Teach
Classroom investigations that leave consequential choices to students.
The materials here are invitations rather than scripts. Each begins with an object students can act on, then offers prompts for moving from observation to mathematical language.
CLASSROOM INVESTIGATION · 45–60 MINUTES
Sixteen tiles, one closed kolam
Suitable for: upper secondary students, undergraduate problem-solving groups, teacher circles and maths clubs.
Central question: Can every binary tile from (0000) to (1111) be used exactly once in a (4) square so that the boundary is closed, adjacent sides match and the drawing is connected?
Begin
Let groups work on the construction board with only the goal visible. Ask them to keep one failed arrangement that taught them something.
Discuss
Which requirements can be checked one edge at a time? Which can only be checked after seeing the whole board? What quantities might be counted before searching?
Extend
Set aside the (0000) tile and permit only sliding moves into its empty position. Ask whether every correct arrangement can now reach every other correct arrangement.
Record
Invite groups to state one conjecture, one piece of evidence and one question their evidence does not settle.
Teacher notes
- Do not introduce the bit order until students need a compact way to record tiles.
- Treat a nearly correct board as data: a boundary leak, mismatched edge or extra component suggests a different invariant.
- When sliding begins, say explicitly that intermediate positions may violate the kolam conditions.
- Separate evidence from proof. A complete computer enumeration can verify a finite claim, but students should still identify what was enumerated and why the program covers every case.