## THE 43rd FIGURE:

Solving a Pattern

**Editor's Note: Will S. ('17)**explains the process by which he found the solution to a tricky geometric pattern.**Pattern**

**Question**

**: HOW MANY CIRCLES ARE IN THE 43rd FIGURE?**

The 43rd figure in this pattern contains 3787 circles. I got this by first noticing that each figure has a box in it. In figure one that box’s dimensions are 1×2, in figure two the box’s dimensions are 2×3, in figure three the dimensions are 3×4, and so on. The boxes I am talking about are the ones boxed in these pictures:

In figure 1, there are two 1×2 rectangles, in figure 2 there are two 2×3 rectangles, and in figure 3 there are two 3×4 rectangles. Also in all of the figures there are 3 additional circles not in the rectangles. Once I noticed this pattern I realized that the height of the lower rectangle and width of the higher one is always equal to the figure number. For example, the height of the bottom rectangle in figure 3 is 3. This allowed me to make an equation for the pattern where n is the figure number: Sum = 2(n(n+1)) + 3. I tested my equation out for the first three figures and it worked, and so when I plugged 43 in for n I got 3787.

Originally, I actually went about finding the sum of the circles a different way. I saw that each figure had a hollow square, but two sides were missing except for one extra circle.I also noticed that besides the square with some missing circles, each figure had a rectangle attached to each side of the square that is there. I saw that the height of the bottom rectangle was always 1 less then the figure number, and the width was always 1 more then the figure number, and opposite for the top rectangle. Based on that information I was able to make an equation for the rectangles on the outside pretty easily and this is what I made for that: 2((n-1)(n+1)), but I realized that the other part of the equation was going to be pretty long and so I decided to try and look for a new pattern that might be simpler.

I thought that this was an appropriately challenging problem to work on because I really like to do things like this involving finding patterns, and this pattern looks pretty complex and interesting. When I was scrolling through the patterns on the website, I clicked on it I instantly started trying to find what the pattern was and it was difficult to hit the nail on the head because there is a lot going on in the pattern, so I decided to do it.

Originally, I actually went about finding the sum of the circles a different way. I saw that each figure had a hollow square, but two sides were missing except for one extra circle.I also noticed that besides the square with some missing circles, each figure had a rectangle attached to each side of the square that is there. I saw that the height of the bottom rectangle was always 1 less then the figure number, and the width was always 1 more then the figure number, and opposite for the top rectangle. Based on that information I was able to make an equation for the rectangles on the outside pretty easily and this is what I made for that: 2((n-1)(n+1)), but I realized that the other part of the equation was going to be pretty long and so I decided to try and look for a new pattern that might be simpler.

I thought that this was an appropriately challenging problem to work on because I really like to do things like this involving finding patterns, and this pattern looks pretty complex and interesting. When I was scrolling through the patterns on the website, I clicked on it I instantly started trying to find what the pattern was and it was difficult to hit the nail on the head because there is a lot going on in the pattern, so I decided to do it.