recreational math

Swap the numbers

Hi,
I am reading a book from Matt Parker now. It is called “Things to make and do in the fourth dimension” and the ~200 pages that I read are quiet amazing. The author is also YouTuber and it seems that he mostly does “Standupmaths” which is cool channel. I got inspired a bit and created this game that I started to call “Swap the numbers”.

I was thinking about battery on my phone and how it is going down and that it would be interesting, if the first and second number swapped with the first after subtraction of 1,2 or 3 or more percent at a time. I wrote down bunch sequences, beginning with 100 and going down by one digit numbers.

It is not finished since I want to find a way to predict how these sequences form and I have not figured it out yet. I will give an example and then show why this game is so peculiar.

Let’s say that we subtract the number four, that is the one that I started with:

100 (subtract four) 96 (swap both digits) 69 (subtract four) 65 (and so on…) 56 52 25 21 12 8 80 76 67 63 36 32 23 19 91 87 78 74 47 43 34 30 3 -1 10 6 60 56 65 61 16 12 21 17 71 67 76 72 27 23 32 28 82 78 87 83 38 34 43 39 93 89 98 94 49 45 54 50 5 1 10 6 60 56

If you quickly go through these numbers you will find out that they repeat. When the “10” appears for the second time it starts to repeat. (I also forgot to say that if there is negative number it will act as positive on the “swap” step.) For some reason, many of these “constants” that I start with, end in lapses of “tens” meaning that after “-1” there is “10” and then that is the cycle until new “10” appears. First few numbers have the length of the cycle or lapse “36” or “12” and so far, there seems to be only “1 and 10” as constants that will pull it down to zero. (Also 100 but that is trivial and I have not checked some that could be obvious.)

I have made a program in Delphi 7 to write for me all the numbers for any given constant, that is useful but I will still have to consider the mechanism itself to start to understand it.

Dragallur

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I have 420 likes on FB and it is completely useless

Hi,
so today I reached 420 likes on my Facebook page. Page that connects directly to this blog. That is almost two times as many followers on the blog itself, after more than two years that is certainly not much but I am surprised how I got those likes on Facebook and how pointless they are anyway.

420

It says that I will soon reach 500 and that I should give them money so that I reach it sooner.

Since I do not post very often here, I am not preparing the posts very long time (like today), I do not get many more followers on WordPress. Most of the views come from some searches on Google and somehow people tend to be very interested in this post because it has about 7 or 8 times as much views as any other one, exactly 1,066 as of 17.7.2017. I guess most schools in USA learn about HClO4 and HClO in winter and that is why I have most views in that time of year even though it was such a long time since I wrote the post.
So, I have got 420 likes and it is completely useless. Why? Because when I share something on the page most people do not even see it on their “walls” (yes, I have an option to “market” my post for money so that more people see it which is completely absurd). Anyway 420 is a number that has something to do with cannabis and it is 42 times 10 which is nice.
420 is these numbers multiplied together: 2;2;3;5;7. Those can be accidentally also written like this:
a_0=2
a_n=a_(n-1)+ceil((ceil(√(n-1.5))^2)/3)
That took me at least half an hour to come up with. The function “ceil” takes the number and rounds it up. I hope it creates the numbers correctly. Well that reminds me that you can create a game from this. I am going to do that in school. Just pick few numbers and then try to write either sequence or equation that fits it. That is actually quite endless game with lot of possibilities and variations. Maybe one could say that they will use only exponents for example though I think that it is much easier with all those other fancy functions that make the numbers “nice” like rounding up.
Well I might try another number another day.
Dragallur

How to! 7) Count to 1,023 on your fingers

Hi,
as promised, weekend post is here! Ok, so I learned this cool thing when I was on seminar in Hamburg. First I thought that someone is pointing middle finger at me for fun (you will see the reason soon) but it was actually counting method. Though you have only 10 fingers you can use them to produce all numbers from zero to 1,023 which is cool.

If you pay close attention and you know something about computers you know that the number 1,023 is very special. It is 1,024 but one smaller (1,024 is actually the special number).

The thing is that 2^10 is 1,024. And in computers you work in binary system with only 0 or 1  ….    on or off and you get the number of combinations that you can arrange binary system if you put 2 to the power of digits you have. On fingers you can not arrange 1,024, you will see why[1]:

Method

Turn you palms towards you. Since in Europe we write from left we will start with left thumb (palms still towards you). Now make fists.. that is number 0

Rise your thumb, that is 1. (1000000000)
Put only your index finger up, that is 2. (010000000)Put your thumb and index finger up, that is 3. (1100000000)  –> the number of digits shows the number of fingers you have.

So basically if you have number lets say 17. You want to transef it into binary. You will do this by subtracting the highest 2^x power which is equal or less to the number itself.

The 2^x numbers go like this: 1,2,4,8,16,32,64,128,256,512,1024…

In the case of 17 you will subtract 16 which is 5th number in the row. 5th finger on your palm is your left pinkie so you will put it up. Then you are left with 1 which you again subtract by the highest 2^x number which is equal or less and this time it is 1. 1 is first number and left thumb on your hands.

If you want 349 you have to subtract 256 (9th number – right index finger)
You are left with 93-64 (7th number – right ring finger)27-16 (5th number – left pinkie)
9-8 (4th number – left ring finger)
1-1 (1st number – left thumb)

Now train a little bit and impress your friends 😉 The key is to remember the first nine numbers and at what positions they are. Also you can show others just counting one by one. Just do not turn your hands towards them, numbers 4,5,128,640 and other could be dangerous 😀

Dragallur

[1]The reason why you can not count to 1,024 is of course that you are starting with 0 not 1 so your right thumb stands for 2^9 and not 2^10. You can produce another binary digit with your tongue 😉

How to! 5) Multiply two digit numbers in your head

Hi,
today I was sitting outside, chilling out and just as that I started to multiply numbers always adding one: 1*1=1 2*2=4 3*3=9 and so on. And I wondered, how far can this go without paper or calculator?

Well I finished with 100*100 without leaving single number and after a while I started to use this kind of fast and intuitive technique which in this post will be used only to multiply two same numbers smaller than 100.

Lets say you need to multiply 47*47.   (40+7)*(40+7)

Fourty will be called “base number”. The first thing you need to do is to multiply base numbers to get your base. This is simple because 40*40 is the same as 4*4 plus two zeros which means 1600.

Base is quite easy to remember so you do not need to concentrate on that much. Now we need to combine base numbers with, lets call them “pariah numbers”.

In our case pariah number is 7. You multiply it with base number to get 40*7=280 or simply 4*7=28 (+ zero from base number). Because we have two same brackets we need to multiply this twice and add it which is the same as multiplying your result by 2.

280*2=560

Now just add this “double number” to your base which should be quite easy:

1600+560=2160

Repeat this one few times and then add the product of two pariah numbers to get the result:

2160+(7*7)=2160+49=2209

Try this few times, and if you are sure in this simple case try any two digit numbers. It is quite nice way to warm up your brain 😉

Dragallur

Crazy large numbers

Hi!
Two or three days ago I learned something about large numbers.
I will talk about numbers small, large and extremely huge numbers.

First of all, small numbers are those which we use in normal live. Every day you use them in your math class or if you want to calculate how much money you spent.
For those numbers you dont need any special way to write them they are quite easy.
One Ten Thousand Million Billion Trillion Quadrillion
That is why in which some states write numbers, it is called short scale because in czech we have:
One Ten Thousand Million Milliard Billion Billiard…. (thats translated)
That is called long scale because there are those “illiards”
http://en.wikipedia.org/wiki/Names_of_large_numbers
Here you can find list of numbers and their names.

Large numbers start to create some problems. If you clicked on the link I posted you probably found that after quintillions you are lost and you dont know how the hell you should remember that.
There is system to write numbers like Unvigintillion. That is ten and sixty six zeros. You probably know this because it is used pretty often: 10^66. I wont explain this for people who dont undestand it because it would be even more boring post than it is now.

At one point this is too small and even if you start to create “towers” of exponents it will look like this: 10^10^10^651682138 which is pretty nasty.
(Btw. e+x means that there is some number of numbers after that number, for example: 153,20e+2 = 153,20
25e+16 = 250 000 000 000 000 000)
So what people created are called Knuth´s up-arrows and they look like this: ↑ (alt+24).
So I will do few examples so you know how it works:
2↑2 = 2^2 = 4
4↑3 = 4^3 = 64
5↑2 = 5^2 = 25
Now you dont get it yet but it gets awesome when i add up one arrow: ↑↑
(I will just remind you that when you have more exponents on more exponents you have to go from right)
2↑↑2 = 2^2^2 = 16
4↑↑3 = 4↑4^4^4 = 4↑256 = 1.34e+154
5↑↑2 = 5^5^5 = 2.9802322e+17
So it means that second number tells us how many times first number will be there
It gets totally crazy with third arrow: ↑↑↑
2↑↑↑2 = 2↑↑2^2 = 2↑↑4 = 2↑2^2^2^2 = 2↑65536 = 2^2^2^2^2^2^2…. 65536 times
4↑↑↑3 = 4↑↑4^4^4 = 4↑↑1.34e+154 = well I hope you get that idea because now it gets like so crazy that I wont continue but if you want to see some other examples go here: http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

Well thats about all hope you get it, if not then ask me below
Dragallur