Welcome to the broken TV remote

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.

The Broken TV Remote

This TV has five channels. Try Using the remote below to navigate to your favourite channel.

TV Screen

Channel # 1

Image with buttons

Initially, the TV appears to be malfunctioning. When a button on the remote is pressed, the TV switches to an unexpected channel.

History

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.

Complex At First, Simple At Last

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…

TV Screen

Channel # 1

Image with buttons

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.

12
1
2
3
4
5
6
7
8
9
10
11

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: \[ (current + requested)\mod 5 = resulting\]

The Lager Pattern

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.

  1. Basic Rule: Channel Range

    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 \(G\).

  2. Symmetry

    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.

  3. Zero Element

    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.

  4. Inverse Element

    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.

  5. Associativity

    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:

  1. The resulting channel is always one of the 5 available channels.
  2. The operation is commutative (order doesn't matter).
  3. There is a zero element (button 5) that leaves the channel unchanged.
  4. Every channel has an inverse channel that returns you to the zero element.
  5. The operation is associative (grouping doesn't matter).

Conclusion

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.