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Corners - Maths Investigation

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Introduction

Corners Draw a grid 5 columns wide, with any number of rows above 2. Select a square of numbers, 2x2, e.g. 7,8,12,13 Multiply together the numbers in opposite corners of the square (e.g. 7*13=91, 8*12=96) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 I shall now begin searching for patterns, and for rules that I will then prove and use to explain patterns. My first step shall be to make examples to compare Ex.1 7 8 12 13 7*13=91 8*12=96 13 14 18 19 13*19=247 14*18=252 The pattern in this case would seem to be a difference of 5. It may be a coincidence that 5 is the number of columns in the grid, and in order to test whether it is in fact a coincidence or not I shall introduce a new letter 'c' which will stand for the number of columns within the entire grid. Algebra n n+1 n+c n+c+1 n(n+c+1)=n�+nc+n (n+1)(n+c)=n�+nc+n+c n�+nc+n+c-(n�+nc+n) = n�+nc+n +c -(n�-nc-n) = c The difference is c, the number of columns within the grid. RULE d = c I shall extend this by varying the size of the square extracted from the grid. ...read more.

Middle

= n�+3cn+3in+9ci� n�+3cn+3in+9ci�-(n�+3cn+3in) n�+3cn+3in +9ci� -(n�+3cn+3in) 9ci� 3x3 square: 4ci� 4x4 square: 9ci� 4 and 9 are both square numbers (similar to the pattern on page 2) 4 and 9 can be replaced with (s-1)�, making the rule: RULE d = (s-1)�ci� The logical way forward seems to be to change the square to a rectangle, for which I already have a theory. I believe that the (s-1)� in the rule, being (s-1) (s-1), is both of the sides of the square - each s represents the length of a side. Based upon this theory, using x and y to represent the horizontal and vertical dimensions of the square, the rule would be d = (x-1)(y-1)ci�. I shall now attempt to prove or disprove this, and if it turns out to be incorrect, I shall endeavour to discover the correct rule. I shall begin testing the theoretical rule by using it to find a solution, then finding the solution without a rule. If I am successful, both of the solutions should match. 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 I shall use a 3x4 rectangle (3 grid squares wide, 4 grid squares tall). ...read more.

Conclusion

is, but this does not give me any algebra to write down. In addition, I performed some preliminary experimentation with sequences of square and triangular numbers, but to no avail. Interesting patterns frequently appeared, however I was unable to explain any of them. I believe that, given sufficient time, a person could and probably has found rules for such sequences. This person, however, is not me. I have since reverted to working with algebra to find other formulae, and noticed that I have a rule for he contents of each corner of the square. Since only the corners are used, (this investigation could extended to use more than just the corners, but in this case has not been) I have therefore discovered each of the formulae for an addition interval (another term I made up, meaning that the sequence of numbers increases by the same number each time) that I believe would be most useful if I was ever to repeat or further extend this investigation. The formulae for the corners are: n n+(x-1)i n+(y-1)ci n+(y-1)ci+(x-1)i A very brief conclusion The overall rule for addition intervals (or whatever they're really called) is d = (x-1)(y-1)ci�. The rule for multiplication intervals (or whatever these are really called) is d = 0 More detailed conclusions are given to each section, and so this will end approximately... now. Appendix ...read more.

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