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Number Puzzles

Cross Sums

Some totals are fussy: two squares that add to 3 can only be 1 and 2. Cross Sums is full of totals like that, and where an across run crosses a down run, one digit has to work for both. The fussy totals are where to begin.

Free sets by level, ages 8–12:

Cross Sums, Level 3: two small grids of white and gray squares, with totals in the gray squares for each run across and down.Level 3: a page from the free set

Before starting

  • Level 3: add three or four one-digit numbers in your head
  • Level 4: add a string of five one-digit numbers, and add and compare two bigger totals
  • Know that digits here run from 1 to 9, with no zero

Free puzzles to print

Each set: a Learn sheet with a worked example, 10 puzzles from easiest to hardest, and the answers.

  • Level 3

    Ages 8–10 · Grades 3–4

    Cross Sums, Level 3: two small grids of white and gray squares, with totals in the gray squares for each run across and down.

    Boards grow from 4 × 4 to 7 × 7 squares (counting the clue row and column), with runs of 2 to 5 squares. In the harder puzzles, the child has to list every way a run can still be made, at least twice.

    On the sheet: “Use digits from 1 to 9. No digit repeats in a run. Each run adds to the clue before it. A run is the white squares in a line, across or down, between gray squares.”

    Print Level 3 (Letter)Other paper size: A4

  • Level 4

    Ages 10–12 · Grades 5–6

    Cross Sums, Level 4: one 8 × 8 grid of white and gray squares, with totals in the gray squares for each run across and down.

    Bigger boards, from 8 × 8 to 10 × 10, with runs up to 5 squares. A new move joins: the block total. Add the across clues of one to three rows and compare them with the down clues that cross those rows. The difference belongs to the squares that stick out.

    On the sheet: “Use digits from 1 to 9. No digit repeats in a run. Each run adds to the clue before it. A run is the white squares in a line, across or down, between gray squares.”

    Print Level 4 (Letter)Other paper size: A4

How it works

The grid of white and gray squares looks like a small crossword. Each white square takes one digit from 1 to 9. A gray square holds the total for the run of white squares next to it, across or down. No digit repeats inside a run.

How to play

  1. Read a clue: the number above the slash is for the run going across, the number below is for the run going down.
  2. Look for runs that can be made only one way. Write their digits small in the squares.
  3. Where two runs cross, keep only a digit that fits both lists.
  4. Fill a square when only one digit is left. Check that no run repeats a digit.

When it gets hard

  • Making 4 in two squares as 2 + 2.

    Hint: No digit repeats inside a run, so 4 in two squares must be 1 + 3.

  • Using the across clue for the run going down.

    Hint: Across is above the slash, down is below. Trace the run with a finger before writing.

  • Writing a digit as soon as one option looks right.

    Hint: Write the options small first. Ink in a digit only when the crossing run agrees.

Tip for grown-ups: Make a small card of the totals that can be made only one way. In two squares these are 3, 4, 16 and 17; in three squares, 6, 7, 23 and 24. Children like spotting them in each new grid.

Frequently asked questions

Is this a crossword with numbers?
Close. The grid looks like a crossword, but the clues are totals, not words. Each answer is a short run of digits that adds to its clue, with no digit used twice in a run.
How would I use these with a grade 4 class?
Start the class on the smaller Level 3 boards and share the one-way totals as a list on the board. Strong solvers can move to Level 4.
Why can't my child use 5 + 5 for 10?
A digit can appear only once in each run. So 10 in two squares has four ways: 1 + 9, 2 + 8, 3 + 7 and 4 + 6.

Questions about printing, levels, answers or membership: see all questions.

Cross Sums, Level 3 Learn sheet with a worked example

Worked example

From the Level 3 Learn sheet

  1. A run is the white squares in a line between gray squares. The clue before it is its total.
  2. Find a run whose total can be made only one way: 3 in two squares is always 1 + 2, and 24 in three is always 7 + 8 + 9. Write those digits small in its squares.
  3. Where an across run meets a down run, the digit must fit both lists. Often only one does.
  4. Write it in, then look at the runs it touches.

A question to ask: “Which digits can make this total in this many squares?”

Free every Sunday A mixed bag of 20 new puzzles, all four levels. Get them by email

The thinking it works on

  • Quick adding

    Adds small groups of digits quickly and accurately.

    Geary, Hoard and Nugent (2012) followed 275 children from grade 1 to 4 as they added small numbers by counting, recalling the sum, or splitting a number. Source

  • Listing every way to make a total

    Lists every set of different digits that makes a total.

    When English (1993) gave 7- to 12-year-olds 'find every combination' problems, the most organized children worked like an odometer, changing one part at a time. Source

  • Reasoning with two crossing clues

    Keeps only a digit that fits both runs where they meet.

    Crone and colleagues (2009) found that 8- to 12-year-olds were much less accurate than adults on matrix puzzles that needed two relations put together. Source

  • Keeping track of options

    Updates pencil notes as squares fill.

    Working with children aged 4 to 11, Alloway, Gathercole and Pickering (2006) found the parts of working memory in place by age 4. There were separate stores for words and for positions on a page. One shared system worked on what was stored. Source

  • Parts and wholes

    Compares the totals of a block of runs to find what sticks out.

    Torbeyns and colleagues (2016) studied 9- and 10-year-olds. On problems where it helped, they used the link 'if a − b = c, then c + b = a' about 40% of the time, with wide differences. Source

Where is this thinking used?

  • Checking a receipt: The items must add to the total printed at the bottom.
  • Crosswords and word games: A solved word gives a letter that helps the word crossing it.
  • Packing a bag: Choosing different items that together stay under a weight limit.
  • Sports scores: Working out which plays could add up to a final score.

For teachers

Curriculum areas
Common Core
Operations and algebraic thinking: sums of one-digit numbers known from memory (Grade 2), used quickly and many times; Number and operations in base ten: adding and subtracting totals within 1,000 for the block-total move (Grade 3)

Math practices: see the box below.

Math practices this puzzle uses

  • Uses: MP1 Make sense of problems and persevere in solving them
  • Not used: MP2 Reason abstractly and quantitatively
  • Not used: MP3 Construct viable arguments and critique the reasoning of others
  • Not used: MP4 Model with mathematics
  • Not used: MP5 Use appropriate tools strategically
  • Not used: MP6 Attend to precision
  • Uses: MP7 Look for and make use of structure
  • Not used: MP8 Look for and express regularity in repeated reasoning

From the Common Core Standards for Mathematical Practice.

More Cross Sums puzzles for members

Members get at least 5 more worksheets at each difficulty for every puzzle, 80 new puzzles every Sunday, and the whole archive. $39 a year or $25 for 6 months.

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Also known as: Kakuro, cross sum puzzles, addition crosswords.