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See how many squares would be needed in order to construct any cross-built up in the way described in the investigation.

Extracts from this document...


Khadija Patel                


Part 1-

Aim: -

My aim is to see how many squares would be needed in order to construct any cross-built up in the way described in the investigation.


  • To use simple techniques first and begin with a simple number.
  • Use examples, diagrams and label them.
  • To record all results found in a table
  • Extend investigation to 3-D


To create a larger cross, surround the shaded squares with white squares. To begin my investigation I will start off with a simple number of squares.

Pattern 1- Number of shaded squares I have chosen is 1.

   Total number of squares = 4 + 1

                                        = 5

White squares = 4

By using the method stated in the investigation, I have constructed a larger cross. By using the same method I will again construct the next sequence of crosses.

Pattern 2-                                 Pattern 3- Total number of squares = 25

                                                        White squares = 12        

Total number of squares = 13

White squares = 8

Pattern 4-

                                                                 Total number = 41

                                                                White squares = 16

I have noticed that the squares going down are odd 1+3+1=5. 1+3+5+3+1=13. In each case 2 has been added to each pattern.1+3+5+7+5+3+1=25, like the previous pattern the addition has always been 2 to each digit.

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So in result the general formula for this would be n x 4 or 4n. I will prove this formula to be correct by choosing a formula at random. I have chosen the number 10. 4n= 4 x 10= 40 white squares.

I am now going to search for the formula the formula for the dark squares. I will the information from the result table and differences (from pg2) I have found that the number of shaded squares equals to the total from the previous squares. To see this look at page 2. I have decided to draw a new table similar to that of the previous page.

Pattern            Total Squares           Shaded Squares             White Squares               Difference in


1                  5                          1                           4                        


2                 13          5                           8                        


3                 2513                           12


4                 41        25                            16

Throughout the investigation I will use the following symbols for the formulae, these will not change. n = pattern number, d= dark squares, w = white squares.

Trying for a formula for shaded squares

I have found out the general formula to find the number of shaded squares which is 2nsquared – 2n + 1, because this is a quadratic form of ax +bx+c, I found a second difference between the pattern of shaded squares.

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Drawing simple diagrams at first can make it easier to find the answer for the investigation. Another good way is using different tables which hold a record of your findings take a part in finding an equation by recording the information on the table to justify your results. Using different techniques to tackle a problem is beneficial because of the reason that I drew a number of diagrams and a variety of assorted result tables to explain what I did. I also used different symbols and a plan to explain what I had done.

Equation 2n – 2n + 1 is very similar to equation 2n + 2n +1. I noticed this immediately after I had found the second equation. The difference between the two is the symbol before 2n. One is – (negative) and the other + (positive). This was because I had to add combine two equations together. The formula for white squares 4n and the formula for shaded squares 2n – 2n + 1, which in result gave 2n + 2n + 1.

The first equation I found was a linear equation, as 4n can be written as 4x. Equation 2 is a quadratic as it is in the form ax + bx+ c. the same with equation 3. I am planning to extend this investigation to 3D.

...read more.

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