How many times have the people in this room blinked in their lives in total? Find an answer to the nearest power of 10.
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?
Evaluate \(a(4,4)\) for the function \(a(m,n)\), which is defined for integers \(m,n\ge0\) by \[\begin{align*} a(0,n)&=n+1\text{, if }n\ge0;\\ a(m,0)&=a(m-1,1)\text{, if }m>0;\\ a(m,n)&=a(m-1,a(m,n-1))\text{, if }m>0\text{, and }n>0. \end{align*}\]
Show that a knight’s tour is impossible on a \(3\times3\) grid.
Show that two queens together can attack every square on a \(4\times4\) 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 \(4\times4\) grid so that none of them attack each other?
Show an knight’s tour on a \(5\times6\) 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.