Rainman

Haha, it is actually my birthday today when you posted this

I think my point still holds, only now the conditions are "magically reversed".

Yep, I got it wrong.

Nice analysis! I wish I had thought of that. The method I used is very similar to differentiation. Differentiation also compares f(x) to some nearby value f(x+h), in order to tell whether the function is growing or declining. However, this particular function is only defined for integers, so it's not differentiable unless we first extend it in some way.

How did you get from the last inequality to the answer? Did you assume that Tn-1 = 1000n and Wn-1 = 12? You might be on the right track with your idea, but you got the recursive functions wrong from the start.

Yeah, and a googolplex = 10^10^100 = 10^1000...

12^12^12 should be interpreted as 12^(12^12) rather than (12^12)^12. So log(12^12^12) = log(12^(12^12)) = (12^12)*log(12) ~ 9 600 000 000 000.

OP doesn't require the numbers 1 through 16, just that the numbers are consecutive.

EDIT: assuming all integers must be positive.

Actually, just one up-arrow is the same as exponentiation. 3^3 = 33 = 27 3^^3 = 3^(3^3) = 3^27 = 7625597484987 3^^^3 = 3^^(3^^3) = 3^^(7625597484987) = 3^3^3^3^3...^3, where the number of threes is 7,625,597,484,987. So 12^^2 = 12^12 = 8916100448256, which is much larger than 1000^^1 = 1000. The inequality doesn't hold for n=1. Generally, 1000^^n is 1000^1000^1000^...^1000, where the number of 1000s is n. You solved puzzle 2 2^^^....^2 always equals 4, no matter how many up-arrows you have.

These problems use Knuth's up-arrow notation (http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation), with ^ being the up-arrow. 1. Find the smallest value for n, such that 1000^^n > 12^^(n+1). 2. We define the sequence g, by assigning g(1) = 3^^^^3, and g(n+1) = 3^^^...^3, where the number of up-arrows is g(n). A famously large number is Graham's number = g(64). Which is larger, Graham's number or 2^^^...^2, where the number of up-arrows is Graham's number?

What kind of time is 00:77:34?