Recently Joanna has become tired of listening to the music in her playlist in the same order over and over again. She still really enjoys the contest of the playlist, but wants to try listening to it in a different order to try to switch things up. Specifically, she wants to try reversing the playlist so that the last song plays first, the second to last song plays second, and so on.

Given Joanna's playlist, can you write a program to reverse the playlist so that she can listen to music in the new ordering?

Arturo realized that he doesn't listen to his vinyl records very often anymore so he decides to hang them up on the wall of his study. He grabs one of his vinyl records which consists of $$$n$$$ concentric circles (circles are concentric if they all share the same center) and decides to paint over the top of the vinyl record to fit the decor of his room. He starts at the outermost ring of the vinyl (a ring is defined as the space between two concentric circles that lie within the circle of a bigger radius and outside the circle with the smaller radius) and paints it red. He then paints the outermost unpainted ring blue and continues this process of alternating colors for each of the rings, each time choosing the outermost one that has not been painted yet.

Given the radii (in centimeters) of the $$$n$$$ concentric circles for a particular vinyl, help Arturo figure out the area (in $$$\text{cm}^{2}$$$) of the vinyl that gets painted red.

As many people know, there are 88 keys on the piano. When a key is pressed, the frequency of the sound produced can be measured. As it turns out, a perfectly tuned piano has as Middle C the frequency of 440 Hz. From there, an octave above is exactly double this, or 880 Hz, and an octave below is exactly 220 Hz. As it turns out, an octave is actually defined mathematically as the note that is double or half of the starting note.

For the notes in between, they happen to be exactly evenly spaced based on the log scale! In fact, given the octave starting at Middle C, the first note to follow Middle C, or D, has a frequency of exactly $$$440 \cdot 2^{\frac{1}{12}}$$$, and the second note has a frequency of exactly $$$440 \cdot 2^{\frac{2}{12}}$$$, and so on and so forth. As it turns out, this pattern holds for all octaves.

Matt has developed a robot that listens to music, and after listening to Alice and Bob's performances of the same song, it has determined the central $$$n$$$ frequencies that make up the melody of both Alice and Bob's performances. Now, given that they performed on an infinitely long piano, and that they both performed for the same time, we can define the difference between the melodies as follows. For every note $$$i$$$, we take the note's difference as $$$b_i - a_i$$$, where $$$b_i$$$ is Bob's note's position on the piano, and $$$a_i$$$ is Alice's note's position on the piano. Taking the sum over all $$$i$$$, we can define the overall difference as the absolute value of that sum.

Given Alice and Bob's melodies in terms of frequencies, determine their difference!

Note: The introduction explaining musical frequencies is the same as part 1 of this problem.

As many people know, there are 88 keys on the piano. When a key is pressed, the frequency of the sound produced can be measured. As it turns out, a perfectly tuned piano has as Middle C the frequency of 440 Hz. From there, an octave above is exactly double this, or 880 Hz, and an octave below is exactly 220 Hz. As it turns out, an octave is actually defined mathematically as the note that is double or half of the starting note.

For the notes in between, they happen to be exactly evenly spaced based on the log scale! In fact, given the octave starting at Middle C, the first note to follow Middle C, or D, has a frequency of exactly $$$440 \cdot 2^{\frac{1}{12}}$$$, and the second note has a frequency of exactly $$$440 \cdot 2^{\frac{2}{12}}$$$, and so on and so forth. As it turns out, this pattern holds for all octaves.

Matt has developed a robot that listens to music, and after listening to Alice and Bob's performances of the same song, it has determined the central $$$n$$$ frequencies that make up the melody of both Alice and Bob's performances. Now, given that they performed on an infinitely long piano, and that they both performed for the same time, we can define the difference between the melodies as follows. For every note $$$i$$$, we take the note's difference as $$$b_i - a_i$$$, where $$$b_i$$$ is Bob's note's position on the piano, and $$$a_i$$$ is Alice's note's position on the piano. Taking the sum over all $$$i$$$, we can define the overall difference as the absolute value of that sum.

Given Alice and Bob's melodies in terms of frequencies, we already know how to determine their difference. However, Eve noticed that perhaps, Alice and Bob are playing in different scales! In other words, if we move all of Alice's notes up or down an integer amount on the piano, then the difference with Bob's performance can be reduced. Given this, determine with the best possible movement for Alice, the minimum difference between the two.

Algo the serial composer absolutely loves to write music and does it all the time. In fact, he has composed so many pieces over the years that now he has grown bored of having so much creative freedom and would like to give himself a challenge. Algo will now limit himself to composing with only the notes from a set of $$$N$$$ chosen note lengths, and he wants to match the overall target length of his song exactly.

For instance, if Algo wants to compose a 1-measure song using only quarter notes, half notes, and whole notes—of beat-lengths 1, 2, and 4 respectively—then there are 6 total possible songs that he could compose:

Algo has just been commissioned to write a song that is $$$M$$$ four-beat measures long. Determine how many unique songs ending at exactly $$$M$$$ measures Algo could possibly write, using only his $$$N$$$ chosen notes!

Ice-T has remained a powerful player in the rap game for almost forty years, ever since his debut in the early 80's. It is a little known secret that Ice-T powers his incredible rapping abilities by amassing points in a 2D plane and harvesting all of the 'T's that he can find among them.

A 'T' is defined to be a set of four points $$$A$$$, $$$B$$$, $$$C$$$, and $$$D$$$ where:

1. Points $$$A$$$ and $$$B$$$ are collinear and form a segment $$$\overline{AB}$$$ that is parallel to either the $$$x$$$ or $$$y$$$ axis
2. Point $$$C$$$ rests on the midpoint exactly equidistant between $$$A$$$ and $$$B$$$
3. point $$$D$$$ forms a line segment perpendicular to the $$$\overline{AB}$$$ segment when connected with $$$C$$$

Help Ice-T become even more powerful! Given a set of $$$N$$$ points in 2D space, find the number of 'T's present.

As an avid musician, Santiago loves to practice his music and work on his fundamentals. One of his favorite exercises is playing through arbitrary scales with notes labelled from $$$1$$$ to $$$n$$$ in order. However, he's recently gotten tired of just playing the same scales over and over again and wants to switch up the order of the notes of the scale.

Before playing each note in the permuted scale, Santiago finds the highest previously played note lower than the current note, $$$l$$$, and the lowest previously played note higher than the current note, $$$r$$$, and defines the goodness of the note as the number of notes in the interval $$$(l, r)$$$ (exclusive). If no such $$$l$$$ exists, Santiago takes all notes equal to or below the current note as part of the interval and if no such $$$r$$$ exists, Santiago takes all notes equal to or above the current note as part of the interval.

Santiago then defines the total goodness of the scale ordering to be the sum of the goodness of the notes played within the scale ordering. Santiago feels he gets higher quality practice from scales with higher goodness. Given a proposed scale ordering, output the total goodness of the corresponding scale so that Santiago knows which scale orderings are worth practicing.