Inspiration |
I dont remember the names of any of the videos. Let me just say that now. All i remember are the messages. The most outstanding of all of these, was probably
"Failure is a learning experience" This is a far cry from my traditional view of "failure is a failure, and you should feel bad for failing." So that kinda helped. Not else really spoke to me even half as much as that video did. It actually changed some thought process, whereas all the others made little to no impact.
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THE SQUARECASE PROBLEM.
The Squarecase problem, as I called it, was a math problem we were assigned. Starting with one square, with every figure the amount of squares grew. How did I solve it?
By every figure, I noticed something. On figure two, two figures were added to the side of the previous figure one. On figure three, three figures were added to the side of the previous figure two. This goes on and on. Knowing this, I came to a conclusion; this object is shaped rather like a triangle; Two equal sides and one unequal side. So I brought up an old skill from my past.
AREA OF AN EQUILATERAL TRIANGLE EQUATION; (X*X)/2.
Not giving up, I counted the area of all the figures up to four, cracking this problem.
When I counted the squares in each figure, I got the following.
Figure Number 2 = 3 Squares
Figure Number 3 = 6 Squares
Figure Number 4 = 10 Squares
When I counted the squares in each figure, I got the following.
Figure Number 2 = 3 Squares
Figure Number 3 = 6 Squares
Figure Number 4 = 10 Squares
I proceeded to use the area of a triangle equation. To do this, I used how many squares composed the sides of the triangle (each time correspondent with the figure!) When I used the area of a triangle equation, I got...
Area Number 2 = 4
Area Number 3 = 4.5
Area Number 4 = 8
I felt rather demoralized. My method was off. I scrapped it for a while and moved to other methods of solving the problem. These methods are eternally lost to the drawing board, as I have no idea how many pieces of paper I destroyed that contained failed attempts. Eventually, about to simply give up, I went back to the area of the triangle and finally.. I noticed something different this time.
Figure Number 2 = 3 Squares - Area Number 2 = 2
Figure Number 3 = 6 Squares - Area Number 3 = 4.5
Figure Number 4 = 10 Squares - Area Number 4 = 8
From a naked eye, there appears to be no correlation. Thats where it is wrong though. Simply out of pure desperation for anything that might work, i decided to divide the figure number by the area number. Heres where things got oddly interesting for me.
Result 2 = 1.5 = 3/2
Result 3 = 1.33333= 4/3
Result 4 = 1.25 = 5/4
All the sudden, all that time I spent in elementary and middle school memorizing what fractions correlated with decimals paid off. I could tell that the decimals that I was recieving in response were all fractions! Converting from decimals to fractions, I noticed something. Each time, the distance between the area number and the figure number decreased by one denominator. So realizing I had finally gotten onto something, I wrestled my way into making it an equation.
First, I started with..
(X*X)/2
This was the equation for the area of a triangle. Then I took into account the changing fraction.
X+1
-----
X
This equation would always end up with the ever-changing fraction, which is the difference between the area number and the figure number. I decided to smash them together into one equation.
Then, we had the Area of a triangle equation to make into one equation. This is represented by "(x*x)*.5", which is identical to the equation; just worded oddly.
Combining this with the "changing fraction" equation I listed above, the solution would be found below.
Area Number 2 = 4
Area Number 3 = 4.5
Area Number 4 = 8
I felt rather demoralized. My method was off. I scrapped it for a while and moved to other methods of solving the problem. These methods are eternally lost to the drawing board, as I have no idea how many pieces of paper I destroyed that contained failed attempts. Eventually, about to simply give up, I went back to the area of the triangle and finally.. I noticed something different this time.
Figure Number 2 = 3 Squares - Area Number 2 = 2
Figure Number 3 = 6 Squares - Area Number 3 = 4.5
Figure Number 4 = 10 Squares - Area Number 4 = 8
From a naked eye, there appears to be no correlation. Thats where it is wrong though. Simply out of pure desperation for anything that might work, i decided to divide the figure number by the area number. Heres where things got oddly interesting for me.
Result 2 = 1.5 = 3/2
Result 3 = 1.33333= 4/3
Result 4 = 1.25 = 5/4
All the sudden, all that time I spent in elementary and middle school memorizing what fractions correlated with decimals paid off. I could tell that the decimals that I was recieving in response were all fractions! Converting from decimals to fractions, I noticed something. Each time, the distance between the area number and the figure number decreased by one denominator. So realizing I had finally gotten onto something, I wrestled my way into making it an equation.
First, I started with..
(X*X)/2
This was the equation for the area of a triangle. Then I took into account the changing fraction.
X+1
-----
X
This equation would always end up with the ever-changing fraction, which is the difference between the area number and the figure number. I decided to smash them together into one equation.
Then, we had the Area of a triangle equation to make into one equation. This is represented by "(x*x)*.5", which is identical to the equation; just worded oddly.
Combining this with the "changing fraction" equation I listed above, the solution would be found below.
And now with this, I solved the questions. What would the figure look like at figure 55? it would have 1540 squares! And at figure 19, how much squares would you have? Exactly 190! The graphs I used in hand with the equation are seen below.