Challenge: Create 64 with two 4's

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Discussion Overview

The discussion revolves around the challenge of creating the number 64 using two instances of the number 4, employing various mathematical operations such as addition, subtraction, multiplication, division, exponentiation, square roots, and factorials. The focus is on exploring creative and unconventional solutions to this problem.

Discussion Character

  • Exploratory, Mathematical reasoning

Main Points Raised

  • One participant proposes a complex solution involving logarithms and nested square roots, suggesting a "Perelman-like" approach.
  • Another participant offers an alternate form of the proposed solution, indicating a preference for a different representation of the mathematical expression.
  • A later reply provides context about the term "Perelman-like," referencing a related problem from Yakov Perelman's work and its connection to a challenge presented at a physicist congress.

Areas of Agreement / Disagreement

Participants express appreciation for the proposed solutions, but there is no consensus on a single preferred method. Multiple approaches are presented without resolving which is definitively better.

Contextual Notes

The discussion includes references to specific mathematical concepts and forms that may not be universally understood, and the term "Perelman-like" is introduced without extensive explanation of its implications.

soroban
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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 :

$\sqrt{\sqrt{\sqrt{4^{4!}}}}$
 
Last edited:
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)
 

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