How to play Daily Sums

The number crossword. Fill each run of squares with different digits that add up to its clue, and let the crossings do the rest.

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Daily Sums is Kakuro, sometimes called cross sums. It looks like a crossword and behaves like one, except the answers are digits and the clues are totals. Nothing is hidden and nothing is random once the board is dealt: every square has exactly one correct digit, and every one of them can be reasoned out.

The board

The grid is a mix of black squares and white ones. White squares are yours to fill. Black squares either separate the runs or carry the clues, and a clue square is split by a diagonal line. The number above the diagonal, on the right, is the total for the run of white squares running across to its right. The number below the diagonal, on the left, is the total for the run of white squares running down beneath it. A black square may carry one clue, both, or none at all.

The two rules of Kakuro

- Every run of white squares must add up to exactly the total in its clue square - No digit may appear twice within the same run

That second rule is the one that does the real work. It means a run of a given length and total can only be made from certain sets of digits, and often from only one.

Reading a run

A run of two squares totalling 3 can only be 1 and 2, in some order. A run of two totalling 4 can only be 1 and 3, because 2 and 2 repeats. A run of two totalling 16 can only be 7 and 9, and one totalling 17 can only be 8 and 9. At the other end, a run of three totalling 6 is 1, 2 and 3, and a run of three totalling 24 is 7, 8 and 9. These are not tricks to memorise so much as arithmetic that repays being noticed. Short runs with totals near the extremes are where every board opens up.

The crossing

A white square is never in one run only. It sits at the intersection of a run going across and a run going down, and it must satisfy both. If the across-run can only be made from 1, 2 and 3, and the down-run can only be made from 3, 8 and 9, then that square is a 3 and there is nothing to weigh up. This is the whole engine of the puzzle, and it is why the sizes here go up rather than the arithmetic getting harder: a bigger grid has more crossings, not bigger sums.

Grades

Boards are graded by the deduction they need, not by how large they are.

- **Gentle** boards fall to crossings alone. Look at what each run allows, cross it with the run through the same square, and squares settle one at a time until the grid is full - **Hard** boards need one step more: sometimes a digit that every possible set for a run insists on can only fit in one of that run's squares, so it goes there even though that square still looks open

A board is only labelled Hard once the harder rule has been shown to be necessary. The generator solves it a second time with that rule switched off, and if the board still falls, it is not Hard and it does not claim to be. The Grade panel says so: it reads Grade when the claim was proved and Aim when it was only sought.

No guessing, ever

Every board is checked before it reaches you by a solver that never guesses. It only ever writes a digit that has been forced, and it has to finish the entire grid or the board is discarded and another is generated. That is a stronger promise than saying the puzzle has a unique answer, because a puzzle can have exactly one answer that no human could find without trial and error. If you are stuck on a board here, there is a run somewhere you have not read closely enough.

The daily board

Everybody gets the same 10x10 board each day, generated from the date in your own browser. It is always a Hard board. Finish it to keep a streak going; the streak survives so long as you do not miss a day. Your times and streaks live in this browser only.

Controls

Tap a white square to select it, then tap a digit on the pad. On a keyboard, type 1-9 to write a digit and Backspace to clear one.

Arrow keys move between white squares
N toggles pencil marks, as does the Notes button
The Size button cycles 8x8, 10x10 and 12x12
Daily switches to today's shared board and back

Strategy tips

Scan for the shortest runs first, and among those for the totals nearest the extremes. A two-square run totalling 3, 4, 16 or 17 has exactly one pair of digits that fits, and so does a three-square run totalling 6, 7, 23 or 24. Those runs are the seams of the board, and everything else opens off them.
## Work the crossings, not the sums
The productive question is never what does this run add to, it is which digits can this square hold. Take the set your across-run allows, take the set your down-run allows, and look at the overlap
When the overlap is a single digit, write it. When it is two, pencil both in and move on: the square that settles them is usually somewhere else
After writing a digit, immediately re-read both runs it belongs to. One digit placed often collapses two runs at once, and that cascade is where the pace of the game comes from
## Use the leftover
When all but one square of a run is filled, the last digit is just the total minus what is there. Check it does not repeat a digit already in the run — if it does, something earlier is wrong
The same trick works from the other side. If a run of four totals 30 and you have written 9 and 8, the remaining two squares must total 13 from the digits you have not used
## Pencil marks earn their keep here
Turn Notes on and mark the candidates for any square you cannot settle. Kakuro rewards this more than most grid puzzles, because a square with two candidates is often resolved by a run three squares away
Do not mark everything. Mark the squares sitting on long runs, where the arithmetic is loosest and your memory will not hold it
## On the Hard boards
When crossings stop yielding, change the question. Instead of asking what this square can hold, ask where in this run a particular digit could possibly go. If every set that still fits the run contains a 7, and only one square in the run can take a 7, that square is a 7
This is the only extra rule the Hard boards need, and it is worth reaching for the moment progress stalls rather than after ten minutes of re-reading
## Pace
On a full board, look for the region with the densest short runs and finish it completely before moving on. Half-solved regions scattered across the grid are harder to hold in your head than one finished corner
A run that turns red is full and wrong. Fix it before going further; a wrong digit propagates through every crossing it touches