The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 are placed in ten slots of the following grid based on the conditions below.
- Numbers in any row appear in an increasing order from left to right.
- Numbers in any column appear in a decreasing order from top to bottom.
- 1 is placed either in the same row or in the same column as 10.
- Neither 2 nor 3 is placed in the same row or in the same column as 10.
- Neither 7 nor 8 is placed in the same row or in the same column as 9.
- 4 and 6 are placed in the same row.
Solution
| Column 1 | Column 2 | Column 3 | Column 4 | |
|---|---|---|---|---|
| Row 1 | 10 | |||
| Row 2 | - | |||
| Row 3 | - | - | ||
| Row 4 | - | - | - |
Clue 1 & 2,
Gives us two conditions-
Numbers in any row appear in an increasing order from left to right. Also, numbers in any column appear in a decreasing order from top to bottom.
Therefore, number 10 must be placed in Row 1, Column 4.
As placing it anywhere else would mean that the number above it or right to it must be greater than 10, which is not possible.
| Column 1 | Column 2 | Column 3 | Column 4 | |
|---|---|---|---|---|
| Row 1 | 10 | |||
| Row 2 | - | |||
| Row 3 | - | - | ||
| Row 4 | - | - | - |
Now clue 3 says,
1 is placed either in the same row or in the same column as 10.
Therefore, we have only two possibilities -
- Row 1, column 1.
- Row 4, column 4.
| Column 1 | Column 2 | Column 3 | Column 4 | |
|---|---|---|---|---|
| Row 1 | 10 | |||
| Row 2 | - | 2/3 | ||
| Row 3 | - | - | 2/3 | |
| Row 4 | - | - | - |
According to clue 4,
Neither 2 nor 3 is placed in the same row or in the same column as 10 therefore, only places for 2 and 3 will be -
- row 2, column 2
- row 2, column 3
- row 3, column 3
But if we put 2 in row 2 column 3 - we will need a number less than it to its left which is not possible.
Similarly, if we put 3 in row 2 column 3, we will have to put 2 in row 2 column 2, which means we will need a number less than 2 in row 1 column 2, which again is not possible.
Therefore, 2 & 3 must be present in row 2 column 2 or row 3 column 3.
| Column 1 | Column 2 | Column 3 | Column 4 | |
|---|---|---|---|---|
| Row 1 | 4 | 6 | 10 | |
| Row 2 | - | 2/3 | ||
| Row 3 | - | - | 2/3 | |
| Row 4 | - | - | - |
Now using clue 6,
4 and 6 are placed in the same row.
Which is only possible for row 1 or row 2.
If we put them in row 2 then 4 has to be in column 3 and 6 in column 4, as we need numbers to be increasing from left to right (clue 1), but then there will be no room for 7, 8, and 9.
Therefore, 4 and 6 have to be in the first row.
| Column 1 | Column 2 | Column 3 | Column 4 | |
|---|---|---|---|---|
| Row 1 | 4 | 6 | 9 | 10 |
| Row 2 | - | 2/3 | ||
| Row 3 | - | - | 2/3 | |
| Row 4 | - | - | - |
As we already know 10's place, there are only two places left for 9 — row 1 column 3 or row 2 column 4.
Consider the second case: if 9 is placed in Row 2 Column 4, 7 and 8 must be placed in row 1 (according to clue 5).
Clue 6 says that 4 and 6 are in the same row, but with 7 and 8 in row 1, no rows are left with two spaces. Hence, 9 cannot be in R2C4.
Therefore, second case is rejected.
| Column 1 | Column 2 | Column 3 | Column 4 | |
|---|---|---|---|---|
| Row 1 | 4 | 6 | 9 | 10 |
| Row 2 | - | 2/3 | 5 | 8 |
| Row 3 | - | - | 2/3 | 7 |
| Row 4 | - | - | - | 1 |
Now 7 and 8 will occupy row 3 column 4 and row 2 column 4 respectively.
And so, 4 and 6 will occupy row 1 column 1 and row 1 column 2 respectively.
Therefore, only place left for 1 is row 4 column 4, and for 5 is row 2 column 3.
Here we won't be able to determine exact positions of 2 and 3, but further data will be provided by the questions of the set, if needed.
| Column 1 | Column 2 | Column 3 | Column 4 | |
|---|---|---|---|---|
| Row 1 | 4 | 6 | 9 | 10 |
| Row 2 | 2/3 | 5 | 8 | |
| Row 3 | 2/3 | 7 | ||
| Row 4 | 1 |
Sum of number placed in column 4 is 10 + 8 + 7 + 1 = 26