Hard Killer Sudoku

Forty-five, and where it comes from

Every row of a finished sudoku holds the digits 1 to 9 exactly once, and those digits add to 45. So does every column, and so does every three-by-three box. That is true of every sudoku ever printed, and on a plain grid it is close to useless because there are no sums to set it against. On a killer grid there is nothing but sums.

The arithmetic scales up as well. Nine rows hold 405 between them, and because the cages tile the grid without overlapping and without leaving anything out, the totals printed on any killer board also add to 405. That is a genuinely useful check if you ever copy a board out by hand: add every cage total, and if the answer is not 405 you have transcribed something wrong.

Innies and outies

Take a box. Look at which cages lie entirely inside it. If those cages between them cover eight of its nine cells, the ninth cell is 45 minus their totals, placed by addition alone with no candidate reasoning of any kind. That leftover cell is an innie.

The reverse works too. If the cages covering a box spill over its edge by exactly one cell, that outside cell is the cages’ combined total minus 45. Outies are much easier to miss, because the cell you end up solving is not in the region you were looking at, and your eye is not on it.

A board reaches this page when it could not be completed without at least one of those moves. That is the entire definition of hard here: nothing above it was needed, and nothing below it was enough.

Why hard boards carry fewer cages

When a freshly generated layout turns out to have more than one answer, the repair is to split a cage in two, which adds a printed total and therefore adds information. Repair a layout often enough and it becomes easy regardless of what was asked for. So a board at this level is allowed two repairs at most and is then thrown away, and the layouts that survive are the sparse ones: usually in the thirties or low forties for cage count, against fifty and upwards on the easy page. Fewer outlines, wider gaps between the anchors, longer stretches with nothing obvious to do.

When the label above the grid says medium

It sometimes will, and that is not a fault. A layout of this kind has to be found rather than manufactured, and the generator has a couple of seconds to find one before the page would start feeling broken. When it runs out of time it serves the closest board it managed and labels it with the grade that board actually earned. A page that says hard and quietly hands over something softer is the same lie as an unsolvable puzzle, just told in the comfortable direction.