Dudeney was born in the town of Mayfield, East Sussex, England. His grandfather, John Dudeney, was well known as a self-taught mathematician and shepherd; his initiative was much admired by his grandson. Dudeney learned to play chess at an early age, and continued to play frequently throughout his life. This led to a marked interest in mathematics and the composition of puzzles. Chess problems in particular, fascinated him during his early years.
Although Dudeney spent his career in the Civil Service, he continued to devise various problems and puzzles. Dudeney's first puzzle contributions were submissions to newspapers and magazines, often under the pseudonym of "Sphinx." Much of this earlier work was a collaboration with American puzzlist Sam Loyd; in 1890, they published a series on article in the English penny weekly Tit-Bits.
Dudeney later contributed puzzles under his real name to publications such as The Weekly Dispatch, The Queen, Blighty, and Cassell's Magazine. For twenty years, he had a successful column, "Perplexities", in the magazine The Strand, edited by the former editor of Tit-Bits, George Newnes. Dudeney continued to exchange puzzles with fellow recreational mathematician Sam Loyd for a while, but broke off the correspondence and accused Loyd of stealing his puzzles and publishing them under his own name.
Some of Dudeney's most famous innovations were his 1903 success at solving the Haberdasher's Puzzle (Cut an equilateral triangle into four pieces that can be rearranged to make a square) and publishing the first known crossnumber puzzle, in 1926. In addition, he has been credited with inventing verbal arithmetic and discovering new applications of digital roots.
In 1884 Dudeney married eighteen-year-old Alice Whiffin (1866-1945). She later became a very well-known writer who published many novels as well as a number of short stories in Harper's Magazine under the name "Mrs. Henry Dudeney". In her day, she was compared to Thomas Hardy for her portrayals of regional life. The income generated by her books was important to the Dudeney household. After losing their first child in infancy, the Dudeneys had one daughter, Margery Dudeney Fulleylove, who emigrated to the United States.
Alice's personal diaries were edited by Diana Crook and published in 1998 under the title A Lewes Diary: 1916-1944. They give a lively picture of her attempts to balance her literary career and her marriage to her brilliant but volatile husband.
In April of 1930, Dudeney died of throat cancer in Lewes, where he and his wife had moved in 1914 after a period of separation to rekindle their marriage. They are both buried in the town cemetery there.
In addition to puzzles, Dudeney had hobbies including billiards, bowling, and especially croquet. He was a skilled pianist and organist, interested in ancient church music and plainsong. Dudeney was a devout Anglican who regularly attended services, studied theology, and on occasion wrote tracts defending church positions.