The history of Sums
Play SumsSums is the crossword with the words taken out. It is older than Sudoku, and for a long time it was the bigger of the two.
An American puzzle
The crossword of numbers grew up in American puzzle magazines, next to the word crosswords it borrows its grid from. By the 1960s it was a regular feature of the magazines that also printed logic problems and number games, under an English name one of their editors gave it.
Japan's favourite
In 1980 the Japanese puzzle publisher Nikoli brought it to Japan, under a name meaning "addition cross", soon shortened. It caught on quickly: for years it was the most popular logic puzzle in Japanese print, and only in 1992 did Sudoku, another American import that Nikoli had renamed, take the top spot. It is still printed in close to a hundred Japanese magazines and newspapers.
Back to the West
When Sudoku swept British newspapers, editors went looking for the next puzzle, and the crossword of numbers was the obvious one: the same small grid, the same digits, and arithmetic on top. The Guardian printed it first, and the Telegraph and the Daily Mail followed.
The arithmetic underneath
The whole puzzle rests on one question: which sets of different digits make a sum? Runs of two to nine cells can show 120 different sums between them, and 34 of those can be made with just one set of digits — the 3s and 17s, the 6s and 24s that every solver looks for first.
Simple as each step is, the puzzle as a whole is not: in 2003, Takayuki Yato and Takahiro Seta showed that solving it in general is NP-complete. As grids grow, no known method can solve every one quickly. A puzzle made for people is the opposite: it always gives way, one certain step at a time.