Pull the chain to release the door for the guide in the room below. He will start the oil on fire which lights up certain hieroglyphs. The lower room with the jeweled chain is to turn off the fires. Go down to the lower room and pick up the shotgun, pull the chain once to turn off the fires and open the doors. Back in the room with the cutouts, you need to jump from to all of the cut-out squares without stepping on any of the other squares.
All jumps are simple jumps.
You must not stray outside of the square including the brown outline. If you fail, go back and turn off the flames in the lower room, and try again. Once you succeed, the door will open, pick up the Timeless Sands, which opens the other door. Go out the door and find yourself at the bottom of the Sphinx. Note where the guide travels.