Sudoku Variants That Are Worth Your Time

Classic sudoku has one constraint repeated three ways: no repeats in a row, a column or a box. Everything else in the genre is an extra constraint bolted onto that skeleton, and the interesting question is which of those additions actually give the solver something new to think about.

There is a simple test. A good variant creates deductions that could not exist in classic sudoku. A bad variant creates rules you have to remember while doing the same deductions you were already doing.

Killer sudoku

Dotted cages, each with a target sum, and no repeated digit within a cage. The grid usually starts completely empty.

This passes the test easily. Killer introduces an entire second layer of reasoning based on arithmetic: a two-cell cage summing to 17 must be 8 and 9, a three-cell cage summing to 6 must be 1, 2 and 3. Combined with the classic constraints, these forced combinations open the grid in ways no amount of classic scanning could.

It also introduces the most satisfying technique in the whole genre, the 45 rule: every row, column and box sums to 45, so a set of cages that almost covers a box tells you the value of the cells left over. That is a genuinely new tool.

A variant earns its name when it teaches you a technique you cannot use anywhere else.

Thermo sudoku

Thermometer shapes on the grid, with digits strictly increasing from the bulb along the stem.

Also excellent, and the most elegant constraint in the family because it is purely ordinal - no arithmetic at all. A five-cell thermometer forces its bulb to be at most 5 and its tip to be at least 5, immediately, before any other reasoning. Long thermometers crossing each other produce chains of inequalities that feel completely unlike classic solving.

Arrow sudoku

A circle and an arrow: the digit in the circle equals the sum of the digits along the arrow.

Good, and underrated. Arrow constraints are weak individually but combine well, and because the sum is bounded by 9 in a single circle, long arrows are heavily constrained from the start. A four-cell arrow must contain small digits, which is a lot of information for one symbol.

Sandwich sudoku

Numbers outside the grid give the sum of the digits sitting between the 1 and the 9 in that row or column.

Divisive but, in my view, worth it. A sandwich clue of 0 means the 1 and 9 are adjacent - enormously restrictive. A clue of 35 means they are far apart with big digits between them. The technique is unusual enough that it takes a few puzzles to develop any intuition, which is exactly the argument for it.

The ones that do not earn their keep

Anti-knight and anti-king constraints - no repeated digit a knight's move or king's move away - are technically valid and mostly tedious. They add checking work rather than deduction, and the checking is exactly the kind humans are bad at and computers are good at.

Diagonal sudoku, where both long diagonals must also contain 1 to 9, is fine but barely a variant. It adds two more of the constraint you already had.

What the variants share

Every good one has the same property: the extra constraint is most informative at the extremes. Cages with very high or very low sums, thermometers that are very long, sandwich clues of 0 or 35. Setters build around those extremes, and solvers should hunt for them first.

That is the practical takeaway. In any variant, before you start scanning, find the two or three clues that are near their limits. They are where the grid opens.

Background reading is easy to come by: the Wikipedia entry on sudoku covers the history and the mathematics, including the proof that the minimum number of clues for a unique classic solution is 17, and the Japanese publisher Nikoli - which gave the puzzle its name - remains the reference source for the wider family of pencil puzzles. If you want a daily deduction habit in a smaller format, the one-clue cryptic and a good riddle archive both scratch a similar itch in under five minutes.