Listing What a Cage Can Hold

The move that makes this a puzzle rather than a latin square with decoration. Three cells at most, one short list, and often a digit written outright.

The list, and what shortens it

Take a cage with two or three cells still empty and write out every arrangement of digits that hits its target. Four filters apply, and applying them in this order keeps the list short: the digits must be within range for the board, they must not repeat where the cage’s own shape forbids it, they must not already be used elsewhere in their row or column, and they must actually produce the target.

What you do with the list is the same whatever it contains. Any cell that holds the same digit in every arrangement is decided — write it in. Any digit that appears in no arrangement for a cell is gone from that cell, even if nothing is decided anywhere. That second half feels like less of a result and is very often the more useful one, because it is what makes the next cage’s list short enough to be worth writing out.

Three cells is the practical limit and the reason is honest rather than arbitrary. Two or three cells produce a list you can hold in your head or scribble in a margin; four or more produce one you would need a second sheet of paper for, and that belongs on a harder rung of this ladder rather than pretending to be an easy move.

Before — a three-cell cage marked 48×, and an L-shaped one
	1	2	3	4
1	cage multiplying to 481234	1234	cage adding to 5234	1234
2	234	cage adding to 9234	single cell, 11	234
3	cage adding to 41234	1234	234	cage adding to 101234
4	1234	1234	234	1234

After — the only set of digits that works, placed
	1	2	3	4
1	cage multiplying to 483	4	cage adding to 5	12
2	4	cage adding to 923	single cell, 11	23
3	cage adding to 412	123	234	cage adding to 101234
4	12	123	234	1234

On a board of side four the largest possible product of three digits is sixty-four, and forty-eight can be reached in exactly one way: four times four times three. The cage is an L, so its two ends sit in different rows and different columns and are entitled to hold the same digit — but the two cells along the top row are not, so the pair of fours has to straddle the corner and the three takes the remaining cell.

what it decides what it wrote

Where the short lists live

The extremes of the range are where to look first. On a board of side six the largest two-cell sum is eleven and it has exactly one pair behind it; so does the smallest, which is three. The same is true at both ends of the products — 30× is five and six and nothing else, 2× is one and two. A cage whose target is near a boundary is usually pinned before you have finished reading it.

Division and subtraction cages are worth a second glance because they behave the opposite way. A large quotient is restrictive — 4÷ on a six leaves only one and four, and two and eight if the board is that big — while a small difference is barely a constraint at all. A 1− cage on a nine has eight pairs behind it and is nearly useless until something else narrows it.

And the shape of the cage decides whether repeats are on the list in the first place. Two cells side by side must differ; two cells in an L may match. That single check, made before writing anything out, removes or adds whole rows of the list, and forgetting it is what leads people to declare a legal cage impossible.

Before — a 5+ cage with one cell it already knows
	1	2	3	4
1	cage multiplying to 483	4	cage adding to 5	12
2	4	cage adding to 923	single cell, 11	23
3	cage adding to 412	123	234	cage adding to 101234
4	12	123	234	1234

After — the other two settled around it
	1	2	3	4
1	cage multiplying to 483	4	cage adding to 5	1
2	4	cage adding to 9	single cell, 11	2
3	cage adding to 412	123	234	cage adding to 1034
4	12	123	234	34

This cage covers two cells of the top row and one below the right-hand end. The cell on the left is already down to a 2, because the column beneath it holds the only 1 on the board so far. Five minus that 2 leaves three to be split between the other pair, and since the second of them shares the row with the 2 it takes the 1 — which puts the remaining 2 below.

Why this is only the second rung

Because it looks at one cage and nothing else. There is no sweep of the board, no memory of what happened elsewhere, and no need to consider what any other cage might want — you read a target, write a short list, and cross things off. It is the cheapest move on this puzzle that involves any thought at all.

It also subsumes the case people meet first without being a different move. A cage with a single cell left over is this argument with a list of one entry: the target minus what is already written. Doing that instinctively and then thinking of the general form as something else is what makes the general form feel harder than it is.

Before — three cells of the top row adding to eight
	1	2	3	4
1	cage with a difference of 21234	cage adding to 81234	1234	1234
2	1234	cage adding to 111234	1234	cage adding to 71234
3	cage multiplying to 121234	1234	1234	1234
4	1234	1234	1234	1234

After — one digit crossed out of all three
	1	2	3	4
1	cage with a difference of 21234	cage adding to 8134	134	134
2	1234	cage adding to 111234	1234	cage adding to 71234
3	cage multiplying to 121234	1234	1234	1234
4	1234	1234	1234	1234

The 8+ cage lies flat along the top row, so all three of its cells are in one line and no digit may repeat between them. Three different digits from one to four adding to eight can only be one, three and four — two is not in any arrangement, so it goes from all three cells. Nothing has been written down, and the board is a good deal more constrained than it was.

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

Which calcudoku cages are worth listing first?

The ones whose target is near the top or the bottom of what the board allows. On a six, 11+ and 30× each have exactly one pair behind them, while a middling target may have four or five.

How large a cage can I list by hand?

Two or three open cells is the practical limit, which is where this technique stops on this site. Four or more is a real piece of work and belongs to the cage interaction rung.

Do I have to include repeated digits in the list?

Only where the cage’s shape allows them. Cells sharing a row or a column must differ; cells that share neither may hold the same digit, and leaving those arrangements out is how legal cages come to look impossible.