Problems

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 $\sum _{n=1}^{N}1/n\ge 100$?

Evaluate $a(4,4)$ for the function $a(m,n)$, which is defined for integers $m,n\ge 0$ by $$\begin{array}{rl}a(0,n)& =n+1\text{, if }n\ge 0;\\ a(m,0)& =a(m-1,1)\text{, if }m>0;\\ a(m,n)& =a(m-1,a(m,n-1))\text{, if }m>0\text{, and }n>0.\end{array}$$

Show that a knight’s tour is impossible on a $3\times 3$ grid.

Show that two queens together can attack every square on a $4\times 4$ grid, but one queen on her own cannot do it. This type of problem is called ‘queen’s domination’.

How many queens can you place on a $4\times 4$ grid so that none of them attack each other?

Show an knight’s tour on a $5\times 6$ chessboard. That is, a path where a knight starts at one square, and then visits every square exactly once, making only moves legal to a knight.