This is for SOMEpi
Have you ever faced a challenging problem?
Few things are more satisfying than discovering a simple solution to such complexities.
Take, for example, a broken TV remote. What might initially seem like a straightforward issue can present a complex problem. Yet, finding a simple fix for it can be incredibly satisfying.
This TV has five channels. Try Using the remote below to navigate to your favourite channel.
Channel # 1
Initially, the TV appears to be malfunctioning. When a button on the remote is pressed, the TV switches to an unexpected channel.
Below is a table displaying your previous button presses on the remote. The last column indicates the result of each button press.
current | requested | resulting |
---|
Try pressing a few buttons. Can you identify the pattern?
Spend a minute analyzing the table to find it.
Despite appearing random at first, there is a straightforward explanation for this seemingly dysfunctional TV remote. Continue realing for the explenation.
If you didn't find a simple explanation for the broken TV remote, try pressing the one button five times.
Looking at the table width just the one presses gives the following.
current | requested | resulting |
---|
Every time you press "1," the TV advances to the next channel.When you reach channel five, it loops back to channel one.
A simple solution at last!
Wouldn't it be great if all the buttons followed this straightforward addition pattern?
But the "5" button behaves a little differently. Try pressing it…
Channel # 1
Nothing happens! The current channel plus five results in… the same channel?
If we want to describe the remote's behaviour using simple arithmetic, a better name for the "5" button would be "0." So, we have:
\[ \text{current channel} + zero = \text{current channel} \] The other buttons work in a similar fashion: \[current + requested = resulting\]
But what if the sum of the current and requested channels exceeds four?
Let's find out! Press the remote to test exceeding four.
Any number greater than four (remember, channel five is now channel zero) loops back to zero.
This is just like a clock that loops back to the beginning after reaching twelve.
Move the clock hand four hours forward. Just like with the remote, the clock loops back and lands on two.
This looping behaviour is known in mathematics as modulo, often written as "mod" or "%." A more precise way to understand modulo is through division.
For example, the result of 7 mod 5 is the remainder of 7 divided by 5: \[ 7 \div 5 = 1 \text{ remainder } 2\] This means: \[ 7\mod 5 = 2\]
If this is new to you, just think of modulo as the same concept as the clock looping around.
This leads us to the final formula for our TV remote: \[ (current + requested) \text{ looping from } 5 = resulting\] Or, even more succinctly: \[ (current + requested)\mod 5 = resulting\]
In mathematics, the Lager Pattern exemplifies a concept known as addition in \(\mathbb{Z}_5\).
This refers to addition performed under modulo 5, where \(\mathbb{Z}\) signifies the use of whole numbers only.
How can you discover this pattern yourself, especially if you're not a math expert? How can you be sure that the remote isn't just randomly selecting channels?
Start with a simpler approach: look for a rule that consistently applies when using the broken remote.
Here's a hint: With just 5 rules, you can fully describe the behavior of the broken remote.
The resulting channel will always be one of the 5 available channels. Let's label these channels \(G\).
This rule might seem obvious, but it has a specific term in mathematics: a binary operation. A binary operation involves two parameters,
the current and the requested channel, and our remote always returns a channel within \(G\).
Another observation reveals that the order of the channels doesn't matter.
For instance, combining channels 1 and 2 yields the same result as combining channels 2 and 1.
This is known as symmetry: the remote operation is commutative.
For button 5, we've identified that it always leaves the current channel unchanged.
This button represents what we call the zero element because,
just like adding zero in arithmetic, it doesn't alter the outcome:
\[ zero element+original=original=original+zero element \]
The presence of such a zero element is another rule of our remote.
The fourth rule states that every channel has an inverse.
You can always return to the zero element (channel 5) by using the appropriate inverse channel.
For a given channel, the inverse can be found as:
\[ inverse\text{ of current }=(5 - current)mod5\]
This means:
\[(current+inverse)mod5=(current+(5-current))mod5=0\]
Channel 0 and channel 5 are considered the same as stated above.
The last rule involves how three channels are combined. The order in which you combine the channels doesn't matter: \[ (request1+request2)+request3=request1+(request2+request3)\] This property is known as associativity.
In summary, here are the 5 rules:
Structures that follow these 5 rules are termed groups.
Despite appearing random, our TV remote adheres to these rules, just as simple addition within whole numbers does.
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:
Here are the 3 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.
You can refer to this table for guidance. The table updates dynamically based on your actions, displaying the outcome of seating a person given the previous arrangement.
E | LL | LR | RL | RR | |
---|---|---|---|---|---|
E | |||||
LL | |||||
LR | |||||
RL | |||||
RR |
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.
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.
Thank you for reading. My Name is Jannes Vanquaillie, a student in Computer Science. I would love your feedback on the article. The code behind it is not that impressive and if you want to seed it or leave feedback on the article you can have a look at the GitHub page Jannaiz/Broken-Objects Be sure to check out the other articles and videos of SOMEpi (I'll place a link later).