Number Paths

Free puzzles to print

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

• Level 1

  Ages 4–6 · Pre-K–K

  

  Joins late, in the harder puzzles with a 3 × 3 grid (1–9) or a 7-cell honeycomb (1–7), then a 4 × 4 (1–16).

  On the sheet: “Write the missing numbers. Each one touches the next.”

  Print Level 1 (Letter)Other paper size: A4

• Level 2

  Ages 6–8 · Grades 1–2

  

  Mixes Number Paths and skip counting. Number Paths: 4 × 4 with 6 to 8 numbers given, then 5 × 5 with fewer numbers given, and a 19-cell honeycomb at the hardest end. Skip counting: by 10s on a 4 × 4, then by 5s and by 2s on a 5 × 5; the hardest are 19-cell honeycombs counting by 2s or 5s.

  On the sheet: “Write the missing numbers. Each number touches the next one.”

  On the sheet: “Write the missing numbers. Each one touches the next.”

  On the sheet: “Count on by the same number. Each number touches the next.”

  Print Level 2 (Letter)Other paper size: A4

• Level 3

  Ages 8–10 · Grades 3–4

  

  Mixes two kinds. Number Paths: 6 × 6 grids, first with easy moves, then needing step counting, then big honeycombs of 25 and 37 cells. Skip counting: by 3s on a 5 × 5, then by 4s or 25s, then by 3s or 4s on a 6 × 6 (up to 144). The hardest skip-count puzzles are honeycombs of 25 and 37 cells. In the hardest puzzles of both kinds, neither the first nor the last number is shown.

  On the sheet: “Write the missing numbers. Each one touches the next.”

  On the sheet: “Count on by the same number. Each number touches the next.”

  Print Level 3 (Letter)Other paper size: A4

• Level 4

  Ages 10–12 · Grades 5–6

  

  Corners touch too, so each square has up to eight neighbors. First a 6 × 6 (1 to 36), then 7 × 7 grids (1 to 49) with fewer numbers given. Then puzzles that need step counting; in the hardest, neither 1 nor 49 is shown.

  On the sheet: “Write the missing numbers. Each one touches the next, side or corner.”

  Print Level 4 (Letter)Other paper size: A4

The thinking it works on

• Counting forward and backward from any number

  Counts on from any written number, and back from it too.

  Secada, Fuson and Hall (1983) found that first graders who counted on could count up from any starting point, among other skills. Most who still counted from one lacked at least one of these skills. Source

• Placing only what must go there

  Places a number only when the squares around it force it.

  In Pillow and Pearson's (2009) study, third graders rated a conclusion that had to follow from what they saw as surer even than a well-supported hunch. Source

• Counting steps between two numbers

  Counts how many steps two numbers need and looks for a route that long.

• Planning a path that fills the space

  Looks ahead so no square is left that the path can never reach.

  Unterrainer and colleagues (2013) followed children from ages 4 and 5 for a year on a planning puzzle. Most solved easy ones, and look-ahead problems got easier with age. Source

• Knowing which squares touch

  Keeps 'touching' to the rule: four neighbors on a grid, six on a honeycomb, eight at Level 4.

• Keeping track and backing out

  Writes lightly, backs out when a stretch runs out of room, and finds the hidden ends.

  Conklin and colleagues (2007) found that tasks needing children to hold and rearrange information in mind kept getting easier through the teen years. Source

Where is this thinking used?

• Routes: Errands or deliveries that visit every stop once.
• Counting on: Finding a seat or a house from a number you can see.
• Cables and paths: Laying a pipe or a garden path that must not cross itself.
• Planning jobs: Every space is used exactly once.