What is Group Theory?

In math, a group is a particular collection of elements. That might be a set of integers, the face of a Rubik’s cube–which we’ll simplify to a 2x2 square for now– or anything, so long as they follow 4 specific rules, or axioms.

Axiom 1: All group operations must be closed, or restricted, to only group elements. So in our square, for any operation you do—like turn it one way or the other—you’ll still wind up with an element of the group. Or for integers, if we add 3 and 2, that gives us 1—4 and 5 aren’t members of the group, so we roll around back to 0, similar to how 2 hours past 11 is 1 o’clock.

Axiom 2: If we regroup the order of the elements in an operation, we get the same result. In other words, if we turn our square right two times, then right once, that’s the same as once, then twice. Or for numbers, 1+(1+1) is the same as (1+1)+1.

Axiom 3: For every operation, there’s an element of our ground called the identity. When we apply it to any other element in our group, we still get that element. So for both turning the square and adding integers, our identity here is 0. Not very exciting.

Axiom 4:  Every group element has an element called its inverse, also in the group. When the two are brought together using group’s addition operation, they result in the identity element, 0. So they can be thought of as cancelling each other out. Here 3 and 1 are each other’s inverses, while 2 and 0 are their own worst enemies.

So that’s all well and good, but what’s the point of any of it? Well, when we get beyond these basic rules, some interesting properties emerge. For example, let’s expand our square back into a full-fledged Rubik’s cube. This is still a group that satisfies all of our axioms, though now with considerably more elements, and more operations—we can turn each row and column of each face.

Each position is called a permutation, and the more elements a group has, the more possible permutations there are. A Rubik’s cube has more than 43 quintillion permutations, so trying to solve it randomly isn’t going to work so well. However, using group theory we can analyze the cube and determine a sequence of permutations that will result in a solution. And, in fact, that’s exactly what most solvers do, even using a group theory notation indicating turns.

From the TED-Ed Lesson Group theory 101: How to play a Rubik’s Cube like a piano - Michael Staff

Animation by Shixie

Cicada 3301.

Cicada 3301 is perhaps the most mysterious seemingly Internet-based organization in the history of online mysteries. Since 2012, the group has posted numerous puzzles online under that name and has made no reference as to what it does or even where it is from. No one knows who runs it, assuming it really is an organization, or even the name of a single member. Cicada 3301 is able to keep its reputation with so little information strictly because of the interesting scavenger hunt of sorts it has released for three years running.

It all started on January 5, 2012 when a 4chan user posted a steganography clue on the “random” board. The image stated that Cicada 3301 was looking for “intelligent” people. In fact, the word used was “recruiting.” For what, no one has said and Cicada 3301 is certainly keeping it a secret. The image hid a clue that takes some level of specialized knowledge to find. It involved a Caesar cipher, which is pretty standard crypto, but the rest took at least some technical knowledge.

From there, and in future puzzles, clues varied greatly in skill-set necessary for solving and even location. Some of the clues were in physical locations, making it necessary for people who could not reach the clues to use posts on the Internet to get further in the hunt. Moreover, some of the references in the clues are pop culture, literature and other non-tech topics.

Some have touted the Cicada 3301 puzzles as unsolvable. This is not true. Several have solved the hunt and have allegedly received emails from the organization. Still, no one has come forward and stated what they were recruited to do, if anything. Judging by the puzzles, it is possible that Cicada 3301 is simply a cyber group like Anonymous. Of course, there is also the possibility that it is really MI6, the CIA or a similar organization. Information security, cryptography and a number of other skills necessary to crack the puzzles offered by Cicada would be helpful to virtually any large organization, which makes it hard to discover who is behind it. In fact, this would not be the first time an organization used such tactics to recruit new members. We have to assume that Cicada succeeded on that front, as it stated that it found the people it needed after the first puzzle. It began all over again on January 5, 2013 and again on January 5, 2014, so it must be an ongoing recruitment effort.

At this juncture, it is impossible to tell when these recruitment efforts from Cicada 3301 will stop, but that is not stopping people from looking forward to the next year’s puzzles. It may not be a very public honor, but it must be satisfying to know you have reached the end of one of the most famous puzzles in Internet history.


A Fascinating Calculation

Dividing by 7 yields this fascinating play of the numbers 1,4,2,8,5 and7. I am not going to spoil the fun by letting you on the pattern that emerges and other captivating properties that you might discover along the way.

Although feel free to write to us if you found anything that marveled you and we would definitely share it with the world.

Who knew division could be this much fun. am i right?

Have fun!

A Monk's Steps

A monk has a very specific ritual for climbing up the steps to the temple. First he climbs up to the middle step and meditates for 1 minute. Then he climbs up 8 steps and faces east until he hears a bird singing. Then he walks down 12 steps and picks up a pebble. He takes one step up and tosses the pebble over his left shoulder. Now, he walks up the remaining steps three at a time which only takes him 9 paces. How many steps are there?

See the solution to this puzzle here!

Example of a “meandering time”: the digits can be passed through entirely by a single winding and unbranching path, with small horizontal lines in the gaps. How many such times does a 00:00-to-23:59 clock have?

GUYS!!!! just wanted everybody to know that ya boi, birdo, u kno, the kid who consistently passes math by like 5 points just got the HIGHEST! FUCKING SCORE!! ON HER MATH LONG TEST!!!!