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• Level: GCSE
• Subject: Maths
• Word count: 2409

I have been asked to investigate the amount of squares protected by a queen in any position on a chessboard.

Extracts from this document...

Introduction

I have been asked to investigate the amount of squares protected by a queen in any position on a chessboard. I have construed this to mean a square chessboard, but for extension work I will investigate this question on a rectangular board. I will use 8x8, 9x9, 10x10, 11x11, 12x12, and 13x13 size boards, because these are useful, practical sizes, and they will give me a range of results. I aim to find out where the queen protects the most squares on a square board, and from this I will try to find an algebraic rule for this piece of data for this shape chessboard. I also aim to find out where the queen protects the most squares on a rectangular chessboard, and to find an algebraic rule for this shape also.

I will also attempt to find out why I get these sort of results, by testing a rook on all boards. I will use a rook because it will allow me to find out if it is diagonals that affect the results as I predict.

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The 12x12 board also follows the same pattern

 37 37 37 37 37 37 37 37 37 37 37 37 37 37 39 39 39 39 39 39 39 39 39 39 39 37 37 39 41 41 41 41 41 41 41 41 41 39 37 37 39 41 43 43 43 43 43 43 43 41 39 37 37 39 41 43 45 45 45 45 45 43 41 39 37 37 39 41 43 45 47 47 47 45 43 41 39 37 37 39 41 43 45 47 49 47 45 43 41 39 37 37 39 41 43 45 47 47 47 45 43 41 39 37 37 39 41 43 45 45 45 45 45 43 41 39 37 37 39 41 43 43 43 43 43 43 43 41 39 37 37 39 41 41 41 41 41 41 41 41 41 39 37 37 39 39 39 39 39 39 39 39 39 39 39 37 37 37 37 37 37 37 37 37 37 37 37 37 37

The 13x13 board also fit the pattern of the other boards; therefore I will ascertain that there is an algebraic rule for a square board.

 Board size Squares protected at rim Squares protected at centre DifferenceAt rim Difference at centre 8x8 22 28 N/A N/A 9x9 25 33 3 5 10x10 28 36 3 3 11x11 31 41 3 5 12x12 34 44 3 3 13x13 37 49 3 5

The differences will be the key to finding the algebraic rule, however, the differences at the centre won't be very useful because they alternate, and, as rules have to be constant in order to be rules, these differences are useless. The differences at the rim are a different story. They are constant and so will be very useful to finding the rule. The rule has to link the board size into it in order for it to become a universally applicable "nth term" rule. Therefore, I will make board size "n".

Since I am trying to find the number of squares protected by the queen, the perimeter squares will be called "p". Now I have the basis of a rule,

n = p

However, this rule still needs work, as, in the first board size, 8 does not equal 22.

Multiplication will make the numbers more similar.

2x8, 16, isn't big enough, 4x8, 32, is too big, 3x8, 24, is much closer, yet is still too high.

Multiplication isn't the answer this time, division would make it too small, yet subtracting 2 would give me the answer.

3n-2 = p

I will test it with the other board sizes.

(3x9) -2 = 25

27-2 = 25

(3x10) -2 = 28

30-2 = 28

(3x11) -2 = 31

33 -2 = 31

(3x12) -2 = 34

36 -2 = 34

(3x13) -2 = 37

39 -2 = 37

This rule does work. Once you have found out the perimeter the rest of the board follows the pattern illustrated.

EXTENSION WORK

REASON FOR RESULTS

I will position a rook in different positions across the chessboard to test if diagonals are the reason for the varied results across the board. I will only test 8x8, 11x11, and 12x12 boards because these will give me useful and varied results with less hassle. I will start on the 8x8 board.

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Conclusion

I will test this pattern on a 10x9 chessboard.

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This board also follows the pattern.

I will now test this on an 11x10 board.

 29 29 29 29 29 29 29 29 29 29 30 30 30 30 30 30 30 30 30 30 30 32 32 32 32 32 32 32 32 30 30 32 34 34 34 34 34 34 32 30 30 32 34 36 36 36 36 34 32 30 30 32 34 36 38 38 36 34 32 30 30 32 34 36 37 37 36 34 32 30 30 32 34 35 35 35 35 34 32 30 30 32 33 33 33 33 33 33 32 30 30 31 31 31 31 31 31 31 31 30 29 29 29 29 29 29 29 29 29 29

The 11x10 board also follows the pattern set by the first board.

 Board size Squares protected at top row Squares protected at centre DifferenceAt rim Difference at centre 9x8 23 30 N/A N/A 10x9 26 34 3 4 11x10 29 38 3 4

These results lead to the possibility of two algebraic rules, but one is more complicated than the other is, so I will go through that one first.

If "n" is the long side, "k" is the short side and "r" is the top row then,

3n +1/2k = r

The complication comes when k is an odd number, because, taking the second board size, 1/2 of 9 is 4.5, but the answer is not 34.5,so you have to round down k to the nearest whole number if it is odd.

Far simpler is this rule if "n" is the short side and "r" is the top row then,

3n -1 = r

which works for all rectangular boards of this shape. Once you have found out the top row the rest of the board follows the pattern illustrated.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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