColor, otherColor)` is called. We provide the same `drawSquare` helper function and you should be able to write your `drawQuilt` function using only that and the turtle functions `forward`, `backward`, `left`, and `right`.

We have set up `quilt.py` to take `quiltSize` and `patchSize` as command line arguments, which means you should supply those numbers when run your program as a script, eg:

```python3 quilt.py 512 128
```

That should make it easy to experiment with different values for those parameters. Warning: It will take a very long time to draw patterns where the patch size is a lot smaller than the quilt size…

As in Task 3, there are lots of variations for your quilt. For example, draw nested squares with your function from Task 2 in the base case. And draw something more interesting in the solid squares on the anti-diagonal:

Experiment a bit and see what you can create!

In your final required task, you will write a recursive function named `shrub` in the file `shrub.py` that draws a tree pattern:

Your function will also return the total branch length of the shrub, as described below. As in the previous two tasks, you are to focus on writing on recursive function, `drawShrub(trunkLength, angle, shrinkFactor, minLength)`, which takes in four parameters:

• `trunkLength` is the length of the vertical branch at the base of the shrub.

• `angle` is the angle between a trunk and its right branch, and between the trunk and its left branch.

• `shrinkFactor` is the length of the right and left branches relative to their trunk. Specifically, the length of the right branch is `shrinkFactor * trunkLength`, and the trunk of the left branch is `shrinkFactor * shrinkFactor * trunkLength`.

• `minLength` is the minimum branch length in the shrub.

The `drawShrub` function (in addition to drawing the shrub) should return a number (float) corresponding to the total length of the branches in the shrub, including the trunk.

Please limit your `shrub` implementation to using only `forward`, `backward`, `left`, and `right` turtle commands.

See the sample invocations of the `drawShrub` function below, with the value after the function call `->` indicating the value returned by that invocation.

`drawShrub(100, 15, 0.8, 60)` ` -> 308.0`

`drawShrub(100, 15, 0.8, 50)` ` -> 461.6`

`drawShrub(100, 15, 0.8, 10)` ` -> 3973.9861913600025`

`drawShrub(100, 30, 0.82, 10)` ` -> 6386.440567704483`

We have set up `shrub.py` to take the four necessary parameters to draw a shrub:

```python3 shrub.py 100 15 0.8 60
```

That should make it easy to experiment with different values for those parameters. Here are some interesting test cases to try, and the expected numeric results:

```python3 shrub.py 100 15 0.8 60   # should print 308.0
python3 shrub.py 100 15 0.8 50   # should print 461.6
python3 shrub.py 100 15 0.8 40   # should print 666.4000000000001
python3 shrub.py 100 30 0.82 40  # should print 707.95128
python3 shrub.py 200 90 0.75 40  # should print 1524.21875
python3 shrub.py 100 15 0.8 10   # should print 3973.9861913600025
python3 shrub.py 100 30 0.82 10  # should print 6386.440567704483
python3 shrub.py 200 90 0.75 10  # should print 5056.675148010254
```

Extra Credit: Fancy Squares¶

In this task, you will combine the drawing of a colorful recursive pattern (as in Task 3) with the returning of tuples (as in Task 4, except we are returning multiple values). In particular, we will draw a recursive pattern of tri-colored squares and count the total number of squares of each color in the pattern:

This pattern contains 386 red squares, 364 blue squares, and 343 cyan squares.

You’ll implement this recursive function as `drawFancySquares(patternSize, minSize, colors)` in `fancy.py`. The parameters are the following:

• `patternSize` denotes the side length of the entire pattern, where the big solid square in the upper left quadrant has side length `patternSize/2`. Recursive subpatterns of side length `patternSize/2` fill the other three quadrants.

• `minSize` is the smallest value of `patternSize` for which a non-empty pattern is drawn. If the `patternSize` is strictly smaller than `minSize`, `recursiveSqCount` draws nothing.

• `colors` is a tuple of three color names. `colors[0]` is the color of the square in the upper left corner of the pattern. `colors[1]` and `colors[2]` are used in the recursive subpatterns, as illustrated below:

`drawFancySquares(512, 1024, ("red","blue","cyan"))` ` -> (0, 0, 0)`

`drawFancySquares(512, 512, ("red","blue","cyan"))` ` -> (1, 0, 0)`

`drawFancySquares(512, 256, ("red","blue","cyan"))` ` -> (1, 1, 2)`

`drawFancySquares(512, 128, ("red","blue","cyan"))` ` -> (6, 4, 3)`

Each calls returns a triple (that is, a 3-item tuple) where:

• the first item is the total number of `color[0]` squares in the pattern

• the second item is the total number of `color[1]` squares in the pattern

• the third item is the total number of `color[2]` squares in the pattern

The rest of the code in `fancy.py` is structured in the same way as in the previous tasks. You should only need to modify the `drawFancySquares` function, but feel free to experiment however you like!

When you’re finished, sign the honor code and stage, commit, and push your work to the server as in previous labs.

• For Task 1, remember that you are required to add a new doctest.

• For Tasks 2, 3, 4, and the extra credit, you do not need to submit any image files. We will generate them as we grade.

Good luck! Do not forget to add, commit, and push your work as it progresses! Test your code often to simplify debugging.

Note that Tasks 0-4 are the only required tasks for this lab. The extra-credit recursive squares is optional. See lab Syllabus for an overview of the rubric used in grading. The file `Gradesheet.txt` gives you specifics of our expectations for this lab.