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Cross sums

Fill white cells with 1 to 9 so every run adds to its clue, with no digit twice in a run — small boards first, growing.

  • Logic
  • Untimed
  • 10 levels

How it plays

Each white cell holds a digit from 1 to 9. A run is an unbroken line of white cells across or down; its digits are all different and add to the clue in the dark cell before it.

Clues read down\across: in "12\7", 12 is the sum of the run below that cell and 7 the run to its right; "\7" has only an across run, "12\" only a down one.

Some sums can only be made one way — 17 in two cells is 8 + 9 — and where an across and a down run cross, the digit must suit both. Levels grow the board. Every puzzle has exactly one solution, checked by a solver.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This practises sum decomposition and crossing constraints. It is practice at these puzzles and at addition facts, not a measured gain in reasoning elsewhere.

Learn

How to get better at cross sums: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Read each clue as down\across. In “12\7”, 12 is the run going down below the cell and 7 the run going right; “\7” has only an across run and “12\” only a down one.
  2. Scan for the runs whose sum can be made only one way — the very small and very large sums for their length. Those runs hold a known set of digits, just not yet in order.
  3. Go to the cells where such a run crosses another run. The cell must take a digit both runs allow; often only one digit is in both sets.
  4. Each time you place a digit, take it off what is left of both its runs: subtract it from the sum and drop it from the digits still allowed. Short leftovers are often forced.

Techniques and tricks

  • The one-way sums

    Learn the sums with a single set of digits. In two cells: 3, 4, 16, 17. In three: 6, 7, 23, 24. In four: 10, 11, 29, 30. They are the smallest and largest few for each length.

    e.g. 4 in two is 1 + 3; 23 in three is 6 + 8 + 9; 11 in four is 1 + 2 + 3 + 5

  • Cross the sets

    Where two runs with known sets meet, the shared cell takes a digit that is in both. Then each run’s other cells take what is left of its set.

    e.g. An across 3 in two (1, 2) crosses a down 4 in two (1, 3): the shared cell is 1, so the across has a 2 and the down a 3

  • No doubles

    A digit may not repeat in a run, so an even sum in two cells can never use its half.

    e.g. 10 in two is 1 + 9, 2 + 8, 3 + 7 or 4 + 6 — never 5 + 5, so no 5 goes there

  • Sum what is left

    Once all but one cell of a run is filled, the last is the clue minus the rest. Do the subtraction before looking for a pattern.

    e.g. A 22 across with 9 and 6 already in it: 22 − 15 = 7

  • Highs and lows

    A cell in a large-sum run cannot hold a small digit, and the other way round. In a two-cell 15, each cell is at least 6, since the other can be at most 9.

    e.g. 15 in two: 6 + 9 or 7 + 8, so a crossing cell that must be 1 to 5 is impossible there

  • Totals agree

    All the across clues add up to the same total as all the down clues, since both count every white cell once. A quick check that you read every clue right.

    e.g. Across 22 and 21 make 43; down 15, 16 and 12 also make 43

As the levels climb

  • The ladder starts with a 2×3 white area and no dark cells inside, so every run is a full row or column. Here the one-way sums and Sum what is left do it all.
  • Next the area grows a row or a column at a time — 3×3, 3×4, 4×4, 4×5, 5×5, 5×6, 6×6, 6×7, 7×7 — and more of it is turned dark, from 15% up to 42%. Dark cells break long runs into short ones and add clues.
  • On the larger boards, the first one-way run may be far from where you are looking. Sweep every clue once before placing anything; the working names a first step when there is one.
  • Where the sums alone allowed a second filling, a digit or two is given in a white cell. Treat a given like a filled cell: it shrinks both its runs straight away.
  • The clock is off unless you turn on Time pressure; then a board gets 60 seconds plus 12 seconds for every open white cell.

Traps to avoid

  • Reading the clue the wrong way round. The number left of the slash goes down, the one right of it goes across. Check the direction before using a sum.
  • Repeating a digit in a run. Two 4s can add to 8, but not in one run — always count the run’s digits as a set with no repeats.
  • Fixing the order too early. A one-way sum tells you which digits, not where they go. Place each only when a crossing run decides it.
  • Leaving two cells that could swap. Every board has exactly one filling, so if two of your digits could trade places without breaking a sum, a crossing run has not been used yet — go back and find it.

How to practise

  • Learn the one-way sums for two and three cells by heart: 3, 4, 16, 17 and 6, 7, 23, 24. Recalling them on sight is most of the speed.
  • Set “Largest board side” to 4 while you learn to cross sets, then raise it a step at a time to 7.
  • Leave Time pressure off at first; turn on Relaxed once 5×5 boards come out right without trying digits at random.
  • When a board goes wrong, find the first step the working names and check whether you used it. Missing the one-way run is the usual reason a board stalls.

Levels

It is untimed unless you turn the clock on. There are 10 levels with the default settings, and the counter stops at the real top. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More logic

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