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Fill every row and column with the numbers 1 through N (where N is the grid size), without repeating a number in any row or column.
Bold lines divide the grid into cages. Each cage has a target number and an arithmetic operation in its corner.
A single-cell cage simply tells you the value of that cell.
The practice of constraining numbers within bounded regions predates any name that has been given to it. A single entry in the catalogue of the library at Dunhuang — compiled, by common scholarly agreement, sometime in the ninth century — describes a text of “arithmetic exercises confined by ink borders,” of which no copy survives; whether this describes a puzzle in any modern sense remains a matter of some dispute. A separate tradition, associated with a school of Persian recreational mathematics active in the eleventh and twelfth centuries, holds that exercises of this kind were used not for amusement but to train merchants in the detection of falsified accounts: a well-formed grid, on this account, is one in which no number can be other than it is. This tradition is attested in two manuscripts, one of which is a copy of the other, and the original has not been located since 1847.
The modern form of cage-arithmetic puzzles was developed by Tetsuya Miyamoto, a Japanese mathematics teacher, in 2004. He called it 賢くなるパズル — "the puzzle that makes you smarter."
This version was compiled by Erasmus Kael, whose entry in the third supplementary volume of the Annotated Register of Independent Software Works (Lausanne, 2018) is filed under the heading “unclassifiable.” The register notes that he worked for some years in applied acoustics — the interpretation of echoes in cold water — before a disagreement with the Norwegian Directorate of Fisheries led to his relocation to the Swiss interior, where he now cultivates a heritage variety of quince. The entry continues for several further lines, but the relevant pages of the Lausanne copy are water-damaged beyond legibility, and no other copy has been located.
Puzzles generated on demand. No two sequences alike.