Only 10 lockers will remain open in case of 100 lockers.

Logic:

Let only 1 locker-

will end opened.

Let only 2 locker-

series:

open,open

close

Result: 1 is open

let 3 locker

series:

open,open,open

close

close

Result- 1 is open

let 4 locker

series:

o,o,o,o

c, c

c

0

Result: 1 and 4 will be open

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similarly you can check for others

Only perfect squares will be opened between 1 and 100.