Ten people take part in a challenge. Each is given a hat, either black or white. Everyone can see the other hats, but not their own.
They speak one at a time. On your turn, you must say black or white. One special person may instead say a whole number between \(0\) and \(777\), and then also guess their colour.
Before the hats are placed, the group may agree on a strategy.
What is the largest number of people who can guarantee a correct guess?
What’s the smallest number of weights we need to weigh any number of grams from \(1\) to \(100\) in a balance scale, if your weights can be placed in any of the two plates of the scale?
Recall that when we write \(n!\) for some natural number \(n\), we mean \(1\times 2\times 3\times \cdots \times n\). You are given that \(20!=243290a0081766bc000,\) for some digits \(a,b,c\). Find those digits. You may want to recall the divisibility rule for \(9\): a number is divisible by \(9\) if and only if the sum of its digits is divisible by \(9\).
On a TV screen, each minute that passes, there is a number that flashes on the screen: first \(5\), then \(55\), then \(555\), and so on. Will any of these numbers be ever divisible by \(495\)? If so, which is the smallest?