Sum Cages

Free puzzles to print

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

• Level 2

  Ages 6–8 · Grades 1–2

  

  4 × 4 grids, adding cages only, with bigger cages in the hardest puzzles.

  On the sheet: “Use 1 to 4 once in each row and column. Each cage adds up to its small number.”

  Print Level 2 (Letter)Other paper size: A4

• Level 3

  Ages 8–10 · Grades 3–4

  

  5 × 5; adding, then subtracting, then multiplication cages; the hardest puzzles use the row total (1 to 5 adds to 15).

  On the sheet: “Use 1 to 5 once in each row and column. Each cage makes its number with its sign. Numbers may repeat in a cage, not in a line.”

  Print Level 3 (Letter)Other paper size: A4

• Level 4

  Ages 10–12 · Grades 5–6

  

  6 × 6; first +, − and × cages, then division joins, then the row total (1 to 6 adds to 21) is needed; the hardest have bigger cages.

  On the sheet: “Use 1 to 6 once in each row and column. Each cage makes its number with its sign. Numbers may repeat in a cage, not in a line.”

  Print Level 4 (Letter)Other paper size: A4

The thinking it works on

• Finding many ways to make a number

  Lists every pair or set of numbers that can make a cage's target.

  Geary, Hoard and Nugent (2012) followed 275 children from grade 1 to 4. They moved from counting to remembering facts and breaking numbers apart (6 + 7 as 6 + 4 + 3). Those with stronger working memory took up breaking numbers apart sooner. Source

• Checking the cage and the row together

  Writes a number only when the cage rule and the row-and-column rule both allow it.

  Crone and colleagues (2004) found that 8- and 9-year-olds stuck to an old sorting rule more often than 11- and 12-year-olds. Keeping to a rule steadily took until 13 to 15. Source

• Pencil notes for every option

  Pencils the options in the corner of each square and crosses them off.

  English (1991) saw children aged 4½ to 9 move by themselves from picking at random toward an orderly way of listing every combination. Source

• Number facts: adding, then all four operations

  Uses adding facts, then subtracting, multiplying and dividing facts.

  Kouba (1989) found that children in grades 1 to 3 solving multiplying and dividing word problems used methods from acting them out with objects to recalling the fact. Source

• Using the structure of the whole row

  Uses the fixed row total (1 to 5 makes 15) to find a last square.

• Keeping track of options

  Holds several open options at once and updates them as squares fill.

  Lee, Bull and Ho (2013) tested 688 children aged 6 to 15 each year and found that tasks for updating and working memory developed in different patterns. Source

Where is this thinking used?

• Budgeting: Several amounts must add to a fixed total.
• Scheduling: No one can be in two places at once.
• Splitting a bill: The parts must make the whole.
• Seating and teams: A few fixed rules decide who goes where.