MHB Challenge: Create 64 with two 4's

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The challenge is to create the number 64 using two 4's with various mathematical operations. Solutions presented include complex expressions involving logarithms and factorials, such as $-\lg \, \log_{\sqrt{4}} \, \underbrace{\sqrt{\sqrt{...\sqrt{4}}}}_{\text{65 times}}$ and $4^{\left(\log \sqrt{\sqrt{\sqrt{e^{4!}}}}\right)}$. An alternate solution is $4^{\left(\sqrt{\sqrt{\sqrt{4!}}}\right)}$, which simplifies the approach without using logarithms. The term "Perelman-like" is used to describe the logarithmic method, referencing a related problem from Yakov Perelman's work. The discussion highlights the creativity and complexity involved in solving mathematical challenges.
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Using any of \{+,\;-,\;\times,\;\div,\;x^y,\;\sqrt{x},\;x!\}

. . create 64 with two 4's.
 
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Re: Challenge

In base 15:

44
 
Re: Challenge

(Clapping)
 
Re: Challenge

A Perelman-like solution suffices :

$-\lg \, \log_{\sqrt{4}} \, \underbrace{\sqrt{\sqrt{...\sqrt{4}}}}_{\text{65 times}}$

Or even,

$4^\left ({\log \sqrt{\sqrt{\sqrt{e^{4!}}}}} \right )$

But as $\log$ is not desired, a twist around the base of logarithm of the latter should do :

$\sqrt{\sqrt{\sqrt{4^{4!}}}}$
 
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Re: Challenge

Hello, mathbalarka!

That's it! . . . Nice reasoning!

Alternate form: .4^{\left(\sqrt{\sqrt{\sqrt{4!}}}\right)}
 
Re: Challenge

Okay, thanks! The other form didn't click to me, really nice!

Now, in my post, the first form using logarithm is referred to as "Perelman-like" as another related, but twisted problem was given in Yakov Perelman's Mathematics Is Fun. He refers a challenger in a congress of physicist in Odessa. I couldn't find further reference, so just named this after him. (a year ago when I found out this kind of approach is in general very doable for these problems)
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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