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Burnside's Lemma Examples
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Counting the number of necklaces with 6 beads, each bead can be one of 2 colors
Applying Burnside's Lemma, calculate the total number of fixed necklaces under all rotations and reflections, then divide by the order of the dihedral group to find the distinct colorings.
Determining the number of unique circular arrangements of 5 different flags around a circle
Use Burnside's Lemma by considering the group of rotations of the circle with 5 flags. Sum up the fixed arrangements for each rotation and divide by the group order.
Counting the distinct colorings of the faces of a cube using three colors
Using Burnside's Lemma, count the number of orbits (distinct colorings) by summing the number of fixed points of each group action (rotations and reflections of the cube), and then divide by the order of the group.
Finding the number of different sticker placements on a standard Rubik's cube face, assuming stickers are all different
Apply Burnside's Lemma by examining the possible rotations of a Rubik's Cube face. For each rotation, count sticker placements that are unchanged and divide by the number of face rotations.
Counting distinct arrangements of 3 red, 3 green, and 3 blue beads on a bracelet (rotation and reflection symmetries)
Using Burnside's Lemma, sum the number of arrangements invariant under each rotation and reflection, then divide by the order of the dihedral group to find the distinct arrangements.
Counting the number of unique ways to color a tetrahedron using 4 colors, with each face a different color
Use Burnside's Lemma, accounting for rotations of the tetrahedron (group actions). Divide the sum of invariant coloring counts for each group action by the number of elements in the tetrahedral group.
How many ways can you paint 4 identical miniatures, using only black or white paint?
Employ Burnside's Lemma by considering the permutation group of 4 objects. Calculate the contribution of each permutation to the number of distinct colorings and divide by 24, the order of the permutation group.
Number of ways to place 4 indistinguishable queens on a 4x4 chessboard so that no two queens attack each other, up to symmetries of the board
Use Burnside's Lemma by identifying all symmetries of the chessboard (group actions), finding the number of placements that remain unchanged under each, and dividing by the total number of symmetries.
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