• This part of the hw is SOLO.  As explained in the course syllabus, you must not discuss these problems or share your code with anyone else except for the course staff and the tutors at Academic Development.  This includes students from former semesters (who may indeed have the answers to these problems!).
   
• This portion is worth 50 pts (4 non-bonus problems worth 12.5 pts each).
   
• Be sure to name your functions exactly as specified, and take exactly the same parameters as specified!
   
• There is no starter file nor any public autograder this week.  Be sure to write your own test functions!!!

1. solvesCryptarithm
   Background:  a cryptarithm is a puzzle where we start with a simple arithmetic statement but then we replace all the digits with letters (where the same digit is replaced by the same letter each time).  We will limit such puzzles to strings the form “A+B=C” (no spaces), where A, B, and C are positive integers.  For example, “SEND+MORE=MONEY” is such a puzzle.  The goal of the puzzle is to find an assignment of digits to the letters to make the math work out properly.  For example, if we assign 0 to “O”, 1 to “M”, 2 to “Y”, 5 to “E”, 6 to “N”, 7 to “D”, 8 to “R”, and 9 to “S” we get:
   
       S E N D       9 5 6 7
     + M O R E     + 1 0 8 5
     ---------     ---------
     M O N E Y     1 0 6 5 2
    
   And so we see that this assignment does in fact solve the problem!   Now, we need a way to encode a possible solution.  For that, we will use a single string where the index of the letter corresponds to the digit it represents.  Thus, the string “OMY--ENDRS" represents the assignments listed above (the dashes are for unassigned digits).
   
   With this in mind, write the function solvesCryptarithm(puzzle, solution) that takes two strings, a puzzle (such as “SEND+MORE=MONEY”) and a proposed solution (such as “OMY--ENDRS").  Your function should return True if substituting the digits from the solution back into the puzzle results in a mathematically correct addition problem, and False otherwise.   You do not have to check whether a letter occurs more than once in the proposed solution, but you do have to verify that all the letters in the puzzle occur somewhere in the solution (of course).  You may not use the eval() function.  Also, you almost surely will want to write at least one well-chosen helper function.
    
2. areaOfPolygon
   Write the function areaOfPolygon that takes a list of (x,y) points that are guaranteed to be in either clockwise or counter-clockwise order around a polygon, and returns the area of that polygon, as described here.  For example (taken from that text), areaOfPolygon([(4,10), (9,7), (11,2), (2,2)]) returns 45.5 (at least the result is almostEqual to 45.5).
    
3. linearRegression
   Write the function linearRegression that takes a list of (x,y) points and finds the line of best fit through those points. Specifically, your function should return a triple of floats (a,b,r) such that y = ax+b is the line of best fit through the given points and r is the correlation coefficient as explained here (and yes, you must follow this exact approach).  For example (taken from the text), linearRegression([(1,3), (2,5), (4,8)]) should return a triple of 3 values approximately equal to (1.6429, 1.5, 0.9972), indicating that the line of best fit is y = 1.6429x + 1.5, with a correlation coefficient of 0.9972 (so it’s a really good fit, too).  Note that the result is approximately equal to these values.  We'll use a large-ish epsilon as noted above.  Also note:  the notes we refer you to do not discuss the meaning of the sign of r, so just return the absolute value |r|.  And: you may ignore the case of a vertical line in your linearRegression code.
    
4. bestScrabbleScore
   Background: In a Scrabble-like game, players each have a hand, which is a list of lowercase letters.  There is also a dictionary, which is a list of legal words (all in lowercase letters).  And there is a list of letterScores, which is length 26, where letterScores[i] contains the point value for the ith character in the alphabet (so letterScores[0] contains the point value for 'a').  Players can use some or all of the tiles in their hand and arrange them in any order to form words.  The point value for a word is 0 if it is not in the dictionary, otherwise it is the sum of the point values of each letter in the word, according to the letterScores list (pretty much as it works in actual Scrabble).
   
   In case you are interested, here is a list of the actual letterScores for Scrabble:
      letterScores = [
      #  a, b, c, d, e, f, g, h, i, j, k, l, m
         1, 3, 3, 2, 1, 4, 2, 4, 1, 8, 5, 1, 3,
      #  n, o, p, q, r, s, t, u, v, w, x, y, z
         1, 1, 3,10, 1, 1, 1, 1, 4, 4, 8, 4,10
      ]
   Note that your function must work for any list of letterScores as is provided by the caller.
   
   With this in mind, write the function bestScrabbleScore(dictionary, letterScores, hand) that takes 3 lists -- dictionary (a list of lowercase words), letterScores (a list of 26 integers), and hand (a list of lowercase characters) -- and returns a tuple of the highest-scoring word in the dictionary that can be formed by some arrangement of some subset of letters in the hand, followed by its score.  In the case of a tie, the first element of the tuple should instead be a list of all such words in the order they appear in the dictionary.  If no such words exist, return None.
   
   Note: there is no fixed dictionary here.  Each time we call the function, we may provide a different dictionary!  It may contain 100 words or perhaps 100,00 words.
    
5. bonus/optional:  nthLuckyPrime [2.5 pts]
   Write the function nthLuckyPrime that takes a non-negative int n and returns the nth number that is both lucky and prime, where a lucky number is as defined here (or, identically, here).  Here are the first several lucky primes:  3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, 211...
   Hint: to do this, you will need to use a list to implement a sieve, similar to how we created the Sieve of Eratosthenes.

Bonus/Optional Graphics:  Read this carefully!  The following problems are bonus/optional.  They are graphical, and you should be certain to place them (and any helper functions they require) below the #ignore_rest line in your hw5.py submission.  Unlike the autograded problems, the bonus graphics problems will be graded only in person on Monday or Tuesday night at office hours.  To get your bonus graphics grade, go to office hours where you should download your submitted code (and it must be just as submitted, with no modifications), run it and show the CA the result.  The CA will grade your bonus at that time, mainly visually but also by looking at the code as needed.

6. Bonus/Optional:  linearRegression Visualization [2.5 pts]
   Write a function that takes a list of (x,y) points, like the linearRegression problem above, only here is generates a graphical visualization showing the points, the coordinate axes, the line of best fit, the actual squares of the errors (the vertical distances from each point to the line), and the total sum of the squares of the errors, as in this picture here (which is part of this web page).  You don't have to match the picture perfectly, but you should be reasonably close.
    
7. Bonus/Optional:  Static Motion AfterEffect [2.5 pts]
   Write a function that draws this picture using only Tkinter primitives (rectangles, ovals, polygons, arcs, etc).

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