The fireplace puzzle in 007 First Light blocks Bond's chase through the Grand Carpathian Hotel during the All The Time In The World mission. A blond suspect slips through a hidden passage behind the dining room fireplace, leaving you to figure out which three of the animal-marked buttons on the wall open it.
Where the puzzle appears
The puzzle sits in the grand dining room on the upper floors of the Slovakian hotel. After chasing the blond man, you arrive to find the room empty and a row of animal symbols mounted above the fireplace. The mechanism is the only way forward.

The clue on the laundry cart
In the center of the dining room, there is a laundry cart with a torn book page resting on it. Inspect the page. The illustration shows a single creature whose body is stitched together from three different animals, which is a chimera in classical mythology, made of a lion, a goat, and a serpent.
Those three animals are the answer. The buttons above the fireplace include several decoy creatures, but only the lion, snake, and ram (goat) match the chimera on the page.
Solving the fireplace puzzle
Step 1: Walk to the laundry cart in the middle of the dining room and interact with the torn page to register the chimera clue.
Step 2: Approach the panel of animal buttons above the fireplace.
Step 3: Press the lion, the snake, and the ram (goat). Order is not required, only the correct three selections.
Once the third correct button is pressed, the fireplace mechanism unlocks, and the hidden passage opens, letting Bond continue his pursuit into the next area.

How to confirm it worked
The back of the fireplace slides away, and a passage is revealed behind it. If nothing happens, you have selected a wrong animal somewhere in the sequence. Re-read the chimera illustration and try again, sticking strictly to lion, snake, and ram.