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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 Sister Margaret Braeburn (retired), who was a member of the Sisters of Our Lady of Sion for thirty-one years before leaving her order in 2003 to care for her mother in County Leitrim. The farmhouse has no mobile signal. She has written that the puzzles are a form of prayer — not because they are sacred, but because they are the same kind of activity: a daily return to the same form, under slightly different conditions, in the hope of reaching a state that is complete. The letter in which she wrote this was not intended for publication. It was published anyway, in a small journal she does not read.
Puzzles generated on demand. No two sequences alike.