I looked on some other Blogger blogs and tried their permalinks. I got mixed results; some worked and others didn't. I looked on Blogger Forum and found that others were having the same problem. Dew's permalinks worked, so I decided to ask him. He gave me a simple overview of how permalinks work, but I was still missing something. After examining my template code I figured out what the problem was.
Titles. Blogger permalinks use titles as an anchor to link back to posts. That's why a link to an old post went directly to the title of the post. I was creating my own titles using the <h4> tag, others whose permalinks didn't work were using the <strong> tag. To make Blogger permalinks work I needed to create a title that would be recognized by Blogger. Following these simple instructions made my permalinks work.
"To enable titles, go to the Settings | Formatting tab. Scroll down to the settings field labeled 'Show title field', and set it to "Yes."
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"When you finish selecting 'Yes' from the drop-down menu, click on the 'Save Changes' button at the bottom of the page. Once saved, you will see a prompt diplayed that reads "Your changes have been saved." in orange at the top of the page. Do not leave this page until you see the message displayed."
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After about an hour and a half I had fixed every post I had ever made, and now my permalinks work.