Another brain teaser... want 10 points?

Question

If you were to construct a 8 x 8 checkered square (i.e., a 8 x 8 chess board), how many squares would there be in total?

Not 64....

thisbrit is on the right track

Answer 1

Can you see the pattern?
There is 1 square of size 8X8
There are 4 squares of size 7X7
There are 9 squares of size 6X6
There are 16 squares of size 5X5
There are 25 squares of size 4X4
There are 36 squares of size 3X3
There are 49 squares of size 2X2
There are 64 squares size 1X1
So, just add them up!
1+4+9+16+25+36+49+64! Yuppy! The answer is 204!

Squares are fun!

Answer 2

1

Answer 3

You will need to click the source link to see exact pics of the diagrams

A checkerboard - which is the same as a chessboard - is an 8x8
arrangement of squares, so it has 64 small squares. Of course, you
could make a "square" comprised of a 2x2 block of the small squares,
or a 3x3 block, etc. If you count all of these "squares," you would
get a much larger number. To count these larger squares, it is easiest
to figure out which points on the checkerboard can be the upper left
corner of a 2x2 square and count them, then do the same for 3x3, 4x4,
etc. When you add up all of these values, you have the total number of
"squares" in a checkerboard. Let's look at a smaller example - a 4x4
"checkerboard." I'll label the points on the board like this:

A---B---C---D---E
| | | | |
F---G---H---I---J
| | | | |
K---L---M---N---O
| | | | |
P---Q---R---S---T
| | | | |
U---V---W---X---Y

There are 16 squares of size 1x1.

Points A, B, C, F, G, H, K, L and M can be the upper left corners of
2x2 squares, for a total of 9. Note that points D, E, I, J, N and O
can't be upper left corners because we can't go 2 squares to the right
of these points. Likewise, points P through Y can't be upper left
corners because we can't go 2 squares down from these points.

Only points A, B, F and G can be the upper left corners of 3x3
squares, for a total of 4. (The reasons the others can't be are
similar to the reasons given above.)

Finally, there is only one 4x4 square, with point A being its upper
left corner. So we have 16 + 9 + 4 + 1 = 30 total squares.

Do you see a pattern in the numbers we were adding in the last step?
Using that pattern, we can determine the total number of squares on an
8x8 checkerboard without counting corners.


As to the relation of this to the game tetris, tetris is a little more
complicated. Not only do you have to consider different shapes (I'd
consider each one separately), but you also have to consider rotations
and reflections of the shapes. For example, the shape:

+---+---+---+
| | | |
+---+---+---+
| |
+---+

can be rotated to three other positions:

+---+ +---+---+
| | | | |
+---+ +---+ +---+---+
| | | | | |
+---+---+ +---+---+---+ +---+
| | | | | | | | |
+---+---+ +---+---+---+ +---+

and can be reflected and rotated to four other positions:

+---+---+---+ +---+ +---+---+
| | | | | | | | |
+---+---+---+ +---+ +---+ +---+---+
| | | | | | | |
+---+ +---+---+ +---+---+---+ +---+
| | | | | | | | |
+---+---+ +---+---+---+ +---+

If we count every way that each of these rotations and reflections
will fit in a 4x4 square, we'll get a much larger number.

Answer 4

64

Answer 5

the sum of n^2 from 1 to 8 = 204.

Answer 6

64 squares

Answer 7

First off, you have your 64, then add one for the biggest one that the 64 fit in, then 4 more for the quartered square, and then add the central square, and on and on, there are dozens, but i do not have time to draw it. . .and count

Answer 8

204

There are

64 1x1
49 2x2
36 3x3
25 4x4
16 5x5
9 6x6
4 7x7
1 8x8

Answer 9

36

Answer 10

65