Problems

A phoenix is a pattern with the interesting property that all of its alive cells die after each generation, yet the pattern as a whole lives indefinitely. Show that if a phoenix is contained in some rectangle at the start, it can never extend more than one cell past this rectangle (i.e: a phoenix can’t expand forever). Below is a picture of a phoenix with period \(2\):

Prove that there is no oscillator of period \(4\) (i.e: the whole pattern repeats every \(4\) generations) which has exactly one cell that also has period \(4\).

A pattern \(P\) is called a garden of Eden if there exists no pattern \(P'\) distinct to \(P\) such that \(P'\) evolves into \(P\) after one generation. Show that a garden of Eden exists. You do not need to provide an example of such a pattern.

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?

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?

In the following puzzle, different animals represent different digits, and your goal is to find which digit each animal represents.

Find all possible solutions.