Problems

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.

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 \(\log_{10}(3)\approx0.4771\), \(\log_{10}(5)\approx0.698\) and \(\log_{10}(6)\approx0.778\) all correct to three or four decimal places (check), show that \(5\times10^{47}<3^{100}<6\times10^{47}\). How many digits does \(3^{100}\) have, and what’s its first digit?

What is the least \(N\) such that \(\sum_{n=1}^N1/n\ge100\)?

Evaluate \(a(4,4)\) for the function \(a(m,n)\), which is defined for integers \(m,n\ge0\) by \[\begin{align*} a(0,n)&=n+1\text{, if }n\ge0;\\ a(m,0)&=a(m-1,1)\text{, if }m>0;\\ a(m,n)&=a(m-1,a(m,n-1))\text{, if }m>0\text{, and }n>0. \end{align*}\]

We bet that some of you play chess and are pretty good. Someone may be better than all of the tutors. Unfortunately for that person, and fortunately for the rest of you, that won’t help too much with the problems today. You’ll need to know how the pieces move and that’s it.

There are various themes of knight’s tours, independence and queens’ domination. We also won’t just look at typical \(8\times8\) chessboards, but grids of different sizes, and even ones that aren’t flat.

Show that a knight’s tour is impossible on a \(3\times3\) grid.