How many squares do you see?
In a square of
2
×
2
there are
1
+
4
squares.
In a square of
3
×
3
there are
1
+
4
+
9
squares.
In a
4
×
4
square there are
1
+
4
+
9
+
1
6
squares.
In a
5
×
5
square there are
1
+
4
+
9
+
1
6
+
2
5
squares.
In a square of 1 0 × 1 0 , how many squares are there?
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Nice solution
[ A w e s o m e ] ( s o l u t i o n s ) .
But one solution is enough!
The number of squares is 1 2 + … + n 2 , for a n × n square.
Putting n = 1 0
1 2 + 2 2 + 3 2 + … + 1 0 2 = 6 1 0 ( 1 1 ) ( 2 1 ) = 3 8 5
Note i = 1 ∑ n i 2 = 6 n ( n + 1 ) ( 2 n + 1 )
It would have been better for the solvers to have figured out the pattern themselves. You made the problem too easy: Level 0.
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solution 1
1+4+9+16+25+36+49+64+81+100=385 .
solution 2 .
i = 1 ∑ n i 2 = 6 n ( n + 1 ) ( 2 n + 1 )
1 2 + 2 2 + 3 2 + … + 1 0 2 = 6 1 0 ( 1 1 ) ( 2 1 ) = 3 8 5