Do offenders admit their crime(s) to their lawyers?

If a person truly commits a crime and is subsequently arrested, does he tell his lawyer the truth, OR, does he deny his crime to his lawyer? Considering the attorney client privilege, I think I would admit it to my lawyer as the more of the truth he knows, the more likely he can prepare strong defense. If I lie, he has to do some guessing which would seem to put him in a weaker position. Take OJ for example, (and assume he committed the crimes); Did he say “Johnny, yep, I did it, here’s where I hid the knife, .....” OR, did he fold his arms in front of him and say “Nope, I didn’t do it. Now defend me.”?