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.
Show how five queens can dominate a standard \(8\times8\) chessboard. That is, each square is attacked by some queen.