How to Write a Solution
Name Your Characters
by Mathew Crawford & Richard Rusczyk

A large thin-shelled vehicle for a young fowl that was created by a huge female bird sat on a wall. The large thin-shelled vehicle for a young fowl that was created by a huge female bird had a great fall. All the horses of the great man who lived in a large castle that ruled over the people in the land and all the men of the great man who lived in a large castle that ruled over the people in the land couldn't put the large thin-shelled vehicle for a young fowl that was created by a huge female bird back together again.

Proofs are a lot like stories. When writing a solution your job is tell a math story in a way your audience will understand and enjoy. Instead of writing about 'A large thin-shelled vehicle for a young fowl that was created by a huge female bird,' we call that big egg 'Humpty-Dumpty' and tell the story. Likewise, a well-written proof often involves naming the important quantities or ideas that play a part in the story of your solution. Naming your characters can also help you find solutions to problems, so it's not something you should wait until proof-writing time to do.

When you do name your characters, you name them simply, clearly, and write up front, so the reader knows exactly where to go to find out exactly who this n person is and what that f(x) function stands for.

Here's an example problem:

The solution below is hard to read because the integers and the sums that are the key to the solution remain unnamed.

The solution below is easy to read because the main characters have names. Specifically, we name the integers in the set and the sums of the elements in subsets that we examine. These names allow us to follow the characters throughout the story. They also allow the writer to describe the characters more completely and succintly.

How to Write the Solution:

Call the 100 integers n₁, n₂,..., n_100.

Let S_k = n₁ + n₂ + ... + n_k for k = 1, 2,..., 100.

Case 1: If S₁, S₂,..., S₁₀₀ are all distinct (mod 100), then exactly one of them must be a multiple of 100.

Case 2: Otherwise, the 100 sums, S_k, have at most 99 distinct residues (mod 100) and by the Pigeonhole principle two of the sums, S_k, have the same residue (mod 100).

This means there exist some integers j and k, 0 < j < k < 101, such that

S_k S_j (mod 100); thus,

S_k - S_j 0 (mod 100).

Now, consider the subset with elements n_j+1, n_j+2,..., n_k. The sum of the elements of this subset is

n_j+1 + n_j+2 + ...+ n_k = (n₁ + n₂ + ... + n_k) - (n₁ + n₂ + ... + n_j)

= S_k - S_j 0 (mod 100).

Thus this sum is a multiple of 100 and we are done.