Hundreds of thousands have tried but so far no one has managed to crack it.
And with the deadline of midnight on 31 January looming, there are fears no one will.
The director of GCHQ, one of Britain's intelligence and security agencies, sent out Christmas cards with a cryptographic challenge.
Participants had to fill in a grid-shading puzzle to unveil a picture in the first of series of increasingly complex challenges.
But on Wednesday GCHQ revealed that although around 30,000 players have reached the final stage, none have successfully completed the puzzle.
:: Download the puzzle here.
GCHQ's website attracted unprecedented levels of visitors eager to take the challenge over the Christmas period.
Nearly 600,000 people successfully completed the opening stage of the puzzle - a nonogram.
When completed correctly it creates a "quick response" code which leads to the next stage of the challenge.
The puzzle was designed to amuse recipients of director Robert Hannigan's card and encourage them to make a donation to the NSPCC, his chosen charity this year.
Evidently taking pity on them, Mr Hannigan has offered the tiniest glimmer of hope to frustrated puzzlers.
"With a few days to go no one has cracked it all yet, so my one and only clue is: it's not as abstract as you think," he said.
Dr Gareth Moore, puzzle author and creator of the website Brained Up, told Sky News the puzzle is "only difficult in appearance" and is just "a question of being patient".
"In actual fact, a picture puzzle like this can actually be solved by considering each row and each column one by one," he said.
The complete solution to the GCHQ puzzle will be published on its website in early February.
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