Problems
Adi and Maxim play a game. There are \(100\) sweets in a bowl, and they each take in turns to take either \(2\), \(3\) or \(4\) sweets. Whoever cannot take any more sweets (since the bowl is empty, or there’s only \(1\) left) loses.
Maxim goes first - who has the winning strategy?
Michelle and Mondo play the following game, with Michelle going first. They start with a regular polygon, and take it in turns to move. A move is to pick two non-adjacent points in one polygon, connect them, and split that polygon into two new polygons. A player wins if their opponent cannot move - which happens if there are only triangles left. See the diagram below for an example game with a pentagon. Prove that Michelle has the winning strategy if they start with a decagon (\(10\)-sided polygon).
Let \(n\) be a positive integer. Show that \(1+3+3^2+...+3^{n-1}+3^n=\frac{3^{n+1}-1}{2}\).
Show that all integers greater than or equal to \(8\) can be written as a sum of some \(3\)s and \(5\)s. e.g. \(11=3+3+5\). Note that there’s no way to write \(7\) in such a way.
Determine all integer solutions to the equation \(24a + 16b = 6\).
John’s father is 28 years older than John and next year he will be exactly three times the age of John. How old is John’s father?
You have an hourglass that measures 8 minutes and an hourglass that measures 12 minutes. How can you measure exactly 44 minutes with them?
Joe has two kinds of weights: 15 grams and 50 grams. He has an infinite supply of each type. Can you help him find a combination that is exactly 310 grams?
Find a formula for \(R(2,k)\), where \(k\) is a natural number.