How do you solve a problem in a HP Laserjet 3600n printer like the cartridges are not engaged

Did you make sure the cartridges clicks in place? There are while plastic locks on each cartridge and hopefully they are not broken. Not expensive to purchase but major teardown to replace.

Could be a broken "cartridge lock" or a bad "cartridge drive motor". Someone is going to have to look at it.

Replace it for a new cartridges. If continues the problem check the laser, clean it.

NO.
All cartridges must be not empty and in working condition to print.

The cartridge is not being seen by the machine....clean the metal contacts on the cartridge and within where the cartridge seat with electrical cleaning solution or isopropyl alcohol...if this is not the problem then look at the front door and make sure is properly aligned if not it can cause this error.

that problem is disaligned main gear check the main gear then clean all the contacts...try this it works for me...

www.fixya.com/.../cartridges_not_engaged_error_10_92_03_hp_3800n

Do replace the yellow cartridge first and see what happens.

defective cartridge or problem whit laser/scaner assy - replace cartridge ( to solve this just turn back your old cartridge and try to print something to be sure before you buy new one) - visit HP servise for laser/scanner assy

check the locking tabs on the cartridge, sounds like they are not locking down the cartridge.