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.
Show that \(R(4,3)\ge9\). That is, there exists a way of colouring the edges of \(K_8\) with no red \(K_4\), nor any blue \(K_3\).
Show that \(R(4,4)\ge18\) - that is, there’s a way of colouring the edges of \(K_{17}\) such that there’s no monochromatic \(K_4\).