Exploring a More Complex Example of Groups: The Broken Train

Now that you've gained an understanding of groups, let's dive into a more challenging scenario.
Earlier, we examined a broken remote that cycled through channels in a predictable pattern. Now, let's consider a different, more intriguing example: a broken train.

Imagine it's been a long day, and you're finally on your way home.
The train is crowded, but you decide to be considerate and let others take the available seats. A woman boards the train and sits on the far left.

Click on the far left seat to see what happens next...

Initially, everything seems normal as the train moves to the next station.
Upon arrival, a man steps onto the train, distracted by his phone. Unaware of the woman already seated, he tries to sit in the same far-left seat. Just as he does, something peculiar occurs.

Click on the far left seat again to see the result.

The woman disappears! The man, instead of sitting on the left, is now seated on the right.
No one else seems to notice, and the man is completely unbothered.

You begin to wonder if you're imagining things, but you decide not to touch any more seats. With growing curiosity, you wait for the next station.

A woman with two children enters. She directs one child to the seat next to the man on the right.

Click on the seat next to the man (the left seat on the right side).

Both the child and the man vanish! You rush over to the mother, concerned, but she looks at you strangely and insists she only had one child.
Though you're certain she entered with two, she quickly finds another seat, seemingly unsettled by your confusion.

You pull out your notebook and start jotting down your observations, eventually formulating three rules:

In summary, here are the 5 rules:

But can we explain these mysterious rules? Do they account for all possible outcomes?
And, more intriguingly, does this broken train form a group like the broken remote?

There are five possible seating arrangements, similar to the five channels on the remote. These are:

You can either write down these possibilities or use the simulation below to test if the train's behaviour forms a group.

Upon inspection, you'll notice that the three rules don't cover all possible scenarios. However, could we perhaps add more rules to form a complete solution? If so, which rules should we add or remove?

Here's a hint: there's only one group with five elements, and it's no coincidence that we studied the broken remote earlier.

Since there's just one group with five elements, we only need to determine whether the train and the remote behave similarly. The key with the remote was the cyclic nature of its one button, which cycled through all the channels. Mathematically, we call this a cyclic element because it covers every possibility when repeated.

If the train operates like the remote, we should start by looking for a cyclic element. Let's begin with LL:

At this point, no rule exists for RR + LL,
so let's introduce two new rules (you can verify that choosing different rules leads to the same final conclusion):

By adding LL, we can generate all possible outcomes. But did we introduce any contradictions by adding these new rules? Let’s check:

If this forms a group, we should be able to add RR + LL first, which gives us RL.

RR + RR = RR + (LL + LL) = (RR + LL) + LL = RL + LL = LR

Now we have two conflicting results:

This contradiction shows that our assumption that the train's behaviour forms a group, even after adding new rules, is false.
You can try other combinations, but you'll find that the train's seats will never behave in the same way as the remote.

Conclusion

By examining simple rules underlying complex behaviours, we discovered a clear pattern with the remote, which is known as a group. However, not all complex behaviours can be simplified in this way, as we saw with the train.