Investigation into Numerical 4 by 4 Squares - page 2
Keywords: Investigation into Numerical 4 by 4 Squares maths Formula, maths sqaure formula
By Mose. on 01/09/2008
Level: GCSE Key Stage 4 (Years 10-11)
Page Number: 2 of 5 pages: 1 2 3 4 5
additional example to prove that my results were correct.
56 57 58 59 60
66 67 68 69 70
76 77 78 79 80
86 87 88 89 90
96 97 98 99 100
56 X 100 = 5600 My results were correct as the difference Is again
60 X 96 = 5760 one hundred and sixty.
I recorded my results in a table, in the event that I will be able to find a formula and trend pattern within my results.
Size of Square Difference between the products
2 x 2 10
3 x 3 40
4 x 4 90
5x 5 160
I can see from my results that there is indeed a pattern occurring. The difference is the square of the previous number before it . For example 2 x 2 its square is 4. In the table if we ignore the zeros, I observed that 2 x 2’s square was the difference for a 3 x 3 square. This continued throughout the table as we see that the square of 3 x3 is indeed the difference for a 4 x 4 square. From my results I predict that the difference for a 6 x 6 square will be two hundred and fifty. I worked this out as 5 x 5 equals twenty five, therefore to follow the pattern of the table I multiplied this number by ten giving me two hundred and fifty.
I am now going to see if my prediction was correct for a 6 x6 sqaure.
5 6 7 8 9 10
15 16 17 18 19 20
25 26 27 28 29 30
35 36 37 38 39 40
45 46 47 48 49 50
55 56 57 58 59 60
5 x 60 = 300 The difference is two hundred and fifty.
10 x 55 = 550
45 46 47 48 49 50
55 56 57 58 59 60
65 66 67 68 69 70
75 76 77 78 79 80
85 86 87 88 89 90
95 96 97 98 99 100
50 x 95 = 4750 The difference is two hundred and fifty.
100 x 45 = 4500
23 24 25 26 27 28
33 34 35 36 37 38
43 44 45 46 47 48
53 54 55 56 57 58
63 64 65 66 67 68
73 74 75 76 77 78
23 x 78 = 1794 The difference is two hundred and fifty.
28 x 73 = 2044
This is clearly what I predicted as the difference is always equal to two hundred and fifty. I attempted another example to back up my results.
1 2 3 4 5 6
11 12 13 14 15 16
21 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 46
51 52 53 54 55 56
1 x 56 = 56 The difference again is always two hundred and fifty.
51 x 6 = 306
To make further investigations quicker. I worked out a formula from my table of results above. With the formula it will enable to see what the differences are for any square without having to work out the products of the top right and bottom left hand number, and the top left and bottom right hand numbers. I could see that the differences of each square is always the size of square penultimate. Therefore I worked out the formula to be this;
(n-1) ² x 10
To ensure that my formula worked I tested it on a 2 x 32
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