Given an array of integers, find and print the maximum number of integers you can select from the array such that the absolute difference between any two of the chosen integers is less than or equal to 1. For example, if your array is a = [1,1,2,2,4,4,5,5,5], you can create two subarrays meeting the criterion: [1,1,2,2] and [4,4,5,5,5]. The maximum length subarray has 5 elements.

An integer d is a divisor of an integer n if the remainder of .

Given an integer, for each digit that makes up the integer determine whether it is a divisor. Count the number of divisors occurring within the integer.

Note: Each digit is considered to be unique, so each occurrence of the same digit should be counted (e.g. for n = 111, 1 is a divisor of 111 each time it occurs so the answer is 3).

Brie’s Drawing teacher asks her class to open their books to a page number. Brie can either start turning pages from the front of the book or from the back of the book. She always turns pages one at a time. When she opens the book, page 1 is always on the right side:

When she flips page 1, she sees pages 2 and 3. Each page except the last page will always be printed on both sides. The last page may only be printed on the front, given the length of the book. If the book is n pages long, and she wants to turn to page p, what is the minimum number of pages she will turn? She can start at the beginning or the end of the book.

Given n and p, find and print the minimum number of pages Brie must turn in order to arrive at page p.

A jail has a number of prisoners and a number of treats to pass out to them. Their jailer decides the fairest way to divide the treats is to seat the prisoners around a circular table in sequentially numbered chairs. A chair number will be drawn from a hat. Beginning with the prisoner in that chair, one candy will be handed to each prisoner sequentially around the table until all have been distributed.

The jailer is playing a little joke, though. The last piece of candy looks like all the others, but it tastes awful. Determine the chair number occupied by the prisoner who will receive that candy.

For example, there are 4 prisoners and 6 pieces of candy. The prisoners arrange themselves in seats numbered 1 to 4. Let’s suppose two is drawn from the hat. Prisoners receive candy at positions 2,3,4,1,2,3. The prisoner to be warned sits in chair number 3.

HackerLand Enterprise is adopting a new viral advertising strategy. When they launch a new product, they advertise it to exactly 5 people on social media.

On the first day, half of those 5 people (i.e., ) like the advertisement and each shares it with 3 of their friends. At the beginning of the second day, people receive the advertisement.

Each day, of the recipients like the advertisement and will share it with 3 friends on the following day. Assuming nobody receives the advertisement twice, determine how many people have liked the ad by the end of a given day, beginning with launch day as day 1.

For example, assume you want to know how many have liked the ad by the end of the 5^th day.

Lily likes to play games with integers. She has created a new game where she determines the difference between a number and its reverse. For instance, given the number 12, its reverse is 21. Their difference is 9. The number 120 reversed is 21, and their difference is 99.

She decides to apply her game to decision making. She will look at a numbered range of days and will only go to a movie on a beautiful day.

Given a range of numbered days, [i…j] and a number k, determine the number of days in the range that are beautiful. Beautiful numbers are defined as numbers where |i - reverse(i)| is evenly divisible by k. If a day’s value is a beautiful number, it is a beautiful day. Print the number of beautiful days in the range.

A Discrete Mathematics professor has a class of students. Frustrated with their lack of discipline, he decides to cancel class if fewer than some number of students are present when class starts. Arrival times go from on time (arrivalTime <= 0) to arrived late (arrivalTime > 0).

Given the arrival time of each student and a threshhold number of attendees, determine if the class is canceled.

Dan is playing a video game in which his character competes in a hurdle race. Hurdles are of varying heights, and Dan has a maximum height he can jump. There is a magic potion he can take that will increase his maximum height by 1 unit for each dose. How many doses of the potion must he take to be able to jump all of the hurdles.

Given an array of hurdle heights height, and an initial maximum height Dan can jump, k, determine the minimum number of doses Dan must take to be able to clear all the hurdles in the race.

For example, if height = [1,2,3,3,2] and Dan can jump 1 unit high naturally, he must take 3 - 1 = 2 doses of potion to be able to jump all of the hurdles.

Gary is an avid hiker. He tracks his hikes meticulously, paying close attention to small details like topography. During his last hike he took exactly n steps. For every step he took, he noted if it was an uphill, U, or a downhill, D step. Gary’s hikes start and end at sea level and each step up or down represents a 1 unit change in altitude. We define the following terms:

• A mountain is a sequence of consecutive steps above sea level, starting with a step up from sea level and ending with a step down to sea level.
• A valley is a sequence of consecutive steps below sea level, starting with a step down from sea level and ending with a step up to sea level.

Given Gary’s sequence of up and down steps during his last hike, find and print the number of valleys he walked through.

For example, if Gary’s path is s = [DDUUUUDD], he first enters a valley 2 units deep. Then he climbs out an up onto a mountain 2 units high. Finally, he returns to sea level and ends his hike.

You have been asked to help study the population of birds migrating across the continent. Each type of bird you are interested in will be identified by an integer value. Each time a particular kind of bird is spotted, its id number will be added to your array of sightings. You would like to be able to find out which type of bird is most common given a list of sightings. Your task is to print the type number of that bird and if two or more types of birds are equally common, choose the type with the smallest ID number.

For example, assume your bird sightings are of types arr = [1,1,2,2,3]. There are two each of types 1 and 2, and one sighting of type 3. Pick the lower of the two types seen twice: type 1.