You begin writing down the letters of the alphabet in this pattern: You write A one time, B two times, C three times, D four times, and so on. What letter is 100th on the list?

To find the 100th letter in this unique sequence, we need to sum the number of times each letter appears. The pattern dictates that A appears once, B twice, C three times, and so on. This creates a cumulative count as we move through the alphabet. For instance, after A and B, we've written 1+2=3 letters. After A, B, and C, we've written 1+2+3=6 letters. We're looking for the letter that encompasses the 100th position in this growing list.

We continue this summation. When we reach the 13th letter of the alphabet, which is M, we have accounted for 1+2+3+...+13 letters. The sum of the first 13 natural numbers can be found using the formula n*(n+1)/2. Plugging in 13, we get 13 * (13 + 1) / 2, which equals 13 * 14 / 2, or 91. This means the 91st letter on the list is the very last M. The next letter, N, is the 14th letter of the alphabet. It will appear 14 times, starting from the 92nd position. Since 100 falls between 92 and 105 (91 + 14), the 100th letter written is N.

This problem beautifully illustrates the mathematical concept of triangular numbers. A triangular number is the sum of all positive integers up to a given integer. For instance, the 3rd triangular number is 1+2+3=6. In our case, the 13th triangular number is 91, and the 14th is 105. These numbers get their name from the fact that they can form an equilateral triangle when represented by dots. They have fascinated mathematicians for centuries, appearing in various areas from combinatorics to the study of polyhedra, and this simple alphabet puzzle taps into a fundamental mathematical sequence.