Write $1681$, using $4$, four times only, and you can use any mathematical operation available within mathematics(except catenation or $4.4$ etc, it should be an operation), like factorial and cube root , $\sqrt{}$, greatest integer etc...

Example- writing $131096=4!+(\sqrt4\times {4^{4!!}})$ (yes for this example I first did the R.H.S!!)

I saw this question in a newspaper as a mathematical puzzle many years ago(in high school), and i solved it then, since then $1681$ is my favourite number, as took me $2$ days then to do it, but now i can't remember how i did it. Any help! I remember I had to use an operation which I was not much familiar with then.

  • $\begingroup$ Catenation is an operation. $\endgroup$ – MJD Oct 2 '14 at 22:06
  • $\begingroup$ If you are willing to use catenation, then one possibility is $(4 \operatorname{cat} (4/4))^{\sqrt{4}}$, where $x \operatorname{cat} y$ denotes $x$ followed by $y$ (in base 10). $\endgroup$ – Eric M. Schmidt Oct 2 '14 at 22:18
  • $\begingroup$ nah, i don't think It should be allowed. Other wise things are easier. I read it somewhere all numbers upto $500$ can be written like this, and I am sure almost every number must be, and must be interesting to find the ones which can't, but for that we must fix a domain of operations first.... Thanks though, Nice if catenation is allowed. $\endgroup$ – Bhaskar Vashishth Oct 2 '14 at 22:22
  • $\begingroup$ @EricM.Schmidt may be if you use base $4$ in catenation, we can allow it, but base $10$, it uses $10$ intrinsicly in mathematical formulation of catenation, no? and that is not allowed $\endgroup$ – Bhaskar Vashishth Oct 2 '14 at 22:25

One way to do it is: $$ \frac{(4 + 4)! + 4!}{4!} $$

  • 1
    $\begingroup$ Nice. I am pretty sure, I didn't do it this way. Mine were uglier operations. Thanks. $\endgroup$ – Bhaskar Vashishth Oct 2 '14 at 22:10
  • 2
    $\begingroup$ Another way by the same idea: $\frac{(4!!)!}{4!}+\frac 44$. $\endgroup$ – mathlove Oct 2 '14 at 22:48

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