Equal Pieces

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

  

  2 pieces of 3 to 5 squares, then 3 pieces, then 4 pieces; every piece faces the same way, so one slides onto the next.

  On the sheet: “Cut the shape into pieces that are all the same.”

  Print Level 2 (Letter)Other paper size: A4

• Level 3

  Ages 8–10 · Grades 3–4

  

  2 pieces of 4 to 6 squares, then 3 pieces, then 4 pieces where at least one is flipped. The hardest are two big halves of 8 to 10 squares, one turned or flipped.

  On the sheet: “Cut the shape into pieces that are all the same.”

  Print Level 3 (Letter)Other paper size: A4

• Level 4

  Ages 10–12 · Grades 5–6

  

  3 or 4 pieces of 6 to 7 squares, then two big halves of 11 to 14 squares. The hardest are 4 pieces of 8 squares packed tight, some flipped.

  On the sheet: “Cut the shape into pieces that are all the same.”

  Print Level 4 (Letter)Other paper size: A4

The thinking it works on

• Seeing that a turned or flipped shape is still the same shape

  Draws one piece, then hunts for the same shape turned or flipped in what's left.

  Kail (1991) found that 9- and 10-year-olds told a letter from its mirror image by turning it in the mind, the same way adults did. Source

• Sharing the squares out equally

  Counts all the squares and divides by the number of pieces.

  Outhred and Mitchelmore (2000) looked at how 115 children in grades 1 to 4 covered rectangles with squares. They found five levels, from coverings with gaps to neat rows and columns. Source

• Planning ahead from the hardest spot

  Starts at an awkward corner, where only a few pieces can fit.

  Albert and Steinberg (2011) found that planning several moves ahead keeps developing from age 10 into the early twenties. Working memory and self-control partly explain this. Source

• Checking and backtracking

  Erases and tries a different first piece when the last ones don't match.

  Checking one's own work (monitoring) gets stronger steadily through childhood and runs alongside self-control skills, according to a review of both research fields. Source

• Taking shapes apart and putting them together

  Sees one big shape as several identical smaller shapes glued together.

  Clements, Wilson and Sarama (2004) mapped a path for children aged 3 to 7. It runs from fitting pieces by trial and error to joining shapes into bigger shapes made of smaller ones. Source

• Sliding a shape in the mind

  Slides one piece onto the next to check every square lines up.

Where is this thinking used?

• Fair shares: Cutting a cake, a pizza or fabric into equal parts.
• Tiling: Identical pieces cover a room with no gaps.
• Packing: Turning an item is often what makes it fit.