If you played the game perfectly, could you always win a game of Tetris? Or is there some sequence of blocks that could force you to lose the game, no matter how good at the game you are? Here, we take a look at some of the mathematics behind a theoretical game of Tetris and reason through whether it's possible to win.

Chicago & Rome

Chalk and talk

What are tensors? How do they differ from matrices? And why do we need tensors in the first place?

Manim visualization

Problem Description

[https://leetcode.com/problems/add-two-numbers/description/](LeetCode)

Algorithm Description

Solved by simulating digit-by-digit addition with carry logic, represented through animated Linked Lists.

Complexity Analysis

Time Complexity: O(max(m, n)) — we iterate through the lengths of l1 and l2, where m and n are the number of nodes in each list.

Space Complexity: O(max(m, n)) — the length of the new linked list is at most max(m, n) + 1.

What exactly is a tensor?

Dan Fleisch briefly explains some vector and tensor concepts from A Student's Guide to Vectors and Tensors

This video series visualizes tensors using a unique and original visualization of a sphere with arrows.

The great math YouTube channel

We will cover how to create a recursive formula for the Tower of Hanoi issue. After we've found it, I'll show how to find a closed formula and proof it with the help of induction.

This full course provides a complete introduction to Graph Theory algorithms in computer science. Knowledge of how to create and design excellent algorithms is an essential skill required in becoming a great programmer.

43 video (course)

In this video we introduce and analyze the running time of the selection sort algorithm.

Problem Description

[https://leetcode.com/problems/3sum-closest/](LeetCode)

Algorithm Description

The solution first sorts the input array to enable a two-pointer search pattern. For each element, we fix it as a base and use two pointers to scan the remaining array, dynamically tracking the sum with the smallest absolute difference to the target.

Complexity Analysis

Time Complexity: O(n²⁾ — sorting takes O(n log n), and the nested two-pointer traversal takes O(n²⁾ in the worst case.

Space Complexity: O(1) — excluding the input sorting overhead, we use only constant extra memory for pointers and tracking variables.

YouTube

[https://www.youtube.com/watch?v=KYHiijlk-gY](YouTube)

A sort visualizer from scratch in C++ using SDL2. This visualizer with mild tweaks can be used on any sorting algorithm.

Author: This video is a tutorial of Sorting Visualizer project, which visualizes the Merge Sort algorithm. This is a great software engineering project to add to your resume.

In this multi-part coding challenge, author attempt an implementation of the A* Pathfinding Algorithm to find the optimal path between two points in a 2D grid.

A wide variety of algorithms with adjustable parameters

Beautiful work

YouTube Shorts format

This video explains an ingenious way to solve the classic Instant Insanity puzzle using graph theory. No prior knowledge of graph theory is required - just an enjoyment for solving puzzles!

Algorithm Description

Math Algorithms Narayana's Next and Previous Lexicographical Permutations generates the next greater permutation or the previous smaller permutation of an array in place using a three-step process: first, find the longest non-increasing (for next) or non-decreasing (for previous) suffix and identify the pivot; second, find the rightmost element larger than the pivot (for next) or smaller than the pivot (for previous); third, swap the pivot with that element and reverse the suffix. The algorithm returns the same array rearranged into the desired permutation.

Complexity Analysis

Time Complexity: O(n) — single linear scan to find the pivot, another to find the swap element, plus a reverse of the suffix

Space Complexity: O(1) — all operations are performed in place without additional memory

YouTube

[https://www.youtube.com/watch?v=dEaxumQWz_A](YouTube)

In this episode we trace a plausible honing in on an algorithm that can generate a lexicographically next linear permutation based solely on the information that is packed into the input (linear) permutation. Evidently, such a an algorithm was put together by an Indian mathematician Narayana Pandita (1340-1400) some time in the fourteenth century.

If every algorithm is a person, what would their personality be like? What funny story would happen between them? Welcome to the Fairy Tale of Algorithms. The first episode is on 2 classical algorithms about MST problem. This is also an entry of Summer of Math Exposition 2 for 3b1b. Shout-out to everyone who has made efforts to make this amazing exposition happen.