On a hot morning in August 1892, Andrew and Abby Borden were killed with a hatchet inside their own home in Fall River, Massachusetts. Their daughter Lizzie Borden was arrested, tried and acquitted - and the case was never solved.

A man knocked down a wall in his basement in 1963 and opened a hidden city cut 85 m into volcanic rock, big enough for 20,000 people. Sixty years later, archaeologists still cannot say who dug the first tunnels.

In January 1931 a stranger left a message sending an insurance agent to an address that did not exist. When he came home, his wife lay dead in the parlour. He was sentenced to hang, then freed, and the Wallace case has never been solved.

Cold War navy microphones on the Pacific sea bed picked up a whale singing at the wrong pitch. Scientists have tracked it across an ocean for decades. Nobody has ever identified the animal making the sound.

Between 1918 and 1919 someone broke into the homes of Italian grocers in New Orleans at night and attacked the sleeping families with their own axes. The police arrested the wrong people, a letter promised to spare every house playing jazz, and the Axeman of New Orleans was never identified.

In 1964 a survey party landed on the one flat ledge of Bouvet Island and found an open boat with no markings, and a pair of oars laid out on the rock 90 m away. No people, no camp, no bodies. Two years later the boat was gone.

Between 1912 and 1914 three brides drowned in their baths, in three towns, with three verdicts of accident. Nobody connected them until a grieving father opened a Sunday paper and recognised the shape of his daughter's death.

In January 1947 a 22-year-old woman was found dead in an empty lot in Los Angeles. The newspapers called her the Black Dahlia. 750 investigators, 500 false confessions and 79 years later, nobody has ever been charged.

In 1898 a Swedish farmer in Minnesota pulled a stone from the roots of a tree. It claimed Norse explorers reached the heart of America in 1362. The experts called it a fake within months. Nobody has ever proven who carved it.

In 1983 a 16-year-old walked to her boyfriend's house in an English village and never arrived. Her murder was the first case ever shown on Crimewatch. 25 years later, a traffic stop of a young man not yet born in 1983 gave the police a name.

In the autumn of 1888, 5 women were murdered in the poorest streets of London. 2,000 people were questioned, 300 were investigated, and the man the newspapers called Jack the Ripper was never caught.

In 1965 a fireball seen in 6 states lit the sky, and soldiers sealed a wood in a small Pennsylvania village. The newspaper said an object had fallen. By the next edition, nothing had been found.

In December 1946 an 18-year-old student walked up a Vermont mountain trail in daylight, was seen by several people, and vanished. Hundreds searched. They found nothing at all.

The biggest manhunt Britain had ever mounted spent 18 months chasing a man who did not exist. It took 26 years and a scrap of dried envelope glue to find out who invented him.

In 1992 Rachel Nickell was killed in broad daylight on a London common while her 2-year-old son watched. For 16 years the police chased the wrong man - while the real killer was already in their files.

Something destroyed 4 square km of a walled capital in a single instant and was heard 150 km away. 400 years later, historians still cannot say what it was.

A cargo ship found a small racing yacht drifting with nobody aboard, and 3 logbooks on the table that explained everything except the one thing that mattered.

Several hundred slabs carrying the same strange message are pressed into the roads of two continents. The method is understood. The hand behind it is not.

500 children arrived for a marching band competition. Within hours, 259 people were in hospital and nobody could say why. Forty-six years later the official file still blames fear, and the babies who collapsed have never been explained.

The Inca ran an empire of millions without writing a single letter. They used knotted cords instead. We cracked the numbers a century ago, and everything else on those cords has resisted us ever since.

A 222 metre ship with 29 men aboard vanished from another vessel's radar in a November storm. There was no distress call and no survivor. Fifty years and two official inquiries later, nobody agrees on what sank her.