1. Remember to let your child do his homework by himself.
2. Remember to let your child do his homework on his own.
3. Remember to let your child do his homework himself.
4. Remember to let your child work things out for themselves.
Do all of them mean same? Identify if their's some difference.
#4 is grammatically incorrect. "Child" is singular, "themselves" is plural.
#1, 2, and 3 all mean the same thing.
As far as I know, in modern English themselves, them, they are used in place of himself/herself., him/her, he/she. Since a child may be a boy or a girl, that's why I've used themselves instead of of a singular pronoun. Anyway, I'd like to refer to the website on which I came across this sentence
Hope you find a little time to take a look at this site.
Number 4 is incorrect. Like "To Imran" said, "themselves" is plural. The one I'd probably use would be #1, although #2 and #3 are okay too.
"Remember to let your child do his homework by himself." (substitute "her" for a girl)
"Remember to let your child do his/her homework by himself/herself."
In everyday speech you'll probably hear something like: "Remember to let your child do their homework by themself." Here we encounter a substitution for the whole "he/she" dilemma, as well as the use of "themself," which technically isn't in the dictionary, but is used.
Here's a dictionary in which "themself" appears but is labled as a shibboleth being "limited to Vulgar English speech or imitations of it" also occurring "as an unfortunate result of trying to avoid using a gender-explicit reflexive pronoun ..." Thus use it at thy peril.
I qouted this sentence from a British website I already mentioned in my last version. Have you gone over it? This thing is pretty confusing to me. Please take a look at that website if you don't mind.