Problems

Two figures are said to be similar if all their corresponding sides are congruent and corresponding angles are equal. By using several copies of the figure below construct a bigger figure that is similar.

Divide a square into several triangles in such a way that every triangle shares a boundary with exactly three other triangles.

There are four numbers written in a row. The first number is \(100\). It is known that if we divide the first number by the second number we will get a prime number as a result, if we the second number by the third number we will get a prime number, and if we divide the third number by the fourth number we will also get a prime number. Can all the resulting prime numbers be distinct?

George claims that he knows two numbers such that their quotient is equal to their product. Can we believe him? Prove him wrong or provide a suitable example.

In the context of Example 14.2 what is the answer if we have five numbers instead of four? (i.e., can we get four distinct prime numbers then?)

Now George is sure he found two numbers with the quotient equal to their sum. And on top of that their product is still equal to the same value. Can it be true?

A maths teacher draws a number of circles on a piece of paper. When she shows this piece of paper to the young mathematician, he claims he can see only five circles. The maths teacher agrees. But when she shows the same piece of paper to another young mathematician, he says that there are exactly eight circles. The teacher confirms that this answer is also correct. How is that possible and how many circles did she originally draw on that piece of paper?

A group of three smugglers is offered to smuggle a chest full of treasures across the dangerous river. The boat they possess is old and frail. It can carry three smugglers without the chest, or it can carry the chest and only two smugglers. The price for this job is extremely high, and the gang is more than interested in completing the job. Think of a strategy the smugglers should follow to successfully transit the chest and themselves to the other shore.

It is easy to construct one equilateral triangle from three identical matches. Can we make four equilateral triangles by adding just three more matches identical to the original ones?

(a) Can one fit 4 letters “T” (see the picture below) in a \(6\times6\) square box?

We do not allow any overlappings to occur.

(b) Can we fit them in a square with smaller side length?