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Hello,
According to the rules, a wizard learns a # of 1st level spells equal to 3 + Int modifier at level 1. Add to that the two spells you learn when you reach level 2, which can be nothing but first level spells, most people will have 5 + Int Modifier 1st level spells that they know.

My question is, I am starting a game at 10th level. My starting Int after racial modifiers was 19 (+4) but has been increased at 4th and 8th levels to 21 (+5). In addition, I purchased the Headband of Vast Intelligence +6 which gives me an Int of 27 (+8). My question is, how many 1st level spells do I know?

You could make a case that I know 13 (5 + 8). However, I did not have the 4th and 8th level ability score increases when I was level 1. Whether I had the Headband is less clear. It's possible it was an heirloom or something that I had at that level and my character started with.

What do you think?

Thanks in advance.