I’ve always enjoyed brain teasers, puzzles and paradoxes. What at first may seem pretty straight forward, can trick us into believing the illogical and outright insane. There are many reasons for this.
Here is a series of puzzles, where the answer may not always be as obvious as it seems.
Using your logic, be your own pedagog.
Click on each picture to reveal the answer, but only do so when you’re ready!
Take your time. Some of these are very difficult!
Video on this puzzle above. This is a difficult one.
This next one is even more difficult. People will come up with all kinds of different answers, most of which are wrong.
This video includes the puzzle above
How many circles do you see in this next picture?
How many times does the letter f occur in the sentence above?
Here is the solution.
The Monty Hall Problem
Suppose you’re on a game show and you’re given the choice to open one of three doors. Behind one door is a ferrari, behind the others doors are goats. You pick a door, let’s say you pick door number 1, and the host -who knows where the ferrari is- opens door number 3, which has a goat behind it.
The host of the show then says to you, “So, do you want to switch to the remaining door, or stick with your original answer?”
Is it to your advantage to switch to door # 2, or does it not matter? This is the Monty Hall Paradox.
Most people would agree that at this point it’s a 50/50 shot….. But what should you really do??
Most people are convinced that after the host opens one of the doors, that it is a 50/50 chance. But that is incorrect. The host of the show knows where the car is, and he’s always going to open a door with a goat behind it. This changes the problem completely. The math nerds call this ‘conditional probability.‘
Another way to look at it is this. In the beginning , you have a 33% chance of winning the car. So if you pick a door and stick with it, how could you odds increase to 50%? They can’t. If you stay with door number 1 you have a 33% chance of winning. And if you switch to door two, you are basically getting a 2 for 1, since the host of the show is always going to open the door with a goat behind it.
Most people don’t want to jinx themselves so they stay with their original decision. Game show host Monty Hall understood this, and probably saved a bunch of money he’d otherwise have to spend on expensive cars, but overall I guess your decision ultimately comes down to if you want to drive home a new car or have a goat’s lifetime of raw cheese and milk.
The Pareto Principle
This is a weird phenomenon that happens whether you like it or not. I’m looking at you, you 1% pharisees. What I mean by that is there are always going to be people who criticize those whom are successful. Think about it. There will, and always should be a certain portion of the population that are more successful than others.
The Occupy Wall Street folks in recent years have coined these people the One Percent, and let me tell you, they are just evil, so freaking evil! (insert sarcasm) They are so evil because they work harder than others in a world where everyone is ostensibly equal in every regard. The only problem, people are not innately equal in every ability. Some are better at this, and others, at that. It’s ridiculous to believe in equality as the liberal media implores us to do so.
This principle proves inequality. For example, at any warehouse, 80% of the workers will be lazy and only accomplish 20% of the work. Conversely, a small fraction of the workers, 20%, will do about 80% of the work. At a used car lot, 20% of the salesmen sell 80% of the cars. Etc..
80% of NBA players will score only 20% of the points, and vice versa.
The Pareto principle is a principle, named after economist Vilfredo Pareto, that specifies an unequal relationship between inputs and outputs. The principle states that 20% of the invested input is responsible for 80% of the results obtained. Put another way, 80% of consequences stem from 20% of the causes. -Investopedia
This can apply to anything really. There are multiple youtube videos that can explain this better than I, but suppose you have a blog called joedubs.com. Only 20% of your subscribers are going to comment on your articles.
Here is another way to look at it; of those 20% of people who actually do comment on the blog, Joe is only going to respond to about 4% (20% of 20%) of the comments.
Jordan Peterson outlines this principle in regards to human productivity. 20% of the population does 80% of the work.
Zipf’s law was popularized by George Zipf, a linguist at Harvard University. It is a discrete form of the continuous Pareto distribution from which we get the Pareto Principle.
According to wikipedia Zipf’s law (/ˈzɪf/) is an empirical law formulated using mathematical statistics blah blah blah, blah, bla ,blah blah bla… Let me just give it to you straight instead of trying to impress you, as every single wikipedia entry seemingly attempts to do. I hate wikipedia sometimes. Just ignore it, and go to the next few entries of google.
For some reason, the amount of times a word is used is just proportional to one over its rank. Word frequency and ranking on a log graph follows a nice straight line. A power-law. This phenomenon is called Zipf’s Law and it doesn’t only apply to English. It also applies to other languages, like, well, all of them. Even ancient languages we haven’t been able to translate yet. And here’s the thing. We have no idea why.
Ok, now even that is not clear. Let me break it down in quantum mechanics using the zygote differendim for π squared. I’m just kidding; that is a bunch of bullshit too.
Benford’s Law, which is also known as the Newcomb-Benford Law, describes the frequency distribution of the leading digits of numbers within sets of numbers. It states that the number “1” appears as the first digit, or leading significant digit, around 30% of the time, the number “2” appears as the first digit 17.6% of the time, down to the number “9” which appears as the first digit less than 5% of the time.
So, to what data sets might Benford’s Law apply? It applies to street addresses, Fibonacci numbers, electricity bills, stock prices, factorials, home prices, population numbers, death rates, powers of 2, the lengths of rivers, and both physical and mathematical constants.
below: Benfords’s law by Numberphile on youtube
Sometimes we are so sure of ourselves; our hubris and ego kicks in full swing. We feel like we can’t possibly be wrong because we’ve thought of everything.
But sometimes, I am wrong. I definitely did not answer these all correctly at first glance. I’ve been wrong before. I bet you have as well.
These conundrums are pedagogic puzzles to understand our own psyche. We’re very seldom correct all the time, or even most of the time. Do the math. The more aware of how unaware we are, the more understanding we have to gain.
“I know that all I know is that I do not know anything” -Platonically attributed to Socrates