Problems

Everyone makes mistakes in maths — and that’s a good thing! Mistakes help us understand what really works and what doesn’t.

The more mistakes you see, the easier they become to spot. And once you can spot them, you’re much less likely to make them yourself.

In this session, we will sharpen our mistake-detecting skills by looking at a collection of fake proofs: arguments that look convincing at first glance, but hide a sneaky error somewhere inside. Your job is to find where things go wrong and explain why.

The diagram shows a triangle drawn on a square grid. The area of the shaded triangle is \(9~\text{cm}^2\). What is the area of one of the little squares of the grid?

The first \(2026\) prime numbers are multiplied. How many zeroes are at the end of this resulting number?

Three skiers—Alice, Bob, and Cynthia—compete in a downhill race. They begin skiing in the following order: first Cynthia, then Bob, and finally Alice.

Each skier starts with \(0\) points. Whenever one skier overtakes another during the race, the overtaking skier gains \(1\) point and the skier being overtaken loses \(1\) point.

At the end of the race, Alice crosses the finish line first, and Bob finishes with \(0\) points.

In what position does Cynthia finish?

On the Problemland Space Station, there are \(1000\) tonnes of air, of which \(99\%\) is oxygen. After an unfortunate asteroid impact, some of the air is vented into space. The hull is quickly repaired, and no further loss occurs.

Afterward, measurements reveal that oxygen now makes up only \(98\%\) of the remaining air, and that only oxygen was lost during the incident.

How many tonnes of oxygen remain on the space station?

Find all solutions of the following puzzle. Each animal represents a different number.

What is the smallest positive whole number whose digits add up to \(2026\)?

The thirteen dwarves sat down around a bonfire in a circle, and their leader Thorin proposed a challenge to pass the time:

Each dwarf would choose an integer so that for every group of three neighboring dwarves, the sum of their numbers must be exactly \(13\).

Could Thorin’s challenge be solved?

In the month of January of a certain year, there are \(4\) Mondays and \(4\) Fridays. What day of the week is the \(20^{\text{th}}\) day of this month?

We wish to place the numbers \(1\) to \(10\) in the circles of the following picture, so that each circle contains exactly one number, in such a way that each line of three circles sums to the same number, can we do this?