Originally posted by lemondropUsing the fact that log(x^n) = n log x we have that (I think there is one too many square roots)
logs are permissible I would assume
as long as it is a mathematical symbol, function, etc
very impressive the solution for 1000
I have no idea how to work that out
4/(log(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(4)))))))))/log(4))-4! = 1000
can be written 4 / log(4^(1/256))/log(4) - 4! = 256*4 / (log(4) / log(4)) - 4! = 1024 - 24 = 1000
Well if we're allowed logs then (as per the wikipedia article for this puzzle) we have
-sqrt(4)log(log(sqrt( ... n times ... (sqrt(4) ... ))/log(4))/log(4)
= -2 log(2^(-n)log(4)/log(4))/log(4)
= 2n log(2)/log(4) = 2n * .5 = n 🙂
and so
-sqrt(4)(log(log(sqrt(4))/log(4)))/log(4) = 1
-sqrt(4)(log(log(sqrt(sqrt(4)))/log(4)))/log(4) = 2
-sqrt(4)(log(log(sqrt(sqrt(sqrt(4))))/log(4)))/log(4) = 3
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Yes, we can even make every integer with just 3 fours (or, with a few small modifications, 3 of almost anything, except -1, 0 and 1):
-(log(log(sqrt(4))/log(4))/log(sqrt(4)) =1
-(log(log(sqrt(sqrt(4)))/log(4))/log(sqrt(4)) = 2
-(log(log(sqrt(sqrt(sqrt(4))))/log(4))/log(sqrt(4)) = 3
and so on
The bit in bold changes, it has n nested square roots, where n is the integer required.
It works because:
sqrt(x) = x^(2^-1)
sqrt(sqrt(x)) = x^(2^-2)
sqrt(sqrt(sqrt(x))) = x^(2^-3)
the maths surrounding the nested sqrts is needed to extract the bold number inside the power of x.
the technique doesn't work for 1 or 0 because sqrt(1) = 1 and sqrt(0) = 0, so the nesting of the sqrts has no effect at all.
Originally posted by iamatigerWell 0 factorial gets you a 1, and if we require an integer greater than one to apply the technique you mentioned above, then noting that 0! + 0! gives you 2 then 3 * 2 = 6. Which is the answer I'm going to shoot for.
Can anyone answer the following bonus question:
How many zeros do you need, combined with any other common mathematical symbols and constants, but no other digits, to make any integer?
Originally posted by AgergHow does 0! = 1? I thought factorial for zero would be zero, zero times one should be zero and one times zero should be zero.
Well 0 factorial gets you a 1, and if we require an integer greater than one to apply the technique you mentioned above, then noting that 0! + 0! gives you 2 then 3 * 2 = 6. Which is the answer I'm going to shoot for.