THE SETUP
A Number Crossword grid uses the same black-and-white cell structure as a word crossword. Across clues are arithmetic expressions; the digits of the result fill the corresponding run of cells. There are no down clues — down runs exist in the grid, but their content is determined entirely by the across expressions that cross through them.
The constraint that makes the puzzle work: where an across run crosses a column, the digit placed in that cell must satisfy every across run that uses it. On a standard Number Crossword, a single cell can appear in only one across run, but that digit must also "look right" when read vertically — forming a valid multi-digit number in the down run.
No leading zeros: a run of three cells cannot start with 0 (just as a three-digit number cannot). This is often the first constraint that eliminates candidate digits.
EVALUATE EACH CLUE BEFORE PLACING
The primary technique: compute the result of every arithmetic expression before you place a single digit. Some clues have a unique result (3 × 47 = 141 — only one answer). Others have a range (a clue might say "a two-digit number times a two-digit number" without specifying which digits, making the result a range of possibilities).
Start with the clues whose result you can calculate directly and completely. Place those runs first. Then look for runs that partially overlap with placed cells — the shared digits constrain what the result can be.
Example: if you've placed 1, 4, 1 in an across run and a down column starts with that 1, the down run must begin with 1 — which may eliminate several candidate expressions for that down position.
DIGIT COUNT IS A HARD FILTER
The length of a run (number of cells) tells you exactly how many digits the result must have. A three-cell run must hold a three-digit number (100–999). A five-cell run must hold a five-digit number (10000–99999). Any expression whose result falls outside the range for its run length is immediately invalid.
This is often the fastest way to eliminate candidate expressions on harder puzzles where the clue isn't a single fixed calculation: compute the minimum and maximum possible result of the expression and check whether the digit count falls in the run's range.
SHARED CELLS PROPAGATE CONSTRAINTS
Where an across run and a down run share a cell, the digit in that cell must satisfy both runs simultaneously. This is the core constraint mechanism of Number Crossword — the same mechanism that makes word crosswords work, now applied to digits.
Once an across run is fully placed, every cell it shares with a down run gives you a fixed digit in that down position. Work down-run constraints after placing each across run: which down numbers are still possible given the fixed digits already placed?
On harder puzzles, this propagation produces a cascade: placing one across run fixes two or three down-run digits, which constrains two or three more across runs, which fix more down digits. Follow the chain.
HOW IT COMPARES TO KAKURO
Both Number Crossword and Kakuro are arithmetic puzzles in a crossword grid. The key difference is the nature of the runs:
- In Kakuro, each run sums to its clue and each digit (1–9) appears at most once in that run. The digits are addends.
- In Number Crossword, each run holds the digits of a single result. The digits form a multi-digit number, not individual addends.
Kakuro is about digit combination and elimination within a run. Number Crossword is about arithmetic evaluation and digit-count matching across runs. The spatial structure is similar; the reasoning is completely different.
THE SOLVE ORDER
- Evaluate every across clue. Note any whose result is fully determined (unique answer). Note the digit count required for each run.
- Place fully determined runs first. These are your anchors.
- For each placed run, propagate shared-cell constraints to crossing runs — fix the digit in the shared position and narrow the crossing run's candidate values.
- Check the no-leading-zero rule: any run whose first cell is now fixed to 0 is invalid — backtrack.
- Repeat until all runs are placed.