Problems

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

What’s $2\uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow 2$?

What’s bigger out of $5^6$ and $2^{14}$?

What’s $3\uparrow \uparrow 3$ 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.

What’s bigger out of $99!$ and ${50}^{99}$?

Let $x$ be the sum of digits of ${4444}^{4444}$. Let $y$ be the sum of digits of $x$. What’s the sum of the digits of $y$?

Using the fact that ${\log }_{10}(3)\approx 0.4771$, ${\log }_{10}(5)\approx 0.698$ and ${\log }_{10}(6)\approx 0.778$ all correct to three or four decimal places (check), show that $5\times {10}^{47}<3^{100}<6\times {10}^{47}$. How many digits does $3^{100}$ have, and what’s its first digit?

What is the least $N$ such that $\sum _{n=1}^{N}1/n\ge 100$?