Problems

After mastering the Caesar shift cypher one may wonder how to generalize it. One possible way is to use Affine cypher. The difference between these two methods can be described as follows:

• In case of Caesar cypher we took a letter with position $n$ from $1$ to $26$ and added to its position a number $d$ obtaining the number $n+d$, then we compute its residue modulo $26$.

• In case of affine cypher we take a letter with position $n$ and consider a number $nx+d$ modulo $26$.

To decipher such code we need to know values $x$ and $d$, then if we have a letter in the code with position $m$, we can find $n$ as $n=(m-d)x^{-1}$ modulo $26$. Here we have to explain what is $x^{-1}$: for a number $x<26$ we are looking for such a number $y$, that $26$ divides $xy-1$.

• Does there always exist a number $x^{-1}$ modulo $26$ for any $x$?

• Using data $x=3$, $d=8$ encrypt the word "SOLUTION".

In the middle of an empty pool there is a square platform of $50\times 50$ cm, split into cells of $10\times 10$ cm. Sunny builds towers of $10\times 10\times 10$cm cubes on the platform cells. After that his friend Margo turns on the water and counts how many towers are still above the water level. They call each visible tower an island.

For example, let’s consider the case when the heights of the towers are as given in the table on the right. Then at the water level of $5$ cm there is $1$ island, at the water level of $15$ cm there are two islands (if the islands have a common corner or don’t intersect at all, they are considered separate islands), and at the water level of $25$ cm, all the towers are covered with water and there are $0$ islands.

Find out how Sunny should build his towers to get the following numbers of islands corresponding to the level of water in the pool: $$\begin{array}{cccccc}\mathit{\text{Water level (cm)}}& 5& 15& 25& 35& 45\\ \mathit{\text{Number of islands}}& 2& 5& 2& 5& 0\end{array}$$

In the solution, write down how many cubes are there composing a tower in each cell as it is done in the example.

On the grid paper, Theresa drew a rectangle $199\times 991$ with all sides on the grid lines and vertices on intersection of grid lines. How many cells of the grid paper are crossed by a diagonal of this rectangle?

The school cafeteria offers three varieties of pancakes and five different toppings. How many different pancakes with toppings can Emmanuel order? He has to have exactly one topping on each pancake.

How many six-digit numbers are there whose digits all have the same parity? That is, either all six digits are even, or all six digits are odd.

Donald’s sister Maggie goes to a nursery. One day the teacher at the nursery asked Maggie and the other children to stand a circle. When Maggie came home she told Donald that it was very funny that in the circle every child held hands with either two girls or two boys. Given that there were five boys standing in the circle, how many girls were standing in the circle?

We say that a figure is convex if a segment connecting any two points lays fully within the figure. On the picture below the pentagon on the left is convex and the one on the right is not.

Is it possible to draw $18$ points inside a convex pentagon so that each of the ten triangles formed by its sides and diagonals contains equal amount of points?

Cambria was building various cuboids from $1\times 1\times 1$ cubes. She initially built one cuboid, then increased its length and width by $1$ and reduced its height by $2$. She then understood that she needs the same number of $1\times 1\times 1$ cubes to build both the original and new cuboids. Prove that the number of cubes used for each of the cuboids is divisible by $3$.

A labyrinth was drawn on a $5\times 5$ grid square with an outer wall and an exit one cell wide, as well as with inner walls running along the grid lines. In the picture, we have hidden all the inner walls from you (We give you several copies to facilitate drawing) 
Please draw how the walls were arranged. Keep in mind that the numbers in the cells represent the smallest number of steps needed to exit the maze, starting from that cell. A step can be taken to any adjacent cell vertically or horizontally, but not diagonally (and only if there is no wall between them, of course).

Is it possible to cut this figure, called "camel"

• a) along the grid lines;

• b) not necessarily along the grid lines;

into $3$ parts, which you can use to build a square?
(We give you several copies to facilitate drawing)