# Probability Musing

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Please let me know if you know the solution to the following problem:

What is the probability of me waking up at 10am?

What additional information should be supplied so as to determine the probability? What do you exactly mean by the probability of this event? Which kind of conditional probability will make sense?

Consider the following comment by Timothy Gowers regarding the model for calculating the probability of an event involving a pair of dice:

Rolling a pair of dice (pp. 6), Mathematics: A very short introduction © Timothy Gowers, 2002 [Source]

I find probability very confusing, for example, this old post.

# Finite Sum & Divisibility

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I wish to discuss a small problem from The USSR Olympiad Problem Book (problem 59) about the finite sum of harmonic series. The problem asks us to prove that

$\displaystyle{\sum_{k=2}^{n} \frac{1}{k}}$  can never be an  integer for any value of $n$.

I myself couldn’t think much about how to prove such a statement. So by reading the solution, I realised that how a simple observation about parity leads to this conclusion.

Firstly, observe that among the natural numbers from 2 to $n$ there is exactly one natural number which has the highest power of 2 as its divisor. Now, while summing up the reciprocals of these natural numbers we will get a fraction as the answer. In that fraction, the denominator will be an even number since it’s the least common multiple of all numbers from 2 to $n$. And the numerator will be an odd number since it’s the sum of $(n-2)$ even numbers with one odd number (corresponding to the reciprocal of the number with the highest power of 2 as the factor). Since under no circumstances an even number can completely divide an odd number, denominator can’t be a factor of the numerator. Hence the fraction can’t be reduced to an integer and the sum can never be an integer.

# Solving Logarithmic Equations

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While reading John Derbyshire’s Prime Obsession I came across the following statement (clearly explained on pp. 74):

Any positive power of $\log(x)$ eventually increases more slowly than any positive power of $x$.

It is easy to prove this (existence) analytically, by taking derivative to compare slopes. But algebraically it implies that (for example):

There are either no real solution or two real solutions of the equation
$\log(x) = x^\varepsilon$
for any given $\varepsilon>0$.

Now the question that arises is “How to find this $x$?” I had no idea about how to solve such logarithmic equations, so I took help of Google and discovered this Mathematic.SE post. So, we put $\log(x)=y$ and re-write the equation as:

$y=e^{y\varepsilon}$

Now to be able to use Lambert W function (also called the product logarithm function) we need to re-write the above equation, but I have failed to do so.

But using WolframAlpha I was able to solve $\log(x)=x^2$ to get $x=e^{\frac{-W(-2)}{2}}$ (which is an imaginary number, i.e. no real solution of this equation) but I was not able to figure out the steps involved. So if you have any idea about the general method or the special case of higher exponents, please let me know.

# Cross Diagonal Cover – II

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I will continue the previous post. Now the question to  be answered is that:

What percentage (or how many) of given $m\times n$ grid can be covered by following Cross Diagonal Cover Algorithm?

I did some calculations, for example:

As per my calculations, I have following conjecture:

The following 3 cases are possible:

1. If $m = n$, then $m$ squares will be covered.
2. If $m>n$ and both $m,n$ are odd, then $m$ squares will be covered.
3. If at least one of $m,n$ is even then $\frac{mn}{2}$ squares will be covered.

As you can see that Case – 1 is quite trivial. I tried to prove other two cases for one week, but failed, so decided to post it as conjecture.

I have figured out that $\sqrt{1- \sqrt{1+ \sqrt{1- \sqrt{\ldots } }}}$ is a diverging series. But now I am struck with another problem: