Taman Shud Case

Who was the Somerton man, how did he die, and what do these strange codes found on a book connected to the man mean? An unidentified male body was found on Somerton Beach in Adelaide, Australia in 1948 wearing a sweater and coat despite the hot day, carrying no identification. There were no clues as to his identity and dental records and fingerprints matched no living person.
Kryptos

It’s like a tease, standing outside the headquarters of the CIA in daily view of some of the nation’s brightest cryptographers yet eluding them for years. The Krytpos monument is a sculpture by artist Jim Sanborn bearing an encrypted message divided into four sections, three of which have been solved since its installation in 1990. With misspellings in the code intact, the first part reads, “Between subtle shading and the absence of light lies the nuance of iqlusion”, and the second part references some invisible buried treasure ostensibly located some 200 feet from the statue itself.
Zodiac 340 Letter

The Zodiac Killer – whoever he is or was – is known just as much for the incredibly complex coded letters he sent to the Bay Area press as for his brutal unsolved murders. While some of his taunting ciphers have been solved, this 340-character message sent in 1969 has never been cracked.
Dorabella Cipher

Sent by cipher enthusiast Edward Elgar to his friend Miss Dora Penny, the Dorabella Cipher seems upon viewing like it might not mean anything at all. But this string of strange symbols, made up of semicircles in various configurations, has been the subject of unfruitful study for over a century. Musicologist Eric Sams claimed to have solved it, but his methods are unproven and his translation is 22 characters longer than the cipher. Another speculation is that the code is not text, but a melody.
Chaocipher

If an autobiography detailing the author’s memories of James Joyce seems like a strange place to find an uncracked cipher, that’s because it is. J.F. Byrne inserted his cryptosystem challenge into the book “Silent Years”, offering $5,000 to whoever solved it. At least three people know how Byrne’s Chaocipher – a machine small enough to fit into a cigar box used to encrypt the message – actually works, but no one has ever solved the code.
RSA Crytographic Challenges
D’Agapeyeff Cipher
Alexander d’Agapeyeff wasn’t even a cryptographer – having previously written a book on cartography, he decided to tackle cryptography in his second book, “Codes and Ciphers”, in 1939. On the last page of the book, he included a modest cryptogram “upon which the reader is invited to test his skill.” But modest or not, d’Agapeyeff’s code has remained uncracked for 70 years, putting this amateur into the same league as the world’s most gifted cryptographers.
The Beale Papers
Shugborough Inscription

Chinese Gold Bar Cipher

Phaistos Disc
“There’s very little that we actually know for sure about the Phaistos Disc. It’s made of clay – check. It dates back to the second millennium B.C.E. – maybe. But its origin, meaning and purpose remain shrouded in mystery. Discovered in Crete, the disc is features i241 impressions of 45 distinct symbols, some of which are easily identifiable as people, tools, plants and animals. But because nothing else like it from the same time period has ever been found, archaeologists haven’t been able to provide a meaningful analysis of its content.”
via:
weburbanist.com