OK, here comes some actual math. I'll make Nightingale rue the day she took my name in vain. ;)
The examples in that video are very clever examples of cautionary tales about math.
We are pattern matching creatures (among other things). Whenever a kid (in an English speaking country) wanted to use the past tense of "go" and said "goed" instead of went, the kid had recognized a pattern, and had not yet been trained that the pattern does not apply to go in the past tense, for some inexplicable reason.
Bilateral symmetry is a basic topological feature. We are extremely good at recognizing bilateral symmetry, as are, I assume, all animals. Why? Because nearly all animals have bilateral symmetry (left and right side of animals are mirror images of each other) and hardly any plants do. It is very useful to be able to identify animals instantly. They might be food. Or, perhaps more importantly, they might think you are food, or at least worth kicking in the head, because you might be a threat.
We take bilateral symmetry so much for granted that we are fascinated by the few animals that do not have it, like most starfish.
We are also quite good at facial recognition, a very complicated process. We can read facial nuances - is that animal looking at me? Does it look like it is about to snarl at me or bite/butt me? These are important questions for survival.
So, we are good at spotting patterns, sometimes even when there is no actual pattern there, like tossing someone into a volcano to insure a good harvest, or rubbing properly blessed olive oil on someone's head to cure disease.
People who choose their own lottery numbers rather than take random picks from the computer believe in patterns that don't exist.
Crossword puzzles are pattern matching games. So is Sudoku.. So are jigsaw puzzles. Most games, particularly one-person games are nearly always exercises in pattern matching. A lot of people quite enjoy the mental exercise.
Mathematics is an exercise in looking for patterns. Many of the patterns are pretty abstract, and, just like playing the piano, or speaking a language, require massive amounts of practice and exposure. There is one common pattern matching game in math that is accessible with pretty minimal background - infinite series of integers. Even children can and do play that game.
The basic rules are, given the beginning of an infinite series of integers, what's the next term in the series, and can you come up with a formula to calculate the nth term without having to calculate all of the previous terms in the sequence.
The second question requires slight knowledge of algebra. Coming up with the next term just requires being familiar with the concept of what a number is. Early grade school kids can do that.
So let's look at a few sequences.
So simple, it looks ridiculous:
1, 1, 1, 1, 1, 1, 1, ...
Next item: 1.
Formula for the last term: 1
Sum of the first n terms (another common sequence question): n
Surprisingly, this sequence is implied in a lot of combinatorics problems, and in a sense, it is the foundation of all integers, since the sequence generated from that third question about the sum of the first n terms is.....
1, 2, 3, 4, 5, 6, 7, 8, 9, ...
Next item: 10
Formula for the nth term: n (duh)
Sum of the first n terms: OK, finally a question hard enough that the answer isn't so obvious that it is insulting to even ask the question. That sequence, of the sums of the integers are called triangular numbers because - well, think bowling pins. Bowling pins are arranged in 4 rows and the total number of pins is ten. Ten is the fourth triangular number: 1+2+3+4=10
Other sequences you have probably seen:
1, 1, 2, 3, 5, 8, 13, 21, ...
Fibonacci sequence: start with 1, 1, then add the two most recent numbers to come up with the next number. It's named after the guy who, in the 1200s, popularized arabic numerals in Europe. It is a sequence with so many interesting properties that a math journal (The Fibonacci Quarterly) started publishing in the 1960s, and is still going strong.
1, 4, 9, 16, 25, 36, ...
The squares of integers, which surprisingly can be generated without multiplication. Simply add consecutive odd numbers to the previous squares.
1+3=4+5=9+7=16+9=25+11=36...
Of course if you need the square of 857, you can add up the first 857 odd numbers. That will work, but it is a hell of a lot easier to come up with a simple formula for generating the square of a number and we all know that: multiply 857 times itself. Preferably with a calculator.
OK, after that huge windup, here, finally, comes the pitch.
The fist example in that video is known as Moser's Circle Problem. If you draw 1, then 2, then 3 etc points on a circle, and then connect all the points with all the other points with straight lines, how many simple regions are created in the circle?
The sequence of answers is 1, 2, 4, 8, 16, ...
That looks for all the world like the consecutive powers of 2, a sequence burned into the memory of every computer science major ever.
The next term? Obviously 32.
But when you actually do it, the answer is 31, though you have to count very carefully, and you will probably have to triple-check your work, because you really want the answer to be 32, and there is an overwhelming temptation to assume you just missed counting one.
But the answer is indeed 31. Quite often, the obvious continuation of math sequence is the correct answer, but not always, and that was the point the author of that version of "How They Fool Ya" was making.
The song lists two other examples where what looks like screamingly obvious predictable extensions of a sequence of values turns out eventually to be wrong. Mathematicians delight in finding things like this, for much the same reason we are fascinated by starfish - they don't fit the pattern. That makes them interesting.
Back in the 1970s, a mathematician named Sloane decided it would be useful to compile a list of integer sequences. All the sequences I listed above were in it, and several thousand more.
Of course people came up with lots more sequences, and this was basically pre-internet, so in the 1990s he came out with a second edition, and a co-author. After that, the internet turned out to be a much better way to make this sort of rapidly changing list available, and it is now know as the Online Encyclopedia of Integer Sequences, or OEIS. Anybody who works in anything even faintly related to combinatorics will recognize the OEIS acronym, and a google search will take you to it.
It started out as a few thousand sequences. It is now closing in on 400,000 sequences. I think it is going up somewhere on the order of ten thousand new sequences a year. That should give you some idea of how many people are absolutely fascinated by integer sequences.
Here is the Opening page:
https://oeis.orgThere is a search box, and if you enter the sequence 1,2,4,8,16,31 it will find Moser's Circle Problem, which it lists as the third of five different problems where that sequence occurs. Here's a shorter search string that will get you there.
https://oeis.org/search?q=a000127&language=english&go=SearchThe above link provides a number of ways to calculate the nth term of the sequence without having to draw dots on a circle, connect all the dots and count the regions. One of the ways left me speechless/amused.
To calculate the nth Moser's Circle number, go to the nth row of Pascal's triangle (google that if you're not familiar) and add up the first 5 terms of the nth row.
That is a serious WTH formula. Adding up part of a row is not the sort of thing usually done with Pascal's triangle. Try it. The first five terms of the 6th row add up to 31. The first 5 terms of the next row add up to 57, which is the next number in the sequence..
Here is a 16 minute explanation of why that formula works. It only requires HS math (no calculus, just algebra II), but if it takes 16 minutes to explain, it must be pretty involved (it is).
https://www.youtube.com/watch?v=YtkIWDE36qUHere is the welcome page for OEIS. I recommend you click on the link for the fibonacci sequence and just skim through it. There are a ton of other interesting links. Explore.
https://oeis.org/wiki/WelcomeOh, and if you think the sequence 1.1.1.1.1.1.1... is trivial, look at it's entry:
https://oeis.org/search?q=1%2C1%2C1%2C1%2C1%2C1%2C1&language=english&go=SearchThere. Next time you will think better than to mock math. ;)
Edited 1 time(s). Last edit at 07/02/2023 05:05PM by Brother Of Jerry.