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The best solution to the three 3s puzzle

Hi,
today I am going to write about “my own” work. Specifically the solution to the three 3s puzzle which I encountered in physics camp (where as a morning routine we solved some problems) and nobody (I guess) knew solution back then.

It goes like this: Using plus, minus, division, multiplication, square roots, exponents, brackets and factorials and three 3s, get the number 1 through 10. All except 10 are quite simple, the solutions go like this (there are more options):

(3+3)/3!=1
(3+3)/3=2
(3/3)*3=3
….

And so on. It is important to remember that while you can use “the square root” you can not use “cubic root” without using 3. Also you can not write 1/3 because you would use one which is not allowed. To explain how my solution works you have to understand the difference between double factorial and factorial used two times. Factorial is defined as follows:

n!=n*(n-1)*(n-2)*…*2*1

If you use the factorial again like this: (n!)! you will get the factorial of the new number, it will grow quickly:

3!=6
(3!)!=6!=720

On the other hand:

n!!=n*(n-2)*(n-4)*4*2 (depends if odd or even)
n!!!=n*(n-3)*(n-6)*4*1 (the last numbers depend on the first again)

This way the factorial is defined is extremely useful, because then you can use it to multiply your number with almost any lower number. This is where my expression comes from:

(((3!)!!)!^43)/((3!)!!!!+(3!)!!!!)=10

Now I need (3!)!!=48 to be multiplied by 5, so I added 43 factorials.

If you search for this riddle on the internet, you will get often solutions which need to use other functions, while some of them are nice: log√√3(3) + 3! = 10. I still find my solution to be the best. Most of those you will find use something obscure like flooring or rounding of numbers which makes it extremely easy. My method only uses the fact that the factorial is defined like that when you stack it on itself, I don’t think that without this property, it is possible to solve the problem.

Dragallur

Blue eyes: logic puzzle

Hi,
about one or two weeks back, while scrolling through XKCD comics I have bumped into Blue Eyes logic puzzle, that Randall (the author of XKCD) heard in Boston from some random guy named Joel. Randall titles the puzzle as the “hardest logic puzzle”. It was not the case for me so I would title it as “most interesting (and hard) logic puzzle”. Here is why:


I wont rewrite the setup for the puzzle, you can just read the original here.

So now you know what this is about… if you want to solve it yourself (it might take some time) stop reading.

Why I find this puzzle so interesting is that from the first view it does not make any sense, it shouldn’t be solvable. What the sear (the one with green eyes) says shouldn’t make any difference right? Because everybody knows that there is other person with blue eyes, they can see it, yet when everybody hears it, it makes a difference and in the induction that proves the solution, this information is totally crucial.

If you start with less people like one, then two and so on, it becomes way more obvious that what the sear says actually does make a difference.

It took me together about 2 hours to solve the puzzle and I think it is perfect example of something quite simple working in non-intuitive ways and how our brains still have hard time grasping it.

Dragallur

First experience with hammock

Hi,

I want to share my experience with sleeping in a hammock for the first time.


So I got hammock for my birthday among other things. After finding some cords to tie it to trees I realized that it will be best to try sleeping in it. Since it is almost November and I happen to live in northern hemisphere, it can get close to 0 degrees (no, not Fahrenheit you dummy). Luckily there seems to be subreddit for anything, including hammocks (or also Flying Whales), one can get some tips there quite easily.

So the pluses after 2.5 nights where one was rainy are: it is extremely comfy, no need to watch out for boughs or stones on the ground, you just need trees. Hammock is very easy to pack and unpack saving ton of space. On the other hand your butt will get cold, I am considering under-quilt which is basically sleeping bag that wraps under the hammock. So far I found it also difficult to find the right trees to be also able to set up the roof but it might just take a little bit more experience to have the right estimate for the separation of the trees.

Dragallur

 

Co\m/eback

Hi,
nobody expected it and yet it happened, nobody has foreseen it and yet it was inevitable, nobody cares and yet I am here once again!


After 5 months, three seasons of Rick and Morty, 2 weeks in CERN and bunch of other stuff, I have returned to strike back with an epic post full of me talking about how I came back while wishing I will write another post after this one.

Now I feel a bit stupid, I have this work that I have to do this semester about GPS. It turned out to be quite complicated and one of the import theoretical facets of my work was to find out how GPS chips differ in various phones. I was googling and googling the phrase “gps chip in mobile” with the false assumption that there is a special chip like that. While gps chips do exist, they are a bit different thing than what I had in mind (costing lets say 150$ or whatever, ready to be implemented to something you build by yourself).

Apparently processors in phones take care of that, like the Snapdragon series, you can find all of the info here.

Everything on one page and I am complete… my final work is not though. It will still require huge time investment and sleepless nights!

Dragallur

What does “x” and “y” even mean?

Hi,
so after quite a break, I am returning with math post. It is going to be simple analytical geometry or whatever this field is called.


After teaching some people about lines, circles, circles in a hydraulic press (read: ellipse) also vectors and all sorts of shapes that you can think of[1] I bumped into a fundamental issue that my “students” missed out something important even after using it (without knowing what it really was) for a long time.

First, let’s talk points. A point on a number line is quite straightforward. Why? Because you do not need to have a lot of information to describe it. Image result for number line

Any point on the number line can be represented using only one number, one value. If you wanted to represent a line on the number line you would just need to set boundaries for the highest and lowest number, something like 3<x<5 [2]. This line would have the length of 2 and you would find it by just drawing all the numbers like 3,3.1,3.2,3.5,4,5 and all the others in between, those are all the possibilities for “x”.

Image result for grid line

Number lines start to get boring on this level so we have to step up on a grid. Here you need more information. As you can see on the map. Your city/town has two coordinates, latitude and longitude which is just a fancy name for the commonly used “x” and “y”. Here the grid is limited for x which needs to be between +180 and -180 and y between +90 and -90. With these two coordinates, you can locate any point on the Earth. The similar thing is with the commonly used grid but there, “x” and “y” do not have any boundaries[3]. But how do we write a line on the grid? Well, we could just ignore the fact that it is a grid and think about it as a number line. We just need to set “x” or “y” to some number and let it be fixed and unchanging while the other one changes and describes our infinite line (without boundaries) or just a segment (with boundaries). If “y” is constant that means that we can move only left and right and the line will be leveled. If “x” is constant we can only move and find the points on the line by moving up and down and the line will be vertical.

I will continue probably next time with lines that go different directions.

Dragallur

[1] Don’t think of too random shapes, they would be too difficult, you still have to think about “nice” shapes.

[2] Any math is really difficult to write in WordPress so what I actually meant was 3<=x<=5 which means that numbers 3 and 5 are also a possibility.

[3] Of course, Earth is a sphere and that is different geometry but it serves as a good example.

Is consciousness an illusion?

Hi,
so apparently we have a problem. Noone knows where consciousness comes from. This is quite a big question since that is what we are, right (?) the mind.


I find this idea of consciousness very interesting, we cannot share what being mindful feels like or see that another creature is aware. We have to take it for granted from what others tell, there is no way you can tell that your friend is not just a machine, that is what I mean.

As the title suggests, I will be asking myself if consciousness is just an illusion. This might seem a bit outrageous which makes sense and I am totally open to it being wrong (which it very easily might be, see disclaimer). I think that awareness will raise as an illusion if there is enough complexity added to the system. By the system I mean of course what we perceive through our senses but also the thoughts. The thoughts might actually be the most important part, making our personality.

I am suggesting that with language and thoughts we create a fake personality, the ego which then creates a sense of existence. Through the sheer amount of information that the body receives and reacts to, arises this illusion.

Have you ever tried to notice how thoughts appear? They do so without somebody telling them anything or controlling them [1], this is why there is no one behind them but a shadow. The thoughts tell themselves that they are real but they are just a reaction to the surroundings. That is why there is no special part of the brain that cares for consciousness, it is just an emergent property of huge complexity and in the end, we truly are only robots.

Dragallur

Disclaimer: I’m not neurologist or expert on this topic, those are just my thoughts on this problem.

[1] If you think there is, think again, because those are just other thoughts.

Three year anniversary

Hi,
apparently, the three year anniversary was already yesterday though I thought that my first post became public on 21.3.2015, I guess I was living in a lie.


What now? I have been very lazy in the past weeks, not posting a lot. I am still planning to continue to write the blog, I have no doubt, definitely.

In the three years I:

Received 2 510 likes
Wrote 126 430 words in a total of 318 posts. (Harry Potter and Prisoner of Azkaban has 107 253 words.)
From my readers, I got 1 250 comments.

This is a small blog, it always was and will be in the future, I am not even really reading anybody else’s posts or trying to get more views. I just let go and write a post about once a week if I feel like it. Whoever reads it, cool, I hope that the posts help or you find them interesting at least sometimes, every like or extra view makes me happy and you are helping with that.

Dragallur

Mnemonics

Hi,
today I will write a short post about the power of mnemonics.


Those are things that help you remember something. One that I used quite a long time ago was Saturn mnemonic. I was not sure if it was Jupiter or Saturn with rings but then I realized that the letter “S” is basically just circle disconnected and changed into the shape with a single movement. One could argue that this does help because “J” could work similarly and it is not foolproof but the beauty in this is that it does not need to make sense, it is enough if it helps you remember something.

Another example would be how in 8th grade at the end of the year we had to remember (in some game) two groups of words, the key to this is for me was visual mnemonic, I simply stacked the objects on top of one another. It went like this: ruler, place, hill, plant sock and the second group was a chandelier, scissors, and map (I do not remember the next two words). The first group was extremely simple because the hill would be kind of roundish in my head, so it would nicely fit on the plate, the plant would grow on top but would be hidden by a sock, does not make sense but did its job.

From my experience, it is best to be able to create your own mnemonics connected with your life. Often they are total nonsense, some kind of absurd story. For example, this is how I remembered what in your skin is responsible for that feeling:

Pacinian corpuscules – Pa like Pascal (not my invention but Pacific as a pressure in the deep ocean would work too)
Kraus corpuscules – From Vampire Diaries, Kraus was quite a cold person (corpuscules responsible for cold)
Ruffini corpuscules – ff like fortissimo in music, fast means hot so corpuscules for heat

Sadly enough I forgot to make one for Meissner corpuscules so I didn’t do that in a test.

Now you should have the idea, I recommend the following tests to try this out or just to test your memory: https://www.humanbenchmark.com/tests/number-memory

Dragallur