Easy warmup problems

Example:

def double_loop_loop(n):
  for i in range(2*n):
    for j in range(i+1, 2*n):
      print(i, j)
double_loop_loop(4)

Problem 1

Make a function that can count how many the word “like” occurs in a sentence you input, and also in the following sentence (case insensitive, i.e. capitalized counts): “Like it’s really easy, like dude what’re you like doing like what is this question, like this seems dumb like; oh man liken this tutorial unto yourselves like get scammed like wow like like like like Ziyong is a pro yah like know?”

Problem 1.5

Make a function that can print out every other word in a string that you input, and also in the sentence from the previous problem.

Problem 2

Make a function that finds the element whose square is biggest in a list (break ties by which one occurs first in the list). e.g. biggestSquare([0,0,0,0]) = 0; biggestSquare([0,1,0,0]) = 1, biggestSquare([0,1,-2,0]) = -2, biggestSquare([1,2,3,4,-4]) = 4

Problem 3

Make a function that gives a random coronavirus related tip (from a list of coronavirus related tips)

Number Theory

Problem -1

Write a program that outputs for each \(i, j \in \{1,2,\ldots, n\}\) computes whether \(i\) and \(j\) are relatively prime (whether they have any common factors) and stores it in a 2d matrix.

Definition: Modular arithmetic. Consider a clock. The time is some number in \([0,12]\) as represented by clock hands. If you are at time \(x\), and you wait for \(5\) hours, you are either at time \(x+5\), or time \(x+5-12\), because in “clock arithmetic” the numbers wrap once you pass \(12\). For example, in clock arithmetic \(7+7 = 2\) (7 hours after 7pm it is 2am).

The mod opperator is very useful for a lot of reasons, which you will soon see. It is defined as \(x \% m = x - km\) where \(k\) is an integer chosen so that \(x-km \in [0,m)\). Basically, you just subtract the mod off of the number until you lie in the right range, exactly like with the clock.

Example:

zodiac_animals = ["Rat", "Ox", "Tiger", "Rabbit", "Dragon", "Snake", "Horse", "Goat", "Monkey", "Rooster", "Dog", "Pig"]
print("hey, want to know your zodiac animal! I can tell you.")
age = int(input("How old are you?"))
print("Your zodiac animal is" + zodiac_animals[age%12])

Problem 0.5 (VERY COOL)

Approximate \(\pi\) with the following method: Draw a bunch of points uniformly at random from the unit square (using np.random.rand) Calculate what fraction are in the unit circle too (satisfy \(x^2 + y^2 \le 1\)). This fraction approahces \(\pi/4\) if you take more and more points.

Problem 0 (VERY COOL)

There is a game called “FLIP FLOP”. \(n\) people sit in a circle. They take turns saying numbers in ascending order. Max says “1”, Alek says “2”, Justin says 3, Madeline says 4, David says 5, Ziyong says 6, and then Leon is like “FLIP!!!” (on any multiple of \(7\) the person says FLIP instead of saying the number. When they say FLIP the direction around the circle changes) Then Ziyong says “FLOP!!!!” (on any multiple of \(8\) the person says FLOP instead of saying the number. When they say FLOP a person is skipped in the current direction) Then David is sad because he was skipped. Madeline says 9 Justin says 10 Alek says 11 Max says 12 Melina says 13 Ethan says FLIP Melina says 15 Max says FLOP Alek feels left out Justin says 17 etc

Now here is the question, lets say there are 100 people in the circle. Bob is 50 seats clockwise of Max Does he ever get a chance to play? SIMULATE IT! once you are done, you can check against my simulation: aleks flipflop

Problem 1 (VERY COOL)

Write a program that computes the last 5 digits of \(2^{123412341234134}\)

Note: if you try to make python compute \(2^{123412341234134}\) it will blow up, this is a ridiculously big number Instead of doing that, use the following mathematical fact (that you should verify!) (a * b)% m = (a%m * b%m)% m Also note, that the number is big, so even a for loop will take a while! Hint: “Exponentiation by squaring” You could also probably find a pattern, stuff repeats in modular arithmetic

Problem -0.5 (COOL)

A magic square is an arrangement of distinct numbers (i.e., each number is used once), usually integers, in a square grid, where the numbers in each row, and in each column, and the numbers in the main and secondary diagonals, all add up to the same number, called the “magic constant.” A magic square has the same number of rows as it has columns, and in conventional math notation, “n” stands for the number of rows (and columns) it has. Thus, a magic square always contains \(n^{2}\) numbers, and its size (the number of rows [and columns] it has) is described as being “of order n”.

First, write a program to check whether an n x n matrix is a magic square. Test it on this square: [[7, 12, 1, 14], [2, 13, 8, 11], [16, 3, 10, 5], [9, 6, 15, 4]]

Next, write a program to create your own magic square. Then check that it is indeed a magic square using the program you wrote above.