Mathematical Black Hole
What is black hole?
Black hole is a place in the universe from which nothing can escape due to infinite gravity
inside it. Anything that goes inside is unable to get out of it.
Same type of black hole exists in mathematics also from where nothing can escape.
There might be some rule which when applied to any number gives a particular number after a
certain number of iterations.
“A mathematical Black Hole is loosely, one element to which other elements are drawn
by some stated process.” Though the no is itself is the star of the show, the real trick is in finding
interesting processes.
Why such mathematical black hole does occur? In general, processes that turn large
inputs into significantly smaller outputs can quickly reduce even an infinite universe of starting
points to a manageable, finite set of cases.
I will now provide some examples that force the given number to be a Black Hole. There
are many such numbers. Some of these are well known by all of you but you have never noticed
that such a hole may exist. Most common example is zero. Let’s see them one by one.
1. Zero (0)
Everyone knows that when any number is multiplied to zero then it produces zero
again. Thus zero can be considered as a mathematical black hole under the rule
multiplication. No any number can escape from zero when applied under this rule.
2. One (1)
In the similar way number 1 can also be considered as a mathematical black hole.
Here rule is that when we take nth power of it, then again 1 is generated. So we are
not escaping out of it.
There is one more way to prove 1 as mathematical black hole.
Take any natural number.
Define [x] as the nearest integer to its square root.
Repeat the same process on the new number formed.
e.g. 123
[125]= 11
[11]= 3
[3]= 2
[2] =1
[1] =1
Hence we can’t escape out of it thereby making 1 as a mathematical black hole.
3. Four (4)
Here is the way to prove 4 as mathematical black hole.
Start with any whole number say 65.
Write it into words. Here, sixty five.
Count total no of characters. In our case, it is 11 (including space).
Repeating the same process, we get:
Eleven
6
Six
3
Three
5
Five
4
Four
If we repeat the same process again with FOUR then we will get four again. That is
we have reached a mathematical black hole from where there is no way to escape
out.
4. 15
In the similar way, the number 15 can also be considered as mathematical black
hole.
Here is the logic to prove it.
Take any natural number greater than 1.
e.g. 84
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Find its all divisors including 1 and number itself.
84= 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Add digits of all number to form a new number.
1+2+3+4+6+7+1+2+1+4+2+1+2+8+4+2+8+4 = 62
Repeat the same process on the new number obtained.
62= 1, 2, 31, 62
1+2+3+1+6+2 = 15
15= 1, 3, 5, 15
1+3+5+1+5 =15
Thus, we got 15 from which we can’t escape. Hence it can also be considered as
mathematical black hole.
5. F(x)=c
We can also define a function f(x)=c to get the simplest mathematical black hole
number.
e.g. f(x)= 10
Now if we put any number in place of x, then we will get 10.
So it may serve as black hole.
6. 123
Here is the most interesting black hole number. I am going to tell the iterative
approach that will prove 123 as black hole.
Start with any natural number like 69534891.
Count the number of even digits, the number of odd digits and total no of digits.
In this case there are 3 evens, 5 odds and total of 8 digits.
Use these three numbers as digits to form a new number, 358.
Repeat the steps with the new number counting even, odds and the total no of digits.
You get 123.
If you perform same set of operations on 123, you get 123 again.
No matter what number you start with, this iterative process always leads to 123.
Therefore 123 can also be considered as mathematical black hole. Once you hit 123,
you can’t escape out of it, just as Black hole of physics doesn’t implies any escape.
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7. 153
The number 153 is more commonly known as Armstrong Number.
The number 153 equals the sum of the cubes of its digits (13 + 33 + 53=153). It is well
known by everyone, but here is an iterative approach that proves it as black hole.
Start with any positive whole no that’s a multiple of 3, one at a time, calculate cube
of each digit.
Add up the cubes to form a new number. Then repeat the process. You’ll eventually
reach 153.
Thus 153 is proved as mathematical black hole.
8. 243
Let us prove 243 as mathematic hole by putting some condition on it.
Take any natural number.
Multiply the digits together.
Stack the number of digits at the end.
Repeat the same process till you get 243.
e.g. 426
4*2*6= 48.
No. of digit=3.
New number formed is:
483
Repeating the same, we get:
963
1623
364
723
423
243
243
243
Thus, 243 can be considered as black hole.
9. 6174 (From Wikipedia)
The number 6174 is called Kaprekar's constant. It may also be proved as mathematical
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black hole using the following property.
i.
ii.
Take any 4 digit number, using at least two different digits. (Leading zeros are allowed).
Arrange the digits in ascending and then in descending order to get two four-digit
numbers, adding leading zeros if necessary.
iii.
iv.
Subtract the smaller number from the bigger number.
Go to step ii.
Eventually, you reach to 6174 in at most 8 iterations.
Exception of this rule is the numbers with recurring digits like 1111, 2222 etc. because on
first iteration only, it subtracts to give 0. So we can’t repeat further.
e.g. 5432 – 2345 = 3087
8730 – 0378 = 8352
8532 – 2358 = 6174
Also,
2111 – 1112 = 0999
9990 – 0999 = 8991 (rather than 999 – 999 = 0)
9981 – 1899 = 8082
8820 – 0288 = 8532
8532 – 2358 = 6174
Please notify me, if you get similar numbers.
In the similar way, we can also black hole in computer terminology. In fact, black hole in
computer is more popularly known as ‘Recursion’. When a function calls itself then it is called
recursion. Sometimes we are trapped in a condition which is never terminated (In programming
languages, this is called infinite loop). We can’t escape from this loop. Programming languages
are only understood by technical users. So we can’t define such black holes for a general user.
But here I am going to define a black hole that is observed in daily life of an end user.
Just follow the following steps:
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1. Open www.google.co.in
2. Type ‘recursion’ in the search field and hit enter.
3. You will see the suggestion given by google in the form of message:
Did you mean recursion?
This suggestion generally comes when wrongly type spelling of any word but here
spelling is correct.
4. Now, again if you click on this suggestion then same page will be opened. In this way,
you have entered into a place from which there is no exit. So, we can say that we have
been trapped by the black hole.
For a technical user or a computer programmer, black hole is nothing but a game of recursion.
For them, I am going to give a few typical black holes.
1. It’s a batch program that makes infinite number of folders inside a folder until your hard disk
gets full.
For writing a batch program, copy this code in notepad and save it as something.bat.
@echo off
:ct
md tips
cd tips
md tricks
cd tricks
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md hacking
cd hacking
goto ct
2. Batch program that opens notepad infinite no. of times.
@echo off
:ct
START %systemroot%\system32\notepad.exe
goto ct
3. Batch program that makes infinite no. of folders on the desktop. It fills your desktop
screen with the unlimited no. of folders.
@echo off
:ct
md %random%
goto ct
by:kamal nayan
kamal.nayan@rocketmail.com
www.tricksntips.in
Kamal.nayan@rocketmail.com
www.tricksntips.in