Question

В квадрате 30×30 закрашено несколько клеток. Известно, что в каждой строчке ровно три закрашенные клетки, а в каждом столбце — либо 4, либо 1 закрашенная клетка. Сколько столбцов, в которых закрашена ровно одна клетка? 

Answer 1

Обозначим число столбцов, в которых имеется одна закрашенная клетка через x, а число столбцов с четырьмя закрашенными клетками через y. В каждой строке имеется по три закрашенные клетки, поэтому общее число клеток будет 30*3 = 90. Тогда по условию получаем систему x + 4y = 30*3 = 90 и x + y = 30. Вычтем из первого уравнения системы второе: x + 4y - x - y = 90- 30 => 3y = 60 => y = 60/3 = 20. Тогда x = 30 - y = 30 - 20 = 10. Следовательно всего 10 столбцов с одной закрашенной клеткой.

Ответ: 10.

Answer 2

Solution for the Number of Columns with Exactly One Shaded Cell in a 30x30 Grid

To solve this problem, we need to find the number of columns in a 30x30 grid where exactly one cell is shaded. We know that in each row, there are exactly three shaded cells, and in each column, there are either four or one shaded cell.

Let's analyze the given information and calculate the number of columns with exactly one shaded cell.

Analysis and Calculation

1. Rows with Exactly Three Shaded Cells: - In each row, there are exactly three shaded cells.

2. Columns with Four or One Shaded Cell: - In each column, there are either four or one shaded cell.

3. Calculating the Number of Columns with Exactly One Shaded Cell: - We can calculate the number of columns with exactly one shaded cell by subtracting the total number of columns with four shaded cells from the total number of columns.

4. Total Number of Columns: - In a 30x30 grid, there are 30 columns.

5. Number of Columns with Four Shaded Cells: - From the given information, we can calculate the number of columns with four shaded cells.

Calculation

The total number of columns is 30. To find the number of columns with exactly one shaded cell, we need to calculate the number of columns with four shaded cells and then subtract it from the total number of columns.

From the given information, we can calculate the number of columns with four shaded cells.

Calculation of Columns with Four Shaded Cells

From the given information: - Each row has exactly three shaded cells. - In each column, there are either four or one shaded cell.

We can use this information to calculate the number of columns with four shaded cells.

Conclusion

After calculating the number of columns with four shaded cells, we can subtract this value from the total number of columns to find the number of columns with exactly one shaded cell in the 30x30 grid.

The number of columns with exactly one shaded cell can be calculated using the given information and the total number of columns in the 30x30 grid.

[[SOURCE #1]]