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Mathematics-Online problems: | ||
Solution to the problem of the (previous) week |
Problem:
1 | 1 | 1 | 1 |
2 | 2 | 3 | 3 |
4 | 2 | 2 | 3 |
4 | 4 | 4 | 3 |
1 | 2 | 3 | 4 |
1 | 3 | 4 | 2 |
3 | 2 | 4 | 1 |
3 | 4 | 2 | 1 |
1 | 2 | 3 | 4 |
2 | 1 | 4 | 3 |
3 | 4 | 1 | 2 |
4 | 3 | 2 | 1 |
Fill in a grid where in addition to the constraints of a) - c) no identical digits are placed along the main diagonals.
Answer:
a) | ||||
b) | ||||
c) |
Solution:
a) Without restriction, the positions of the four ones can be chosen from 16 possibilities, i.e., there are
b) Each row can contain a random permutation of the four digits. This means that in each row there are
c) After permutation of rows and columns of the grid (
possibilities), we can presume that the digits in the first row and first column are placed in ascending order:
1 | 2 | 3 | 4 |
2 | |||
3 | |||
4 |
Starting by a placement of the digit 4 in the second row, we obtain the following four possibilities:
|
There is a total of
Below, you can see one of the possible arrangements without repetition along the main diagonals.
1 | 3 | 4 | 2 |
4 | 2 | 1 | 3 |
2 | 4 | 3 | 1 |
3 | 1 | 2 | 4 |