The following set contains four questions related to Logical Reasoning. Choose the best answer to each question.
You are given an n×n square matrix to be ﬁlled with numerals so that no two adjacent cells have the same numeral. Two cells are called adjacent if they touch each other horizontally, vertically or diagonally. So a cell in one of the four corners has three cells adjacent to it, and a cell in the ﬁrst or last row or column which is not in the corner has five cells adjacent to it. Any other cell has eight cells adjacent to it.
1) What is the minimum number of different numerals needed to ﬁll a 3×3 square matrix?
Answer: 4
Solution:
As per the given definition, in a the following are the cases of adjacent cells.
or
or
In all the three cases X
_{1 }and X
_{2 }are adjacent to each other.
As per the question n = 3 so the matrix will look like
As four corners are not adjacent to each other so there can be same numerals . similarly for other squares the arrangement can be as bellow
Thus final arrangement of minimum numerals satisfying all the conditions will be
Where a,b,c and d are different numerals .
So minimum number of numerals required = 4
2) What is the minimum number of different numerals needed to ﬁll a 5×5 square matrix?
Answer: 4
Solution:
As per the given definition, in an x n metrix the following are the cases of adjacent cells.
Or
Or
In all the three cases X
_{1 }and X
_{2 }are adjacent to each other.
As per the question n = 5 so the matrix will look like
As every alternate squares are not adjacent to each other so there can be same numerals . similarly for other squares the arrangement can be as bellow
a |
b |
a |
b |
a |
c |
d |
c |
d |
c |
a |
b |
a |
b |
a |
c |
d |
c |
d |
c |
a |
b |
a |
b |
a |
Where a,b,c and d are different numerals.
So minimum number of numerals required = 4
3) Suppose you are allowed to make one mistake, that is, one pair of adjacent cells can have the same numeral. What is the minimum number of different numerals required to ﬁll a 5×5 matrix?
a) 4
b) 9
c) 25
d) 16
Solution:
As per the solution of previous question if we don’t make any mistakes , matrix will look like
a |
b |
a |
b |
a |
c |
d |
c |
d |
c |
a |
b |
a |
b |
a |
c |
d |
c |
d |
c |
a |
b |
a |
b |
a |
As we can see each numerals have been used at least 4 times. So even if one mistake is allowed, then also there won’t be any change in the solution given in previous question .
So minimum number of numerals required = 4
4) Suppose that all the cells adjacent to any particular cell must have different numerals. What is the minimum number of different numerals needed to fill a 5×5 square matrix?
a) 9
b) 25
c) 16
d) 4
Solution:
As per the previous two questions 5 matrix will look like
a |
b |
a |
b |
a |
c |
d |
c |
d |
c |
a |
b |
a |
b |
a |
c |
d |
c |
d |
c |
a |
b |
a |
b |
a |
But in this question b can not have 2 a’s or 2 d’s in it’s adjacent cell and it is also true for all other numerals as well. This is satisfied only when there are at least 9 numerals.
1 |
2 |
5 |
4 |
7 |
4 |
3 |
6 |
1 |
8 |
9 |
7 |
8 |
9 |
3 |
1 |
4 |
2 |
7 |
5 |
5 |
3 |
6 |
1 |
4 |
So minimum number of numerals required = 9
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