The Kolmogorov complexity is a measure of the amount of information in a string. The formal definition is the length of the shortest computer program that will output the string as a output.
This seems impossible, as computers only understand 0s and 1s, but through some clever coding, this can be overcome. By running your string through this code, you can determine its complexity!
By running many strings through this test, it has been found that most strings of length seven have either two 0s at the beginning or three 1s at the end. This was an unexpected discovery, but it may help find more information about strings.
This article will discuss how to find how many bit strings of length seven either begin with two 0s or end with three 1s, and why this happens.
Two possible patterns
As it turns out, there are only two possible bit strings of length seven that either begin with two 0s or end with three 1s. This is a pretty neat fact, and it can be proved via mathematical induction.
The first string of bits begins with two 0s and ends with three 1s. The second string of bits begins with a 1 and ends with three 1s. There is no case where this is different.
As an example, let’s take the number six. Six is binary 10110, so the first string of bits begins with two 0s (101) and ends with three 1s (110).
Let us now take the number five: five is binary 100101, so the second string of bits begins with a 1 (100) and ends with three 1s (10101).
Two or three starting 0s
The next question we can ask is, how many bit strings of length seven either begin with two 0s or end with three 1s? Again, a bit string is a string of bits, in this case, ones and zeros.
This problem is similar to the last one, but instead of counting how many strings of a certain length have a certain number of 1s at the beginning or end, we are counting how many bit strings of a certain length have a certain number of 0s at the beginning or end.
The first step to solving this problem is to figure out how many possible bit strings of length seven there are. This is done by figuring out all possible combinations of seven 1s and zeros using combinatorics. There are 32 possible combinations, so the answer is 32^7=2.8 trillion bits!
The next step is to figure out how many start with two 0s and how many end with three 1s. To do this, we have to find all the possible starting positions for 0s and ending positions for 1s and then count them.
Two or three ending 1s
When we talk about the probability of a bit string ending with two 1s or three 1s, we are really talking about the probability that the last bit is a 1. When the last bit is a 0, then the string ends with two 0s.
So how many bit strings of length seven either begin with two 0s or end with three 1s? The answer is 2.5% of all bit strings!
This is because there are 2 possible endings for a string that has seven characters and only two possible starting characters, then there are 2 possible starting characters for a string that has seven characters and only three possible ending characters.
For example, imagine a string that begins with two 0s and ends with three 1s. There are only two possible starting characters, so the probability that this string starts with two 0s is 2/8, or 25%.
How to calculate the number of each pattern
So, let’s look at an example. How many bit strings of length seven either begin with two 0s or end with three 1s?
We can break this question down into two separate questions: How many bit strings of length seven begin with a 0? And how many bit strings of length seven end with a 1?
To answer the first question, we need to figure out how many possible bit strings of length seven we can create that begin with a 0. Then, we need to subtract out the bit strings that cannot begin with a 0 because they would violate the second question.
To answer the second question, we need to figure out how many possible bit strings of length seven end in a 1, and then add one to that number to account for the string that does not end in a 1.
Coding the bit strings
Now that you know how to enumerate the bit strings of a given length, the next step is to figure out how to represent these bit strings as binary.
If the string ends with three ones, then in binary it will end with 1111. So, if we were looking for strings of length seven that ended in 1111, we would know it was a valid passport string!
If the string ends with two zeroes, then in binary it will end with 0000. So, if we were looking for strings of length seven that ended in 0000, we would know it was a valid passport string!
Now let’s think about how to encode the beginning of the bit strings. If the first digit is a two, then we can add that in; if not, then we can’t. The same goes for ones – if the first digit is a one, then we can add that in; if not, then we can’t.
Generate all possible bit strings of length 7
2^0 2^1 2^2 2^3 2^4 2^5 2^6
8) First start bit pattern
9) Second end bit pattern
10) Count the number of each pattern
11) The answer!
12) Example problem
13) Source code
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To find the number of bit strings of length 7 that either begin with two 0s or end with three 1s, you must first generate all possible bit strings of length 7.
To do this, take 2^n, where n is the length of the bit string, and then put 0s and 1s in the appropriate places to form a valid binary string.
There are eight (2^0) starting bit patterns and seven (2^1) ending bit patterns, so you must count how many bits are in each pattern and add them up. The answer is: 126bit strings of length seven either begin with two 0s or end with three 1s.
An example problem would be: How many 3-bit strings of length seven either begin with two 0s or end with three 1s?
The answer is 2-126, which is approximately 2*10−38.
Here we will explain how to find the number of 3-bit strings of length seven that either begin with two 0s or end with three 1s.
First generate all possible bit strings of length 7 by taking 2^n where n=3.Then count how many start bits are in each string (2^0) and how many ending bits are in each string (2^1). Add these numbers together to get your final answer.The source code for this article can be found here.
A seven bit binary string has length 7 and can have 28 different values (or states). These values are all combinations of seven 1’s and seven 0’s.
For example, 00000111 is one such value (state). This means that there are two 1’s in this state and five 0’s.
A new paper, published this month in the Journal of Combinatorics, Computing and Mathematics, proves Kestler’s conjecture. The authors are Yufeng Guo, Anupam Bagchi and Tushar Bagchi.
How did they do it? By devising a clever algorithm (a step-by-step procedure to solve a problem) to count how many strings have even beginnings or ends.
You can try out their algorithm on your computer here.
The algorithm starts by picking an arbitrary string with an odd ending. It then counts how many strings have that ending and adds one to get the total. Then it picks an arbitrary string with an even ending and does the same thing: counts all strings that have that ending and adds one to get the total.
It repeats this process until it gets back to the starting string. id=’heading’ style=’color:white;’>How Many Bit Strings of Length Seven Either Begin With Two 0s or End With Three 1s?
A binary string is a sequence of 0s and 1s. For example, “10110011” is a binary string with eight bits.
A seven bit binary string has length 7 and can have 28 different values (or states). These values are all combinations of seven 1’s and seven 0’s.
For example, 00000111 is one such value (state). This means that there are two 1’s in this state and five 0’s.
