Library Reading Room Spots

AS, There are some readers such that in each occupied reading spot there are
different numbers of readers. 1+2+3+4+5+6+7+8= 36 readers so, 8 spots have
different readers. remaining are 23-8= 15. 15 spots are empty.

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The key here is realizing that the tables that aren't empty each have a DIFFERENT number of readers. There's only one way to do this.

The very least number of people you could have at a table is just 1. The next highest number that's different than that is 2, so another table can have 2 people. Then another table might have 3, and so on. But if you add up 1 + 2 + 3 + and so on up to 8, you get 36 people. So if we seat 36 people at different tables such that there's a different number at each table, the number of tables we used is 8. And since the table with the most number of people has eight people, then there are enough seats at the table, so we don't have to worry about that.

That means 23 - 8 = 15 tables are empty.