Problems

Look back at Problem 3. You will now prove that actually we can do better! Show that given \(5\) distinct whole numbers - not necessarily consecutive - we can also find three of them such that their sum is divisible by \(3\).

On the questioner’s planet, every alien is either a Crick or a Goop. A Crick can only ask questions whose answer is “yes,” while a Goop can only ask questions whose answer is “no.”

An alien stands on each cell of a \(4\times 4\) chessboard. Every alien asks the same question:

“Do I have an equal number of Cricks and Goops among my neighbours?”

(Here, a neighbour means any alien on a horizontally or vertically adjacent cell.)

How many Cricks and how many Goops could be on the chessboard?

Downtown MathHattan has a grid pattern, with \(4\) streets going east-west and \(6\) streets south-north. You take a taxi from School (A) to cinema (point B), but you would like to stop by an ice cream shop first. In how many ways can a taxi get you there if you don’t want to take a route that is longer than necessary?

Gabby the Gnome has \(3\) cloaks of different colours: blue, green, and brown. He also has \(5\) different hats: \(3\) yellow and \(2\) red. Finally, he owns \(6\) different pairs of shoes: \(2\) yellow, and \(4\) red. Gabby is selecting an outfit: a cloak, a hat, and a pair of shoes. In how many ways can he do it if he wants the colour of his shoes to match the colour of the hat?

You have an \(n\times m\) chocolate bar. You break the bar into two pieces along a line between its squares, then your friend and you take turns (your friend starts) choosing one of the pieces and breaking it again along a line between its squares. The player who cannot make a move loses. For which values of \(n\) and \(m\) do you win?