Hey, it’s April Fool’s Day! Let’s mess with our brains a little by tripping them up with some wicked paradoxes.

The great thinkers of history have come across a few puzzling ideas that looked good on paper but just didn’t jive with common sense. These contradictory or ambiguous ideas are known as paradoxes. This video quickly highlights some of the most well-known paradoxes. There are many others.

For more in-depth explanations of paradoxical fun check out Jim Al Khalili’s book, Paradox: The Nine Greatest Enigmas in Physics. This book includes a brilliant explanation of the dreaded Monty Hall paradox.

Here is a fun introduction to Monty Hall presented by mathematician, Marcus du Sautoy:

Back in 1981, the big toy of the year was the Rubik’s cube. We –the kids of olden times of yore– went bananas over this thing. We fiddled with them constantly and everywhere to the dismay of our parents, teachers, and that poor little old lady we accidentally knocked over because we weren’t paying attention to where we were going on the sidewalk. (Sorry again, Mrs. Theibault.)

As you can see from the picture above it was a simple cube made up of what appeared to be 3x3x3 equal cubes with one cube always hidden in the middle. The exterior surfaces of each of the cubes had colored stickers on them. When the cube was fresh out of the box all of the squares on each side of the whole cube matched with different colors on each facet. The three layers of the cube could be turned independently in all directions. Within a few turns and flips of the cube you were able to mix up the blocks of colors until you had shuffled the colors randomly around the cube. Then it was time to solve the puzzle by twisting the cube until all of the colors matched on all sides.

If you haven’t played with a Rubik’s Cube before, give it a try. Beware! It can be a little addicting. Puzzle it over for a few weeks. Remember that if this toy was actually a 3x3x3 cube of cubes there is another imaginary cube in the center that you can’t see and imagine how that is spinning around in there too.

If you are lucky, one of the pieces will fall off and you will get a glimpse of how the mechanism inside makes it work. You’ll probably want to deconstruct and reconstruct the whole thing. If so, take a look at these amazing mods:

Some people can solve the Rubik’s Cube without cheating. I never solved the Rubik’s Cube analytically. I solved it sort-of-by-accident two or three times. At best, I developed a sense that you had to get one layer solved to improve your chances.

If you have struggled with your Rubik’s Cube for a few weeks and it’s starting to pop its parts, I encourage you to cheat and watch one of the solution videos on YouTube. Why should you cheat? Because knowing how to solve it, helps you understand how to think and plan in 3-D.

Check out RuBot. It was programmed to solve the Rubik’s Cube:

(There is newer version of RuBot with a face and cheesy robot noises, but it creeps me out.)

If you enjoy the original 3x3x3 Rubik’s Cube, you will love Jaap’s Puzzle Page. It is a huge site devoted these kinds of spatial puzzle games that will continue to challenge you.

Now, I will blow your mind. A square is a 2-dimensional shape. A cube is a 3-dimensional shape. Imagine, if you can, a cube in four dimensions. This is what is known as a hypercube. Here is a pathetically inadequate two-dimensional animation that gives the impression of what a hypercube is sort of like, but not really:

We have a hard time imagining hypercubes because our brains evolved to live and survive in three dimensions. Fortunately, computers don’t trip over their own brains and can compute geometries in other dimensions for us.

Here is a YouTube video uploaded by drag0nfur of what the programmer calls “A 3D depiction of a 4D rubiks cube being solved by a computer.”

Did you catch the text at the end that said “There are actually 8 3x3x3 cubes, one is hidden in a non-visible dimension. Please don’t ask me why it’s hidden, brains will splode if you do.”

My brain already popped its parts at the mere thought of a Rubik’s Hypercube, but thanks for the warning.

Flatland: A Romance of Many Dimensions by Edwin A. Abbott

Published in 1884, Edwin Abott Abbott’s Flatland is a hilarious romp through a rigidly structured two-dimensional society populated by lines and other geometrical shapes. The storyline is built with a sublime, intuitive exploration of the mathematical concepts of the first through third dimensions. In the end, along with the characters, we are invited to conceptually grasp beyond those first three dimensions.

Carl Sagan gives the best explanation of Flatland in this clip from his classic series, Cosmos.

Written

on 04/01/2013