Synonyms for Puzzle out:

n.

get through, overcome, cope, shrug off, surmount, handle, manage, carry off. figure out (noun)
decipher.
puzzle out (noun)
figure out, solve, lick, work out, work.

v.

clear up (verb)
solve, decipher, figure out.
do (verb)
work out.

Other synonyms:

decipher. clear up
figure out.