Algorithmic Puzzles @algopuzzles
Problems in graph theory, combinatorics and algorithms. Contests are held on Algo Muse website. algomuse.net Joined April 2009-
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Show that a map formed on a plane by finitely many circles is 2-colourable.
Prove that a graph with n nodes either has a path of length k or has O(kn) edges.
Consider the set of n vertices of a convex polygon. Count the number of non-crossing paths that visit all the vertices. [Src: 11011110, David Eppstein] #exportober
You are given an array A[1..n] in which each element is at most k positions away from the position it would have been in if A was sorted. You don't know the value of k. Design an algorithm to sort the array in O(n log k) time. #exportober
A tiling problem with wireframes. [Source: Micheal Brand]. #exportober
Consider the following variant of chess: In the first move only, white can opt in to make a single chess move or two consecutive chess moves; after that it is usual chess. Let's call this variant mhess. Theorem: In mhess, white can always force at least a draw.
Take an nxn grid. Colour each square white or black in such a way that each 2x2 grid has exactly 2 white and 2 black squares. How many ways can this be done? Like I said, it is accessible to high school students and teachers. Of course counting options one by one is suicidal.
Tricky math puzzle: Suppose we have a disk of diameter 20 and strips of width 1 and length 20. We can cover the disk using 20 strips by laying them side-by-side. Can we do it with 19? No cutting is allowed. My next tweet is the answer/solution. 1/6
Suppose A is the adjacency matrix of an undirected graph. Prove that the absolute value of any eigenvalue of A is less than the max. degree of the graph. #exportober
On an nxn grid a few nodes are set on fire at t=0. The fire spreads like this: if at least two neighboring nodes of node x are lit, then node x will also catch fire in the next time step. What's the minimum number of nodes that have to be lit to burn the entire grid? #exportober
We are given as input the vertices of a polygon P. Determine in near-linear time if P is a star. Specifically, determine if there is a point q inside P such that for every point p on the boundary of P, the line segment pq lies in the interior of P. (Src: David Mount) #exportober
We have a campus in the shape of a perfect disk of radius 1 km. We need to open 7 coffee shops so as to minimize the farthest distance anybody has to travel to get coffee. Where should we open the shops? [Src: Puzzle Toad] #exportober
In a directed graph, a king is defined as a node from which all the other nodes are reachable in at most two steps. A tournament is a directed graph that has exactly one edge between any distinct pair of vertices. Prove that every tournament has a king. #exportober
An nxn matrix maps every vector v to a vector that is orthogonal to v. What can you say about n? #exportober
Show that a finite field cannot be algebraically closed. #exportober
Neeldhara 🐦|🐘 @neeldhara
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Puzzle Pot @puzzlepot
36 Followers 587 Following welcome to my channel @PuzzlePot Hello, viewers! Today, we have an exciting and brain-teasing challenge for you: the "Jumbled Word Clothes Items Puzzle." This
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When water is poured into a cube and the three primary colors on each side mix ...
@sp_monte_carlo Have you come across Pollard's stuff? stat.yale.edu/~pollard/Books/ Looks like he would appreciate help (update Oct 23) editing his new book: stat.yale.edu/~pollard/Books…
A LEGO paper airplane making machine [📹 kenken_lego]
Here are some instructions for how to draw a perfect drop of water 📸Wes Modes
When water is sprayed in the opposite direction, icreating low-pressure area around the fire. This draws in surrounding air, which can deprive the fire of oxygen, making it difficult for the flames to sustain themselves, related to the Venturi effect
This broke me. I'm sobbing uncontrollably right now.
Karnataka streamlined functioning of 5,700 libraries and brought them under the governance of respective gram panchayats. Around 4,675 libraries have been renovated already and over 300 new gram panchayat libraries have been established. thehindu.com/news/national/…
(2/5) We assigned this as a HW exercise in a class I taught as a grad student at MIT circa 1991 Coincidentally, it was written up around the same time by C. Mallows in "Another comment on O'Cinneide" The American Statistician, 45-3 Proof is easy using Jensen's inequality twice:
Here, in full directly on Twitter, is "A Hackers' Guide to Language Models". This 90 minute tutorial is designed to be the one place I point coders at when they ask "hey, tell me everything I need to know about LLMs!" It covers both @OpenAI models and open source ones in depth.
The Tesla AI team has pulled off a miracle and doesn't get enough credit. I just say where I want to go and my car takes me there with just computer vision and 0 human input, routinely. $TSLA Watch in 4K: youtu.be/qduHwGuS0Lw
When bananas ripen, their stems release ethylene gas, and when it spreads to the rest of the fruit, it will quickly cause the fruit to rot. That's why the plastic wrap around the stem keeps the ethylene gas contained so the bananas stay safe
Hi morning folks! During July I'm going to lead a learning experience for current K-12 teachers called "21st Century Mathematics". Its goal is to help teachers to connect the math they teach with some math that has been discovered in the 21st century. justinlanier.org/21st-century-m…
It’s been nearly 5 months of being completely bedridden. I’ve made very little progress. I can still barely move, or think or talk. I want to be out in the world living a life. My body hurts. But my heart hurts more. I wish I had a more positive update.
Elsevier is losing editors
One of my favorite exercises on measure theory 😈
Navigating a scientific career is never easy. One way to enjoy its progress is to observe and respect the commas, semicolons and fullstops. Here's an insider's view of the process
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