20. Application Before the invention of the
telephone, Samuel Morse (1791-1872)
developed an efficient system for sending
messages as a series of dots and dashes
(short or long pulses). International code, a
modified version of Morse code, is still
widely used.
a) How many different characters can the
international code represent with one to
four pulses?
b) How many pulses would be necessary
to represent the 72 letters of the
Cambodian alphabet using a system
like Morse code?


Let's tackle each part of the problem step by step.

### Part (a) - Number of Different Characters with 1 to 4 Pulses

In Morse code, each character is represented by a sequence of dots and dashes. For this problem, consider each pulse (dot or dash) as a binary choice.

1. **1 pulse:** There are 2 possibilities (dot or dash).
2. **2 pulses:** There are \(2^2 = 4\) possibilities (dot-dot, dot-dash, dash-dot, dash-dash).
3. **3 pulses:** There are \(2^3 = 8\) possibilities.
4. **4 pulses:** There are \(2^4 = 16\) possibilities.

To find the total number of different characters that can be represented with 1 to 4 pulses, we sum the possibilities:

\[
2 + 4 + 8 + 16 = 30
\]

So, **30 different characters** can be represented with 1 to 4 pulses.

### Part (b) - Number of Pulses Necessary for 72 Characters

Here, we need to determine the minimum number of pulses required to represent 72 different characters.

1. **1 pulse:** Represents 2 characters.
2. **2 pulses:** Represents 4 characters.
3. **3 pulses:** Represents 8 characters.
4. **4 pulses:** Represents 16 characters.
5. **5 pulses:** Represents \(2^5 = 32\) characters.
6. **6 pulses:** Represents \(2^6 = 64\) characters.
7. **7 pulses:** Represents \(2^7 = 128\) characters.

Since 72 characters fall between 64 and 128, **7 pulses** are necessary to represent all 72 characters.

### Final Answers:
a) The international code can represent **30 different characters** with one to four pulses.

b) **7 pulses** would be necessary to represent the 72 letters of the Cambodian alphabet using a system like Morse code.

Would you like more details, or do you have any questions? Here are some related questions:

1. How does Morse code differ from binary code?
2. How can we calculate the number of characters represented by more than 4 pulses?
3. What is the mathematical formula to calculate the number of possibilities for any number of pulses?
4. How does increasing the number of pulses impact the complexity of decoding messages?
5. Can Morse code be optimized to represent non-alphabetic symbols?

**Tip:** When dealing with binary choices, powers of 2 are a powerful tool for counting possibilities!

Explore how Morse code represents characters using pulses, including the number of characters represented by 1 to 4 pulses and the pulses required for 72 Cambodian alphabet letters. Suitable for Grades 6-8.

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