Thinking about the Pythagorean Number System is a step in the right direction. Writing in the snow proved to be problematic though. The wind was still blowing just enough to consistently blow snow over your thinking and fill it in. Though, apparently, that doesn't stop you from cracking the code.

Had you been talking out loud during this, Sam's eyes would have lit up at the point THE FIRE SIGNS was revealed, being the one that was good at Astronomy and would have immediately been able to tell you, if she had the ability to speak in this frozen world, which constellations to select. However, as you think your way through everything in silence, she too remains silent, shivering and getting colder and colder.

Something wasn't right though. Touching the appropriate constellations on the lock with your finger does nothing. Sam remains clocked in her cold prison. Was there something you missed? Or did you need to touch the constellations with something else?