Problems

Every even number is not prime. The number \(9\) is not an even number, therefore it is not not prime, i.e: the number \(9\) is prime.

In a school there are three sports clubs, which we call \(A\), \(B\), and \(C\).

A student argues as follows:

“To find how many people attend at least one club, we can add the number of people in each club. However, students who attend all three clubs get counted three times. To fix this, we should subtract them twice. Therefore, the number of people who attend at least one club is \[\text{people in }A+\text{people in }B+\text{people in }C -2\times(\text{people in all three clubs}).\]”

Is this reasoning correct?

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?