# Pretty Pyramids

Expert problem-solvers are always on the lookout for patterns. Sometimes, just thinking about a problem in the right way can make a tricky formula or proof obvious. But seeing a pattern usually isn't a matter of luck - it a matter of learning how to hunt for regularity and mathematical order purposefully. Some common and famous patterns will appear again and again, so be on the look out for them: arithmetic and geometric sequences, perfect squares, cubes, and other powers, prime numbers and the Fibonacci sequence are a few examples. Hunting down patterns is a skill, and like any other skill, the secret to becoming a master is practice, practice practice!

Here are some tips to get you started:

- When given a formula or pattern, experiment with it. Don’t be afraid to try different things and test some smaller cases.
- Keep track of as much information as you can about every case you work on, and keep your notes well-organized.
- Look for numerical patterns in your notes and diagrams such as arithmetical and geometric sequences.
- When you spot a pattern, figure out why it exists.
- As you practice with many problems, you'll gradually build up a tool kit of both types of patterns and problem-solving techniques for finding those patterns hidden in challenging problems. If you get stumped on a particular problem, look at the solution and be sure to study the technique so that you can add it to your toolkit.

Evaluate:

\[ 1 + 2 + 3 + \dots + 8 + 9 + 8 + \dots + 3 + 2 + 1. \]

This problem can be solved by route, but it can also be solved with a pattern that will also work for much larger cases of our pyramid.

Look at the symmetry in \( 1 + 2 + 3 + \dots + 8 + 9 + 8 + \dots + 3 + 2 + 1 \) -- we have only one 9, but 2 copies of every other number less than 9, so maybe we can pair them up. Also, notice that \( 1, 2, 3 \dots\) is a simple arithmetic sequence, increasing by 1 at each step. Because of this, it can be "folded over" on itself in pairs that add up to the same value \((9)\) in each column:

\[ \begin{array} {c c c c c c c c c c c c c c c c } & 1 & + & 2 & + & 3 & + & 4 & + & 5 & + & 6 & + & 7 & + & 8 \\ + & 8 & + & 7 & + & 6 & + & 5 & + & 4 & + & 3 & + & 2 & + & 1 \\ \hline & 9 & + & 9 & + & 9 & + & 9 & + & 9 & + & 9 & + & 9 & + & 9 \\ \end{array} \]

In total, there are 9 sets of 9 (including the original central column of the pyramid). Therefore, the answer is \(9^2 = \fbox{81}.\)

In Basic Mathematics, you will learn about:

- Recognizing Patterns: This is the bread and butter of being a mathematician. A quick recognition of patterns allows you to form hypothesis and ideas, paving a route to attacking the problem.
- General Term Pattern Recognition: After you find a pattern, the next step is to describe it rigorously, and in such a way that you can easily calculate even the cases of the pattern that have very large numerical values.

These basics will prepare you for more advanced study of mathematical patterns such as:
- Arithmetic and Geometric Progressions
- Perfect Squares, Cubes and Powers
- Prime Numbers and their distribution

- The Fibonacci Sequence