Robinson Crusoe and Friday are playing cards. Friday takes \(9\) cards numbered \(1\) to \(9\) and shuffles them. Then he lays them out in a row, making a \(9\)-digit number. Robinson notices something surprising: this number is divisible by \(9\). Was this a coincidence, or will it always be divisible by \(9\)?
At Willy Wonka’s chocolate factory, sweets are always packed into boxes of \(3\).
One day, Charlie, Veruca and Augustus each bring a bag of sweets to the factory. Charlie has some number of sweets, Veruca has one more sweet than Charlie, and Augustus has one more than Veruca.
When they put all their sweets together, will they be able to pack them perfectly into boxes of \(3\), with no sweets left over?
After some playing with the \(3\times 3\) board, Sam guessed that there were \(900\) different light patterns that could be obtained by playing on this board. Was he right?