Problems

On a bus out of the Hattius kingdom, you sit behind two wizards. The first says to the second “I’ve a positive integer number of children, all of whose ages are positive integers. The sum of their ages is the bus number, and their product is my age."

The second wizard replies “If you told me your age and how many children you had, would I be able to work out their individual ages?"

The first wizard says “No, unfortunately not." To which the second says “Aha! Now I know how old you are." What was the bus number?

Let \(n\) be a positive integer. Show that \(3^{2n+4}-4^n\) is always divisible by \(5\).

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.