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 there is a way of placing two queens on the board such that between them they attack every square on a \(4\times4\) grid. But also show that one queen on her own cannot do it. This type of problem is called ‘queen’s domination’.
A set of chess pieces is called independent if none of them can attack each other. How many independent queens can you place on a \(4\times4\) grid?