Procedure

Interactive Necklace Equivalence Exploration:

Setup and Configuration:

• Open the simulation interface which displays an interactive necklace with colored beads arranged in a circle.
• Configure experiment parameters using the settings panel:
  • Adjust the number of beads (6-12 beads) using the bead count slider
  • Modify the number of colors (2-8 colors) using the color count slider
  • Choose difficulty mode: Easy (direct colors) or Hard (color mapping challenge)

Understanding Necklace Equivalence:

• Two necklaces are considered equivalent if one can be transformed into the other through:
  • Rotations: Moving beads cyclically around the circle (any number of positions)
  • Reflections: Creating a mirror image of the necklace pattern
  • Combinations: Applying both rotations and reflections in sequence
• The necklace pattern is represented as a string showing the sequence of colored beads
• Position numbers on beads help track transformations and understand the equivalence concept

Interactive Exploration:

1. Practice Mode:

   • Use the "Rotate →" button to rotate the necklace one position clockwise
   • Use the "Reflect ↔" button to create a mirror image of the current pattern
   • Observe how the string representation changes with each transformation
   • Monitor the transformation log to track your operations

2. Challenge Mode:

   • Click "New Challenge" to generate two necklaces (Target and Current)
   • Analyze both patterns carefully, considering possible rotations and reflections
   • Determine if the necklaces are equivalent using your understanding of transformations
   • Answer "Yes, Equivalent" or "No, Not Equivalent" based on your analysis
   • Receive immediate feedback with explanations

Learning Progression:

• Start with Easy mode to focus on equivalence concepts without color confusion
• Progress to Hard mode where colors are mapped differently, requiring deeper analysis
• Use the color mapping guide in Hard mode to decode the actual colors
• Challenge yourself with different bead counts and color combinations
• Track your understanding through the scoring system and accuracy feedback

Mathematical Concepts Reinforced:

• Equivalence relation properties: reflexive, symmetric, and transitive
• Group theory applications through dihedral group transformations
• Burnside's lemma for counting distinct equivalence classes
• Pattern recognition and systematic analysis of mathematical structures