I recently found an excellent presentation on Vishal Lama’s blog by Timothy Gowers regarding the importance of Mathematics. From my experience, the general public has a really skewed view of Mathematics. The thought of doing Mathematics for recreation seems absolutely ridiculous to most people. This lecture excellently portrays the beauty of Mathematics and applicability of all areas of Mathematics.

If you want to see the presentation, you can see it on YouTube.

I am a bit late posting a solution to this problem because I was on holidays. However, I am now back to my regular routine. As for finding the area of the Koch Snowflake, Reasonable Deviations has already posted a solution to this problem. Also, Rod has posted a variation of the problem which finds the perimeter of a Koch Snowflake. Since the problem has already been solved, I will only do a quick overview of the solution.

Answer: The easiest way to solve solve this problem is to calculate the area added to the Koch Snowflake after each iteration.

Area of the first iteration can easily be calculated by using Pythagoras.

Using the solution for the first iteration, we can easily calculate the area of the second iteration. The area will be the area of the original equilateral triangle plus the area of 3 smaller equilateral triangles.

The area of the third iteration would, therefore, be:

If we continue doing this, we will get the following summation.

Since the summation is a geometric series, we know that when .

Therefore,

Question: A Koch Snowflake is a fractal which can be built by starting with an equilateral triangle, removing the inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process indefinitely.

Find the area of a Koch Snowflake when the sides of the starting equilateral triangle has the length .

Question: If we take the 12 days of Christmas literally, what is the total number of items you will have received at the end of the twelve days? For example, on the first day you have received one item (1 partridge), on the second day you will receive three items (1 partridge, and 2 doves), on the third day you would receive six items (1 partridge, 2 doves, and 3 french hens), etc.

Solution: From my analysis, I have come up with three different ways to count the total number of items you will have received at the end of the twelve days.

1) The first method to solve this problem is to use brute force. On the first day, we will receive item. On the second day, we will receive items. On the third day, we will receive items. Therefore, on nth day, we will receive items. Since we want the number of items received for all 12 days, we need to calculate the following summation: . We can easily calculate the answer with the following perl script:

$soln = 0;
 
for ($i = 1; $i < 13; $i++) {
   for ($j = 1; $j < $i+1; $j++) {
      $soln = $soln + $j;
   }
}
print $soln;

2) The second method to solve this problem is to use recursion. On the nth day, you are going to get n items plus the number of items you got the day before. Therefore, if is the number of items you receive on the nth day, will follow this recursive pattern:

In order to calculate the number of items received on all 12 days, we need to calculate the following summation: . The solution can be calculated with the following script:

$soln = 0;
$prevDay = 0;
 
for ($i = 1; $i < 13; $i++) {
   $prevDay = $prevDay + $i;
   $soln = $soln + $prevDay;
}
 
print $soln;

3) The third method to solve this problem is to group items by type instead of days. For example, you will receive 1 partridge for twelve days, 2 doves for eleven days, 3 french hens for ten days, etc. In general, this means that you will receive n items for (13 - n) days. This would mean that the total number items you would receive would be . This summation can be calculated by the following script:

$soln = 0;
 
for ($i = 1; $i < 13; $i++) {
   $soln = $soln + $i*(13- $i);
}
print $soln;

In the end, however, we come back to the same solution which is 364.

I am finally on Christmas vacation which has gotten me in the Christmas spirit. I thought it would be fun to have a problem related to Christmas. The problem is very easy… which is great because I’m on vacation. In order to solve this problem, you will need to be familiar with the lyrics to the 12 days of Christmas.

Question: If we take the 12 days of Christmas literally, what is the total number of items you will have received at the end of the twelve days? For example, on the first day you have received one item (1 partridge), on the second day you will receive three items (1 partridge, and 2 doves), on the third day you would receive six items (1 partridge, 2 doves, and 3 french hens), etc.