Question

Вася вырезал из волейбольной сетки кусок 31 на 31 ячеек. Петя красит эту сетку следующим образом. Он выбирает любой квадрат (из ячеек сетки) и красит его границу. Какое наименьшее число квадратов он должен покрасить, чтобы вся сетка оказалась покрашенной? Выбираемый квадрат может быть любого размера. Разрешается красить куски сетки несколько раз.

Answer 1

Решение:

Рассмотрим (4*30) сторон единичных квадратиков. Любой квадрат содержит не более двух из указанных сторон. Поэтому необходимо покрасить:

2(31−1) = 60 квадратов размером 15×15 достаточно, если разместить так, чтобы все они примыкали к сторонам большого квадрата.

Answer 2

Problem Analysis

Vasya has cut out a piece of a volleyball net that consists of 31 by 31 cells. Petya wants to paint the net in such a way that every cell is painted. He can choose any square from the cells of the net and paint its border. The goal is to determine the minimum number of squares Petya needs to paint in order to paint the entire net.

Solution

To solve this problem, we can start by considering the smallest possible square, which is a 1x1 square. In this case, Petya would need to paint all 31x31 cells individually, resulting in a total of 31x31 = 961 squares.

However, we can optimize this process by noticing that we can combine adjacent cells into larger squares. By doing so, we can reduce the number of squares that need to be painted.

To find the minimum number of squares Petya needs to paint, we can follow these steps:

1. Start with the entire net consisting of 31x31 cells. 2. Divide the net into four equal quadrants. 3. Check if each quadrant can be combined into a larger square. If so, combine them. 4. Repeat steps 2 and 3 for each resulting square until no further combinations are possible. 5. Count the number of squares obtained after all possible combinations.

Let's go through an example to illustrate this process:

1. Start with the entire net consisting of 31x31 cells. 2. Divide the net into four equal quadrants, each consisting of 15x15 cells. 3. Check if each quadrant can be combined into a larger square. In this case, it is not possible to combine them. 4. Count the number of squares obtained after all possible combinations. In this case, we have four 15x15 squares.

Therefore, the minimum number of squares Petya needs to paint is four.

Answer

Petya needs to paint a minimum of four squares in order to paint the entire 31x31 cell net.