What can you see?
In the giving image, we can clearly see a large square containing 12 pink triangles, 12 blue triangles with one pink square in the middle of the square.
How do you think the pattern was made?
The pattern is made by joining the midpoint of the sides of the smallest square in the image, to create a smaller square. Each odd square is coloured pink and each even square is coloured blue.
What proportion of the total area of the square is taken up by the small square in the middle?
To solve this, we can make use of simple algebra. If we consider one side of the square as x, the total area of that square would be x^2. To find the area of the second largest square, we can use the Pythagoras theorem ( a^2 + b^2 = c^2), as we can take c as the length of the side of the square. We can infer that a and b is equal to x/2, as they are both half of the side of the largest square. Once we calculate this, we see that the area is (x^2)/2. We can continue to do this to find the area of the third largest square, which will be x^2/4. When we look at the pattern forming, we can see that the area is getting halved each time. Hence the total areas for each of the squares will be:
x^2, (x^2)/2, (x^2)/4, (x^2)/8, (x^2)/16, (x^2)/32 and (x^2)/64
The smallest square in the middle has an area of (x^2)/64. Hence the proportion of the total area that is taken by the smallest square in the middle is 1:64, or it takes up a 64th of the total area of the square.
If this process could be continued forever, what proportion of the image would be coloured blue?
To solve this problem, I first tried finding the proportion of blue at each stage of forming the image (using the interactive feature). For example, at the 0th stage, the proportion of blue in the image would be zero. In the first stage (i.e when the first blue square is formed), the proportion of blue would be 1/2 of the total image as the area of that blue square is (x^2)/2. However, I realized that the proportion of blue decreases with the addition of every new pink square, and hence I had to subtract the area of the new square from the proportion of blue. Whenever a new blue square is added, we must add the area of that square to the proportion of blue that existed in the previous stage. The structure of the pattern then became:
0, 1/2, 1/2 - 1/4, (1/2 - 1/4) + 1/8, ...
We can infer that the previous value of the pattern is always being subtracted or added by a particular number. This number can is always 1/4, 1/8, 1/16, ... Therefore we can see the number being added or subtracted is always increasing in the negative powers of 2.
However, the negative powers of 2 are either being added OR subtracted. To represent this each time, we can use; -1^n.
When we add all these elements to generalize the proportion of blue at each stage, we get:
(n-1) + (-1)^t-1 x (2)^-t
Where t is the term or stage of the pattern
And n is the value of t at each stage
Let's see if this works:
for the first stage, where t = 1, n - 1 = 0
0 + (-1)^0 x 2^-1
=1 x 1/2
=1/2
For the second stage, where t = 2, n-1 = 1/2
1/2 + (-1)^1 x 2^-2
=1/2 -(1) x (1/4)
=1/2 -1/4
=1/4
For the third stage, where t = 3, n-1 = 1/4
1/4 + (-1)^2 x (2)^-3
=1/4 + (1)(1/8)
=1/4 + 1/8
=3/8
Therefore, this formula works and shows the correct proportion of blue at each stage of the formation of the image.