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Doomsday, as told in The Last Olympian, ends up taking place exactly on Percy's birthday (whatever that is, i don't have the books in front of me).

However, due to the 5 days lost in the Lotus Casino (in The Lightning Thief), shouldn't Percy have been 15 years and 360 days old, and not exactly 16?

Were did the extra 5 days come from?

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The Lotus Casino only prevents the passage of time for those inside it. Outside, time is still passing at the normal rate. Therefore, Percy's birthday would still be on the same date regardless.

For instance, if Percy had stayed inside the casino for 100 years, he would still be 100 years older when he comes out - he just wouldn't have aged 100 years. Essentially, he would have had (and missed out on) 100 birthdays in that time without realizing it.

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