Another puzzle the was e-mailed to me v this website. Mine instinct was the the prize was simply a lot, but I thought about it and the systems is actually relatively simple...
You are watching: How many squares are in a 6x6 grid
Before analysis the answer deserve to I interest you in a clue?The an initial thing is why the price is not simply 64...


1x1 | 8 | 8 | 64 |
2x2 | 7 | 7 | 49 |
3x3 | 6 | 6 | 36 |
4x4 | 5 | 5 | 25 |
5x5 | 4 | 4 | 16 |
6x6 | 3 | 3 | 9 |
7x7 | 2 | 2 | 4 |
8x8 | 1 | 1 | 1 |
total | 204
See more: The Type Of Element Is Determined By, The Periodic Table
Formula for n x n Chessboard?
It"s clear from the analysis over that the solution in the instance of n x n is the amount of the squares indigenous n2 come 12 the is to say n2 + (n-1)2 + (n-2)2 ... ... 22 + 12Mathematically the is written as follows:
Can you extend your an approach to calculation the number of rectangles on a chessboard?
Below room some instances of feasible rectangles...
Dimensions | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||
Positions | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | ||
1 | 8 | 64 | 56 | 48 | 40 | 32 | 24 | 16 | 8 | |
2 | 7 | 56 | 49 | 42 | 35 | 28 | 21 | 14 | 7 | |
3 | 6 | 48 | 42 | 36 | 30 | 24 | 18 | 12 | 6 | |
4 | 5 | 40 | 35 | 30 | 25 | 20 | 15 | 10 | 5 | |
5 | 4 | 32 | 28 | 24 | 20 | 16 | 12 | 8 | 4 | |
6 | 3 | 24 | 21 | 18 | 15 | 12 | 9 | 6 | 3 | |
7 | 2 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | |
8 | 1 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | |
1296 |
Elegant strategy to rectangles, consider the vertices and diagonals.


n x n or n x m?
The n x n (eg. 9x9,) or n x m (eg 10x15,) troubles can now be calculated. The variety of vertices being given by (n + 1)2 and also (n + 1).(m + 1) respectively. Thus the final solutions are as follows.n x n: (n + 1)2 x n2 / 4n x m: (n + 1) x (m + 1) x (n x m) / 4Which can obviously be arranged right into something much more complicated.Rectangles in Maths Nomenclature
It"s constantly my intentionally to explain the difficulties without formal maths nomenclature, v reasoning and also common sense. Yet there is quite a succinct solution here if you carry out know about combinations, as in permutations and combinations. Horizontally we are choosing 2 vertices from the 9 available. The order does not matter so it"s combinations fairly than permutations. And also the exact same vertically. Therefore the answer come the rectangle trouble can it is in answered by:9C2•9C2 = 362 = 1296PayPalI constantly think it"s arrogant to include a donate button, but it has actually been requested. If I help you obtain a task though, you can buy me a pint! - nigel



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