Harder Four Fours

Number Theory Level 2

Here is a list of the first few four fours:

$0=44-44$
$1=44÷44$
$2=(4÷4)+(4÷4)$
$3=(4+4+4) \div 4$
$\cdots$

Is it possible to get all the numbers from 0-100 using only four fours and any operations? (This includes square roots, decimals, factorials Etc.

No Yes

3 solutions

Jonathan Tse
May 12, 2019

Answer credit: https://www.slideshare.net/sbishop2/four-fours-solutions

There are infinitely many ways to make one and all real numbers can be composed of ones

Rio Schillmoeller - 2 years, 1 month ago

Thanks for updating the problem's wording - I've deleted my report now.

Chris Lewis - 2 years ago

Jason Carrier
May 16, 2019

$sec(arctan(0))=1$

$sec(arctan(1))=\sqrt{2}$

$sec(arctan(\sqrt{2}))=\sqrt{3}$

$sec(arctan(\sqrt{3}))=\sqrt{4}=2$

In this way, given a way to make 0, we can inductively produce any integer, by stringing together $n^2$ instances of $sec(arctan(\dots))$ Since we have 0=44-44, we are done.

Very nice!

Chris Lewis - 2 years ago

A Former Brilliant Member
May 13, 2019

when you mention "any operations", the answer can hardly be "no". Cause you would not be able to prove that a certain number can not be written, using 4 fours. In order for the question to be more meaningful, first you need to have a limited agreed-upon set of operations.

Harder Four Fours

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