Problems

McDonald’s used to sell Chicken McNuggets in boxes of 6, 9 or 20 in the UK before they introduced the Happy Meal. What is the largest number of Chicken McNuggets that cannot be bought? For example, you cannot buy 8 Chicken McNuggets, but you can buy \(21 = 6+6+9\) Chicken McNuggets.

Show that the equation \(x^4+y^4=z^4\) cannot satisfied by integers \(x,y,z\) if none of them is 0.

A regular polygon has integer side lengths and its perimeter is 60. How many sides can it have?

Find positive integers \(x,y,z\) such that \(28x+30y+31z = 365\).

Given a piece of paper, we are allowed to cut it into 8 or 12 pieces. Can we get exactly 60 pieces of paper starting with a single piece?

Pascal’s triangle, (known as Khayyam’s triangle in Iran and Yang Hui’s triangle in China) is seen below.

Well, what does it mean? We start with diagonal lines of 1s. Then every other number in the interior is the sum of the two numbers above it. With this simple rule, the triangle shows lots of cool structure.

In Pascal’s triangle, what are the numbers in the diagonal next to the diagonal of ones?

In Pascal’s triangle, what is the sum of the entries in each row?

Oliver throws a fair coin three times. What are his chances of getting three heads, two heads and one tail, one head and two tails, or three tails?

In Pascal’s triangle, what numbers appear in the diagonal next to the positive integers?