When Something Runs Out of Room

Two readings of the same bookkeeping, and nobody learns one without the other. This is what medium means on this site.

The cell with one candidate left

Every digit written strikes itself out of the rest of its row and the rest of its column, and every cage worked out strikes more out on its own account. Do that honestly and eventually some cell has a single survivor. Write it in; there is nothing else it can be.

The reason this is worth naming as a technique rather than treating as bookkeeping is that it needs the crossing-out to have been recorded somewhere. A solver holding candidates in their head will not see a cell run down to one, because the moment of arrival is invisible unless the marks are on the paper. This is where the pencil starts to earn its keep, and the board here draws the small digits for you.

It is also the point at which cage arithmetic and line arithmetic start working together rather than in turns. A cell might be down to two candidates because of its cage and lose the second because of its column, and neither half would have decided anything alone. That is why this sits above the cage listing rather than beside it: the cheap moves narrow, and this is where the narrowing pays.

Before — the 3 has one place left in the first column
	1	2	3	4
1	cage with a quotient of 414	cage adding to 111234	1234	1234
2	14	1234	cage adding to 111234	cage adding to 71234
3	1234	1234	1234	1234
4	cage multiplying to 824	24	1234	1234

After — it is written there
	1	2	3	4
1	cage with a quotient of 414	cage adding to 111234	1234	1234
2	14	1234	cage adding to 111234	cage adding to 71234
3	3	124	124	124
4	cage multiplying to 824	24	1234	1234

The 4÷ cage at the top of the first column can only be a one and a four, and the 8× cage at the bottom can only be a two and a four. Between them those cages have claimed every cell of the column except one, and none of them can hold a 3 — so the remaining cell is the only home the 3 has left, even though nobody has decided what the other three cells are.

what the reasoning read what it decides what it wrote

The digit with one cell left

The other way round, and the harder of the two to see. Take a row and pick a digit that is not in it yet. Look at every empty cell in that row and ask whether the digit is still possible there. If exactly one cell survives the question, that is where the digit goes — and the cell may well still have three or four other candidates of its own, which is precisely why nobody notices.

That is what makes this move worth hunting deliberately rather than waiting for. A cell down to one candidate announces itself; a digit down to one home does not announce anything at all, and finding it means asking the question of each missing digit in turn rather than looking at the cells. Six digits and six rows is thirty-six small questions on a six, most of them answered instantly.

The payoff is that it reaches into places nothing cheaper does. A cell with four candidates is invisible to every rung below this one, and this argument can settle it outright — which on a medium board is very often the move that restarts everything.

Before — a cell of the 8× cage with one candidate remaining
	1	2	3	4
1	cage with a quotient of 414	cage adding to 111234	1234	1234
2	14	1234	cage adding to 1123	cage adding to 71234
3	3	124	14	124
4	cage multiplying to 82		134	134

After — the last survivor written in
	1	2	3	4
1	cage with a quotient of 414	cage adding to 11123	1234	1234
2	14	123	cage adding to 1123	cage adding to 71234
3	3	12	14	124
4	cage multiplying to 82	4	13	13

The 8× cage in the bottom-left holds a two and a four in some order, and the previous move settled that the two goes on the left. That leaves its partner with a single candidate, which is the whole argument — no cage arithmetic was repeated and no line was counted, the cell simply had one digit left standing in it.

What makes a board medium, precisely

A medium board here is one where the solver, always taking the cheapest available move, was at some point forced into one of these two. No cage anywhere had a short enough list, no line was one gap from finished, and the board did not move until a cell or a digit was noticed to have run out of room.

It is a real threshold rather than a gradual one. An easy board never needs it even once; a medium board contains at least one moment where nothing at all happens until somebody looks at the small digits they have been writing and reads what they say.

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 is a hidden single in calcudoku?

A digit that has only one cell left in its row or column where it is still possible. The cell may have several other candidates, which is why the move is easy to walk past.

Do I really need to write candidates in the cells?

For anything above an easy board, yes. Both halves of this technique are readings of the crossing-out you have already done, and neither is visible if the crossing-out only happened in your head.

Which of the two should I look for first?

The cell with one candidate, because it announces itself as you scan. Hunt for the digit with one home only when scanning has stopped producing anything.