Count 'em All 9!
Count the total number of quadrilaterals in the 6 × 5 grid above.
Clarifications:
- Quadrilateral is a polygon that has 4 sides.
This is one part of Quadrilatorics .
The answer is 315.
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2 solutions
But isn't that only counting the number of rectangles in the grid? I think it makes more sense to count how many ways you can choose 4 points out of the 42 points i.e. 42C4
Careful counting shows that the above figure with
6
columns and
1
row has
6
+
5
+
4
+
3
+
2
+
1
=
2
1
quadrilaterals.
Careful counting shows that the above figure with
6
columns and
2
rows has
1
2
+
1
0
+
8
+
6
+
4
+
2
+
6
+
5
+
4
+
3
+
2
+
1
=
6
3
quadrilaterals.
Careful counting shows that the above figure with
6
columns and
3
rows has
1
8
+
1
5
+
1
2
+
9
+
6
+
3
+
1
2
+
1
0
+
8
+
6
+
4
+
2
+
6
+
5
+
4
+
3
+
2
+
1
=
1
2
6
quadrilaterals.
Careful counting shows that the above figure with
6
columns and
4
rows has
2
4
+
2
0
+
1
6
+
1
2
+
8
+
4
+
1
8
+
1
5
+
1
2
+
9
+
6
+
3
+
1
2
+
1
0
+
8
+
6
+
4
+
2
+
6
+
5
+
4
+
3
+
2
+
1
=
2
1
0
quadrilaterals.
So, the figure which follows the same pattern and has
6
columns and
5
rows(i.e. the figure without the cut) should have
3
0
+
2
5
+
2
0
+
1
5
+
1
0
+
5
+
2
4
+
2
0
+
1
6
+
1
2
+
8
+
4
+
1
8
+
1
5
+
1
2
+
9
+
6
+
3
+
1
2
+
1
0
+
8
+
6
+
4
+
2
+
6
+
5
+
4
+
3
+
2
+
1
=
[
(
2
1
×
5
)
+
(
2
1
×
4
)
+
(
2
1
×
3
)
+
(
2
1
×
2
)
+
(
2
1
×
1
)
]
=
[
2
1
×
(
5
+
4
+
3
+
2
+
1
)
]
=
[
2
1
×
1
5
]
=
3
1
5
□
Moderator note:
This solution is tedious and unnecessarily complicated.
The answer is directly ( 2 7 ) × ( 2 6 ) .
Do you see why?
From this note , we know that the number of quadrilaterals in an a × b grid is ( 2 a + 1 ) × ( 2 b + 1 ) = 4 a b ( a + 1 ) ( b + 1 )
Now, by substituting a = 6 and b = 5 , we would get 3 1 5 .