Challenge: Create 64 with two 4's

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In summary, the conversation discusses various solutions to a challenge involving the use of mathematical operations such as addition, subtraction, multiplication, division, exponentiation, square root, and factorial to create the number 64 using only two 4's. One solution involves using logarithms, while another involves a twist on the base of the logarithm. The conversation also mentions a similar problem presented by Yakov Perelman in his book "Mathematics Is Fun."
  • #1
soroban
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Using any of [tex]\{+,\;-,\;\times,\;\div,\;x^y,\;\sqrt{x},\;x!\}[/tex]

. . create 64 with two 4's.
 
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  • #2
Re: Challenge

In base 15:

44
 
  • #3
Re: Challenge

(Clapping)
 
  • #4
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:
  • #5
Re: Challenge

Hello, mathbalarka!

That's it! . . . Nice reasoning!

Alternate form: .[tex]4^{\left(\sqrt{\sqrt{\sqrt{4!}}}\right)}[/tex]
 
  • #6
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)
 

FAQ: Challenge: Create 64 with two 4's

How is it possible to create 64 with only two 4's?

It may seem impossible at first, but with a little bit of creativity and mathematical manipulation, it is possible to create 64 using only two 4's.

What are the rules for this challenge?

The only rule for this challenge is that you can only use two 4's and any mathematical operations to create the number 64.

Can fractions or decimals be used in this challenge?

Yes, fractions and decimals can be used in this challenge as long as they are created using the two 4's and any mathematical operations.

Is there only one solution to this challenge?

No, there are multiple ways to create 64 using only two 4's. Some solutions may be more complex than others, but as long as the end result is 64, it is a valid solution.

How can this challenge be beneficial for learning?

This challenge can help improve problem-solving skills, critical thinking, and mathematical creativity. It also highlights the importance of understanding the properties of numbers and how they can be manipulated to reach a desired result.

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