Problems

Let \(s_n\) be the product of the numbers in the \(n^{\text{th}}\) row. (e.g. \(s_3=1\cdot3\cdot3\cdot1=9\)) What’s the limit \[\lim_{n\to\infty}\frac{s_{n-1}s_{n+1}}{s_n^2}?\]

Other than \(1\), does any number appear more than eight times in Pascal’s triangle?

What’s \(3\uparrow\uparrow3\)?

Is \(4^{15}\) more or less than a billion?

Approximately how many footsteps do I take in a year? (estimate to the nearest power of \(10\))

What’s \(2\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow2\)?

What’s bigger out of \(5^6\) and \(2^{14}\)?

What’s \(3\uparrow\uparrow3\) as a power?

Is \(45^{45}\) bigger or smaller than \(10^{80}\)?

How many times have the people in this room blinked in their lives in total? Find an answer to the nearest power of 10.