Group Theory Puzzles

In this post, we'll look at some puzzles based on the presentations of finite groups that students will approach as a puzzle of turning one string into another according to some rule set.

Setup

A broad collection of manipulatives could work for this activity. The main requirement is that one type be easily discernible from one another but an abundance of copies of each type. Here are a few possibilities:

• Scrabble tiles (1000 for $20)

• Fruit counters (108 for $23)

• Animal and fruit erasers (300 for $8)

I'll frame this in terms of letters. While I'll use distinct sets of letters for each puzzle just for the sake of having the puzzles appear more distinct at a glance, if using Scrabble tiles you'll likely want to recast these using \(\textrm{E, A, I, O, N, R, T}\) so that each student has as many of each as possible.

The puzzles will take the form of trying to turn one word into another, if possible, by following certain rules. The rules concern when one string can be replaced another. For example, the rule \(\textrm{HI} = \textrm{LO}\) would mean that \(\textrm{CHIVE} = \textrm{CLOVE}\) and \(\textrm{LORE} = \textrm{HIRE}\). We will use \( \square \) to mean an empty string of symbols, which serves as a way to insert or delete letters. For example, \(\textrm{ARE} = \square\) means \(\textrm{CAREFUL} = \textrm{CFUL}\) and \(\textrm{SOD} = \textrm{SOARED}\).

Challenges

In each puzzle set below, four or five are possible and one or two are impossible.

AB

Rules:

1. \(\textrm{AA} = \textrm{BB} = \textrm{ABAB} = \square\)

Puzzles:

1. \(\textrm{ABABA} \stackrel{?}{=} \textrm{ABBAA}\)

2. \(\textrm{AB} \stackrel{?}{=} \textrm{BA}\)

3. \(\textrm{AAABBB} \stackrel{?}{=} \textrm{BBBAAA}\)

4. \(\textrm{BABA} \stackrel{?}{=} \square\)

5. \(\textrm{BAB} \stackrel{?}{=} \textrm{ABABABA}\)

6. \(\textrm{BAB} \stackrel{?}{=} \textrm{BABABAB}\)

CDE

Rules:

1. \(\textrm{CD} = \textrm{DE} = \textrm{EC}\)

2. \(\textrm{CC} = \textrm{DD} = \textrm{EE} = \square\)

Puzzles:

1. \(\textrm{CDCDCD} \stackrel{?}{=} \square\)

2. \(\textrm{CECE} \stackrel{?}{=} \textrm{EC}\)

3. \(\textrm{EDE} \stackrel{?}{=} \textrm{C}\)

4. \(\textrm{EDC} \stackrel{?}{=} \textrm{D}\)

5. \(\textrm{CD} \stackrel{?}{=} \textrm{DC}\)

6. \(\textrm{ED} \stackrel{?}{=} \textrm{CE}\)

FG

Rules:

1. \(\textrm{FFFF} = \textrm{GG} = \textrm{FGFG} = \square\)

Puzzles:

1. \(\textrm{FG} \stackrel{?}{=} \textrm{GF}\)

2. \(\textrm{FGF} \stackrel{?}{=} \textrm{G}\)

3. \(\textrm{GFG} \stackrel{?}{=} \textrm{F}\)

4. \(\textrm{FFFG} \stackrel{?}{=} \textrm{GF}\)

5. \(\textrm{FFG} \stackrel{?}{=} \textrm{GFF}\)

6. \(\textrm{GFGF} \stackrel{?}{=} \square\)

HIJK

Rules:

1. \(\textrm{HH} = \square\)

2. \(\textrm{II} = \textrm{JJ} = \textrm{KK} = \textrm{IJK} = \textrm{H}\)

Puzzles:

1. \(\textrm{IH} \stackrel{?}{=} \textrm{HI}\)

2. \(\textrm{IJ} \stackrel{?}{=} \textrm{JI}\)

3. \(\textrm{IKJ} \stackrel{?}{=} \textrm{H}\)

4. \(\textrm{KIJ} \stackrel{?}{=} \textrm{H}\)

5. \(\textrm{JK} \stackrel{?}{=} \textrm{I}\)

6. \(\textrm{JKJK} \stackrel{?}{=} \textrm{H}\)

LM

Rules:

1. \(\textrm{LLL} = \textrm{MM} = \textrm{MLMLML} = \square\)

Puzzles:

1. \(\textrm{LMLMLM} \stackrel{?}{=} \square\)

2. \(\textrm{LM} \stackrel{?}{=} \textrm{ML}\)

3. \(\textrm{MLLM} \stackrel{?}{=} \textrm{LML}\)

4. \(\textrm{LLMLL} \stackrel{?}{=} \textrm{MLM}\)

5. \(\textrm{LMLML} \stackrel{?}{=} \textrm{M}\)

6. \(\textrm{MLMLM} \stackrel{?}{=} \textrm{L}\)

Notes

• Each of the steps are reversible. This means if you want to show \(\textrm{X} \rightarrow \textrm{Y}\), you can show \(\textrm{X} \rightarrow \textrm{Z}\) and \(\textrm{Y} \rightarrow \textrm{Z}\) for the same \(\textrm{Z}\), then reverse the steps of the last procedure to get \(\textrm{X} \rightarrow \textrm{Z} \rightarrow \textrm{Y}\). Since having a way to get from \(\textrm{X}\) to \(\textrm{Y}\) gives us a way backwards, we might as well be writing \(\textrm{X} = \textrm{Y}\), as our notation has been suggesting.

• As students work through, they can apply their previous work in the next steps. For example, in \(\textrm{AB}\), if they show \(\textrm{AB} = \textrm{BA}\), they will find later puzzles easier if they use this new rule. This also means that finding nice rules that aren't explicitly given as puzzles can be useful. For example, they are only asked to decide if \(\textrm{IH} = \textrm{HI}\) or \(\textrm{IJ} = \textrm{JI}\), but there are four similar rules that are good to explore: whether \(\textrm{JH} = \textrm{HJ}\), \(\textrm{KH} = \textrm{HK}\), \(\textrm{JK} = \textrm{KJ}\), or \(\textrm{IK} = \textrm{KI}\).

• These puzzles come from presentations of finite groups:

• \(\textrm{AB}\) is the Klein four-group \(Z_2 \oplus Z_2\)

• \(\textrm{CDE}\) is the symmetric group \(S_3 \cong D_3\)

• \(\textrm{FG}\) is the dihedral group \(D_4\)

• \(\textrm{HIJK}\) is the quaternion group \(Q_8\)

• \(\textrm{LM}\) is the alternating group \(A_4\)

Some of them can be thought of as symmetries of nice geometric objects, which students may pick up on.

While it is natural to think of certain procedures while working in a group theoretic setting, there are analogs that students might discover here. For example, if two strings are equal, appending the same string to the left ends of both gives another pair of equal springs. (Similarly with appending on the right.)

Extra Challenges

For each of the rule sets above, students might like thinking of other puzzles for each rule set and posing them to each other. If those are understood, here are some more rule sets students might like to explore:

• \(\textrm{NOP}\): \(\qquad \textrm{NN} = \textrm{OOO} = \textrm{PPPP} = \textrm{NOP} = \square\)

• \(\textrm{QR}\): \(\qquad \textrm{QQQ} = \textrm{RRRR} = \textrm{QQRQRRR} = \square\)

• \(\textrm{ST}\): \(\qquad \textrm{SS} = \textrm{TTTT} = \textrm{STSTST} = \square\)

• \(\textrm{UV}\): \(\qquad \textrm{UUU} = \textrm{VVV} = \textrm{UVUV} = \square\)

• \(\textrm{WXYZ}\): \(\qquad \textrm{WW} = \textrm{XY} = \textrm{YZ} = \textrm{ZX}\), \(\qquad \textrm{WWW} = \textrm{XXX} = \textrm{YYY} = \textrm{ZZZ} = \square\)

Students might also enjoy coming up with their own rule sets. Since they likely aren't burdened with the onus of group axioms, they might come up with some wild things, pushing more into MU puzzle territory. Once they have a rule set, they could take a word, turn it into another through a complicated procedure, and then hand the puzzle of getting from one to the other to a friend.

This post was sponsored by the Julia Robinson Mathematics Festival