Cells That Answer Themselves

Both moves cost nothing at all. Between them they finish every easy board on this site, and they open every board at every level.

The printed digit

A cage of one cell has a number in its corner and no sign beside it, and that number is the digit. There is nothing to work out, nothing to check against anything else, and no way to get it wrong. Copy it in.

It is worth noticing how much a single one of these is worth, because their scarcity is most of what makes a board hard. Writing a digit removes it from five other cells on a six and from eight on a nine, and every one of those removals shortens the list of arrangements for whatever cage that cell belongs to. One printed digit near the middle of a board routinely settles three or four cages that were previously wide open.

That is also why the levels on this site differ so much in how many of them appear. Nothing chooses the number directly; the carve merges cages while the solver can still follow it, and a gentler cap means it stops earlier, leaving more single cells behind. The free digits are an outcome of the difficulty rather than a dial aimed at it.

Before — a cage of one cell, with no sign beside its number
	1	2	3	4
1	cage multiplying to 61234	cage adding to 131234	1234	1234
2	1234	1234	1234	cage adding to 71234
3	cage adding to 121234	1234	single cell, 21234	1234
4	1234	1234	1234	1234

After — the digit copied straight in
	1	2	3	4
1	cage multiplying to 61234	cage adding to 131234	134	1234
2	1234	1234	134	cage adding to 71234
3	cage adding to 12134	134	single cell, 22	134
4	1234	1234	134	1234

The cell at row 3, column 3 is a cage all by itself, marked 2 with no operation attached. A cage of one cell cannot do arithmetic with anything, so its number is simply the digit that belongs there. Nothing was deduced and nothing needed to be; the board came with this one already answered.

what it decides what it wrote

The last gap in a line

The mirror image, and the move that mops up. When a row has one empty cell left, the digit that goes in it is whichever of the full set is not already there — five written digits on a six leave exactly one candidate, and no arithmetic is involved at any point. The same applies to a column, and both are worth checking after every digit you write.

This is the step people skip because it feels like it does not count, and skipping it is why boards feel stuck for no reason. Each gap you close writes a digit, and each digit written strikes itself out of a row and a column elsewhere on the board, which very often turns a cage with four arrangements into a cage with one. The chains that make this puzzle satisfying are largely made of exactly this.

On paper the discipline is to sweep rather than to hunt. Run your eye down all the rows and then all the columns after every few moves, rather than checking the line you happen to be looking at. A gap sitting in a column you have not glanced at for ten moves is the single most common thing left undone on a stalled easy board.

Before — the top row is one cell short of complete
	1	2	3	4
1	cage multiplying to 61	cage adding to 132		4
2	23		4	cage adding to 7123
3	cage adding to 1234	134	single cell, 22	13
4	234	134	13	123

After — the missing digit has nowhere else to go
	1	2	3	4
1	cage multiplying to 61	cage adding to 132	3	4
2	23		4	cage adding to 7123
3	cage adding to 1234	134	single cell, 22	13
4	234	134		123

By this point the top row holds a 1, a 2 and a 4, and one cell is still empty. On a board of side four the row has to contain each of 1, 2, 3 and 4 exactly once, so the gap takes the 3 — and no cage was consulted to work that out. The row rule alone decided it.

Why these two carry an easy square

An easy board on this site is one the solver finishes using nothing above the second rung, and in practice most of its moves are these two. That is a definition rather than an approximation of one: the carve stopped merging cages the moment a merge would have needed anything harder.

Which also means an easy board is guaranteed finishable by a reader who knows only what is on this page and the next one. If yours has stalled, the overwhelmingly likely cause is a line with one gap in it that nobody has looked at, rather than anything requiring a technique you have not met.

Where this sits on the ladder

Rung 1Free cells
A one-cell cage is its own answer, and a row with one gap left has no choice about it.
Rung 2Cage combinations
List every way a small cage can be filled and keep whatever all of them agree on.
Rung 3Singles
A cell with one digit left, or a digit with one cell left in its row or column.
Rung 4Unit totals
Every row adds up to the same known number, so the cages inside it price the rest.
Rung 5Cage interaction
What a group of cells can hold between them, whether they share a cage or only a row.
Rung 6Assumption
Write a digit, follow it until something breaks, and rub it out again.

Common questions

What does a cage with no sign mean?

It is a single cell, and the number in its corner is the digit that goes there. With one cell there is no operation to perform, so no sign is printed.

Why bother filling the last gap in a row straight away?

Because the digit you write is immediately unavailable everywhere else in its column, which shortens the arrangements for every cage that column touches. It is free progress that creates more free progress.

Will these two moves finish a square on their own?

On an easy one here they very nearly do, with the cage combinations filling the gaps between them. From medium upwards they stall somewhere and a harder argument has to take over.