Solving medium killer sudoku

Medium killer sudoku is the tier where the 45-rule starts firing on multiple axes and the cage-sum patterns extend beyond the few singletons that solve easy. Most medium killer puzzles ship with no givens at all — the cages and sums carry the entire constraint structure, and the solve depends on reading both the arithmetic (which digits can sum to N) and the geometry (where the cage sits relative to rows, columns, and boxes) in parallel.

Killer pairs and killer triples expand

At easy, only a handful of cage sums had a single valid digit set: 3, 4, 16, 17 for two-cell cages, 6 and 24 for three-cell cages. Medium killer brings the next layer of cage-sum patterns into play. A 2-cell cage summing to 5 can be 4 or 3. A 3-cell cage summing to 7 must be 4. A 3-cell cage summing to 23 must be 9.

The full table of cage-sum digit sets is worth memorising at medium. Most solvers print or sketch the table and reference it for the first hundred or so medium killer puzzles, until the patterns become visual habit. Spotting them at sight is the difference between a 20-minute medium killer solve and a 35-minute one.

Killer pairs and killer triples at medium often eliminate digits not just from inside the cage but from the cage's row, column, and box. A 3-cell cage in the middle of box 5 with sum 7 must contain 4, so 1, 2, and 4 are eliminated from every other cell in box 5. That single inference frequently knocks three or four cells into place.

The 45-rule on both axes

At easy, the 45-rule mostly fired on whichever row or column had its cages most cleanly contained. At medium, the rule applies systematically to every row and every column, and the puzzles are designed to reward solvers who scan both axes equally.

The mechanic is unchanged: for any region (row, column, or box), add up the sums of every cage entirely contained in the region. Subtract from 45. The remainder must equal the sum of the cells from cages that span the region's boundary. With a single boundary-crossing cage and a single innie or outie cell, the calculation gives you that cell's value directly.

Box-level 45 applications start appearing at medium too, though they remain the dominant move at hard. When a row-axis or column-axis 45 calculation doesn't narrow anything, the box-axis version sometimes does — boxes have more cage-edges than rows do, so they often have cleaner innie/outie configurations.

Pencil marks at this tier

Killer puzzles need pencil marks earlier than classic puzzles, because the cage constraints generate eliminations that aren't visible at the cell level. At medium, most solvers pencil-mark every empty cell with its full candidate set, plus the possible digit sets for each unresolved cage written near the cage sum. The piece on pencil marks without the clutter covers the lightweight notation that works for classic; killer adds the cage-set notation on top.

Track which cage-sum patterns are still possible for each cage, especially for cages with several valid digit sets. As placements eliminate possibilities, narrow the cage's digit set notation accordingly. The cage notation often resolves to a single set before the cells inside the cage do — and once the digit set is locked, the cells inside are constrained even before they're placed.

Sum-and-position pairs

The piece on naked and hidden pairs and triples covers the classic versions of these techniques; at killer, they apply with the additional cage constraint. A naked pair at killer is two cells in the same unit whose pencil marks contain only the same two digits — but at killer, those two cells might also share a cage, in which case the cage's sum confirms the pair from a second direction. Pairs in killer often appear in cages and benefit from the cross-check.

When medium killer stalls

Medium killer stalls in two characteristic ways.

Single-axis 45-rule. Most solvers learn the 45-rule on rows and apply it routinely. Many never apply it to columns with the same discipline. When a medium killer puzzle stalls and the row-axis scan finds nothing, switching to the column axis is often the next move.

Sum-only thinking. Spending all the attention on "what digits sum to N" without checking which of those digit combinations actually fit given the cage's row, column, and box position. A 3-cell cage summing to 14 has many valid digit sets, but most are eliminated by what's already in the surrounding cells. Joint constraints — sum AND grid position — are the move.

Two habits to drop

Skipping the cage-pattern memorisation. The full table of cage-sum digit sets fits on half a page. Memorising it pays back across every killer puzzle from medium onward.

Treating each cage as independent. Cages interact through the rows, columns, and boxes they share. When two cages share a unit, the digits in one cage are constrained by the digits in the other. Reading the cages collectively rather than one at a time surfaces eliminations that a one-cage-at-a-time scan misses.

When the cage-sum patterns fire on first inspection and the 45-rule feels automatic on both axes, you're ready for hard killer, where innies and outies start carrying real weight and box-level 45-rule applications become the daily move. Medium killer is the tier where the cage-as-constraint instinct settles into something that doesn't require conscious effort. That settling is most of what makes the higher tiers possible.