Posted in Science & Nature

Folding Paper

Take any piece of paper and fold it in half. Then fold it in half again. Chances are, you will not be able to fold the paper more than seven times. Try it. No matter how thin the piece of paper is, it is extremely difficult to fold a piece of paper in half more than seven times. The reason? Mathematics.

A standard sheet of office paper is less than 0.1mm thick. By folding it in half, the thickness doubles and becomes 0.2mm. Another fold increases it to 0.4mm. Already, the problem can be seen. Folding a paper in half doubles the thickness, meaning every fold increases the thickness exponentially (2ⁿ). By seven folds, the thickness is 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128 times the original thickness. This makes the piece of paper so thick that it is “unfoldable”.

Another limitation is that folding the paper using the traditional method means the area also halves, decreasing exponentially. With a standard piece of paper, the area of the paper is so small after seven folds that it is mechanically impossible to fold it. Furthermore, the distortion caused by the folds is too great for you to apply enough leverage for folding the paper.

Could these limitations be overcome by using a larger piece of paper? Sadly, no matter how large the piece of paper, it is impossible (or at least extremely difficult) to fold a piece of paper over seven times. This has been a mathematical conundrum for ages, until it was solved in 2002 by a high school student named Britney Gallivan. Gallivan demonstrated that using maths, she could fold a piece of paper 12 times. The solution was not simple though. To fold the paper 12 times, she had to use a special, single piece of toilet paper 1200m in length. She calculated that instead of folding in half every other direction (the traditional way), the least volume of paper to get 12 folds would be to fold in the same direction using a very long sheet of paper.

Mathematics, along with science, is what makes something that seems so simple, impossible.

Posted in Science & Nature

Common Side-Blotched Lizard

Rock-paper-scissors is a fun game that is played by people of all ages and nationalities. But there is also a species of lizards that plays this game, albeit in a rather strange way.

Male common side-blotched lizards, also known as Uta stansburiana, have a mating strategy based on the game, where the chances of “winning” is equal and one type has an advantage over another type while being disadvantaged against another type. The males come in three types, differing in the colour of their necks: orange, blue and yellow.

  • Orange-throated males are the strongest but do not like to form a bond with the female (i.e. do not want a relationship). They can easily win over a fight against the blue-throated males to win the female, but yellow-throated males can sneak in and win over the female instead. Orange beats blue but loses against yellow.
  • Blue-throated males are middle-sized but do form strong bonds with females. They lose in a fight against orange-throated males, but can easily defend against yellow-throated males as they are always with their female. Blue beats yellow but loses against orange.
  • Yellow-throated males are smallest but can mimic females, letting them approach females near orange-throated males. They mate with the females while the orange-throated male is distracted, but this strategy does not work with blue-throated males as they have stronger bonds with the females. Yellow beats orange but loses against blue.

Interestingly, although the proportion of the three types average out to be similar over the long run (much like the probability of a person playing a certain hand), in the short term the preferred strategy tends to fluctuate. For example, orange-throated males may strive with their masculine strength for four or five years, but then the trend will slowly switch to yellow-throated males and their mimicking, female-stealing strategy. After another four or five years, blue-throated males will make a comeback as they win over females with their strong bonding.


Posted in Science & Nature


Rock-paper-scissors is a game with a long history. The earliest example of the game is a Chinese game called huoquan, which follows a cyclic rule where the frog eats the slug, the slug dissolves the snake and the snake eats the frog. The reason why rock-paper-scissors has been saved throughout history is because of the uncertainty it contains. Any hand you choose, the chance of winning is the same. Ergo, there is no single best choice and there is no move that will always win. But this is still a game played by people. It is not a game played by emotionless machines, meaning that you can use human psychology, the surfacing of emotion and specific signs and movements to help deduce your opponent’s hand. Mentalist Derren Brown can read tiny flickering of muscles in the opponent and microexpressions to pull off his “undefeatable rock-paper-scissors trick”, but this is near impossible for a normal person to try. However, you can use the following strategies to improve your odds.

  1. Use paper on a beginner: Statistically, people prefer using rock. Males especially have a strong tendency to play rock.
  2. Use scissors on an experienced player: People who know the first trick can be defeated by going one step further.
  3. Use a hand that loses to the hand your opponent played: This uses the psychology of the opponent wanting to mix up hands and wanting to beat the hand you last played (which is the same as theirs as you drew).
  4. Say what you will play and play that hand: In a competitive situation like rock-paper-scissors, people tend not to trust others. Thus, if you say you will play a certain hand, they will think is a trap and not play the hand that defeats that hand. For example, if you said you will play scissors, the opponent will play paper or scissors and you will either win or draw.
  5. Do not give the opponent a chance to think: People have a subconscious tendency to play a hand that beats the hand that they played before. Without time to think, the subconscious takes action meaning that you can predict their move. If you do the same as strategy 3 and play a hand that loses against the opponent’s previous hand, you will win.
  6. Suggest a certain hand: This is a form of hypnosis where you suggest something to the opponent’s subconscious. To use this trick, pretend to go over the rules by saying “rock, paper, scissors” then play a certain hand. The opponent will likely play the hand that the subconscious last saw.
  7. If you keep drawing, use paper: This is the same as strategy 1.

Unfortunately, rock-paper-scissors has an equal probability of a win and a draw, meaning draws are rather common. Thus, a computer engineer called Samuel Kass devised a game where two additional hands are added: rock-paper-scissors-lizard-Spock. Lizard is played by making your hand into the shape of an animal’s head, while Spock is played using the Vulcan Salute from the science fiction show Star Trek, where you make a V-shape with two fingers on each side. The rules are as follows.

Scissors cut paper. Paper covers rock. Rock crushes lizard. Lizard poisons Spock. Spock smashes scissors. Scissors decapitate lizard. Lizard eats paper. Paper disproves Spock. Spock vaporizes rock. Rock crushes scissors.

As each hand has two ways of winning, the odds of winning is 10/25, or 2/5 and the odds of drawing is 5/25, or 1/5. As you can see, you have double the chance of winning compared to drawing, making the game much faster to play than the original game.

Posted in Science & Nature

Dimensions: Flatland

As we live in a three-dimensional world, it is difficult to imagine that there are higher dimensions. To illustrate this, the thought experiment of the hypothetical “Flatland” can be considered. Let us assume that there is a two-dimensional world called Flatland. Here, the concept of depth does not exist. Only forwards, backwards, left and right exist; there is no up and down. Everything that happens here would look like it was drawn on paper.

Now let us interact with Flatworld. If we were to touch Flatworld with our finger, it would be like poking your finger through a newspaper. The inhabitants of Flatworld would see a circle suddenly appear out of nowhere that grows larger and larger. A person would appear as if they were being seen through a CT scanner – in sections. The concept that things can be above or below would sound crazy to a Flatlander, even though to us it appears as a simple concept.

Let us take an ant walking along a piece of paper as an example of a “2D object”. If the ant wishes to go from one edge of the paper to the opposite edge, it must walk along the 2D plane. However, with our 3D powers, we can fold the paper into a cylinder; now the ant can walk to the other point in an instant (across the fold). To another ant on the other side, the ant would look as if it teleported and suddenly appeared out of nowhere.

In another experiment, we make a Mobius strip (a ribbon is twisted once then its two sides are joined) and make an ant walk along it. Although the ant would think that it was walking in a straight line along a two-dimensional surface, it would have walked on both sides of the strip – a three-dimensional concept. If the Mobius strip concept is confusing, think of a garden hose instead: an ant walking along a straight garden hose is walking in: 1D (straight line), 2D (hose is actually a flat surface) and 3D (the ant can walk in a corkscrew pattern along the hose).

If we were to tell that ant that it had just travelled in a higher dimension, that ant would either scoff at us or be genuinely terrified of the experience. To it, we (or the giant pink circle that it sees our finger as) would look like some omnipotent being that can see everything going on in its world and teleport from one place to another. And although the concept of depth would initially intimidate the ant, it would bring the level of the ant’s understanding of the world up one dimension. For if we see what we only know, then how can anyone see anything new? The only way to truly learn and understand new things would be to jump out of the box and see everything from the outside – just like an ant seeing the piece of paper it was on from a higher ground.

Although we may laugh at the foolishness of the Flatlanders (and the ant), to a being of the 4th dimension, we would appear just as stupid and naive. By applying what we learned from the world of Flatland to our three-dimensional world, we can expand our horizon of knowledge and understand what the fourth-dimension is.

(This post is part of a series exploring the concepts of dimensions. Read all of them here: