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Punoxysm comments on What false beliefs have you held and why were you wrong? - Less Wrong Discussion
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I once believed that six times one is one.
I don't remember how it came up in conversation, but for whatever reason numbers became relevant and I clearly and directly stated my false belief. It was late, we were driving back from a long hard chess tournament, and I evidently wasn't thinking clearly. I said the words "because of course six times one is one." Everyone thought for a second and someone said "no it's not." Predictable reactions occurred from there.
The reason I like the anecdote is because I reacted exactly the same way I would today if someone corrected me when I said that six times one is six. I thought the person who corrected me must be joking; he knows math and couldn't possibly be wrong about something that obvious. A second person said that he's definitely not joking. I thought back to the sequences, specifically the thing about evidence to convince me I'm wrong about basic arithmetic. I ran through some math terminology in my head: of course six times one is one; any number times one is one. That's what a multiplicative identity means. In my head, it was absolutely clear that 6x1=1, this is required for what I know of math to fit together, and anything else is completely logically impossible.
It probably took a good fifteen seconds from me being called out on it before I got appropriately embarrassed.
This anecdote is now my favorite example of the important lesson that from the inside, being wrong feels exactly like being right.
One of the smartest people in my high school spent a class arguing that a there were 4^20 possibilities for a sequence of 4 amino acids, when in fact it was 20^4. Not quite as elementary as yours, but our brains all play tricks on us.
Wikipedia says there are 500 known amino acids. I take it you were talking about a domain involving fewer potential amino acids?
The 20 amino acids involved in protein creation.
Presumably the slightly greater than twenty that form the universal basis of biological protein polymers. In other words, the amino acids explicitly coded for by DNA.
I once spent about 20 minutes in class trying to justify the claim that a cubic meter of water weighs one ton, not 100 kilograms.
As you may note, this is not a false belief; it's relevant because I spent a significant fraction of this time wracking my brain trying to jog it free to see if I was being dense (pun intended) like you're describing (I have pretty strong reasons to believe I didn't remember this incident backwards as a case of self-justification by retcon)
You just made me sufficiently confused to quickly google the volume of a cubic meter and the weight of a litre of water. I suspect this is a good habit to reinforce but I can't help feeling silly.