Given a pointer pointing to the header of a linked list, how to detect whether the linked list has a loop?

An additional question: how can math knowledge help here?

Given an string as input, find the length of the longest palindrome it contains. A palindrome is a word which stays the same if reading backwards. Examples include madam, Hannah. 

Determine whether a given integer can be expressed as the sum of two perfect squares.

(Coin Change Problem) Given a set of $N$ possible denominations of coins, find the number of different combinations to pay for a desired sum.

Given an array of integers nums and an integer target, return indices of the two numbers such that they add up to target. You may assume that each input would have exactly one solution, and you may not use the same element twice.

You are given an integer array height of length $n$. There are $n$ vertical lines drawn such that the two endpoints of the $i$th line are $(i, 0)$ and ($i$, height[$i$]). Find two lines that together with the $x$-axis form a container, such that the container contains the most water. Return the maximum amount of water a container can store.

Input: height = $[1,8,6,2,5,4,8,3,7]$

Output: $49$

Explanation: The above vertical lines are represented by array $[1,8,6,2,5,4,8,3,7]$. In this case, the max area of water (blue section) the container can contain is $49$.

A numeric array of length $N$ is given. We need to design a function that finds all positive numbers in the array that have their opposites in it as well. Describe approaches for solving optimal worst case and optimal average case performance, respectively.

Design an algorithm that finds the number of ways in which you can traverse $N$ meters by doing jumps of $1$, $2$, $3$, $4$, or $5$ meter lengths.

On a $M\times N$ board, some cells are occupied. Find the size of the largest square of unoccupied cells. 

Given $n$ non-negative integers representing an elevation map where the width of each bar is $1$, compute how much water it can trap after raining.

Example $1$:

Input: height = $[0,1,0,2,1,0,1,3,2,1,2,1]$

Output: $6$

Explanation: The above elevation map (black section) is represented by array $[0,1,0,2,1,0,1,3,2,1,2,1]$. In this case, $6$ units of rain water (blue section) are being trapped.

Example $2$:

Input: height = $[4,2,0,3,2,5]4

Output: $9$

Given two strings s and t, return true if they are equal when both are typed into empty text editors. '#' means a backspace character. (Note that after backspacing an empty text, the text will continue empty.)

Example 1:

Input: s = "ab#c", t = "ad#c"

Output: true

Explanation: Both s and t become "ac".

Example 2:

Input: s = "ab##", t = "c#d#"

Output: true

Explanation: Both s and t become "".

Example 3:

Input: s = "a#c", t = "b"

Output: false

Explanation: s becomes "c" while t becomes "b".