> For the complete documentation index, see [llms.txt](https://wenbo-notes.gitbook.io/cs1010-notes/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://wenbo-notes.gitbook.io/cs1010-notes/past-year-exam/midterm-pe/pe1-ay18-19.md).

# PE1 (AY18/19)

## Problems

### 1. Vote

This is a "give-away" question.

### 2. Newton

In `math.h` library in C, there is a function called `fabs(x)`, which will return the absolute value of a floating-point argument `x`.

We may need the idea of getting $$\epsilon$$ when we want to judge whether two `double` are **equal**. However, if we only want to know which `double` is bigger enough, we can just use the operators $$>,<,\geq,\leq$$, that's enough.

### 3. Goldbach\*

The good way to write `prime`.

{% code title="prime.c" lineNumbers="true" %}

```c
#include "cs1010.h"

bool is_prime(long n)
{
    for (int i = 2; i * i <= n; i += 1)
    {
        if (n % i == 0)
        {
            return false;
        }
    }
    return true;
}
```

{% endcode %}

Note that using this method (`i * i <= n`) can save us from the trouble of dealing with type casting when using the `sqrt()` from `math.h` library.

However, below is mine version of `prime.c`, which may be a little bit faster.

{% code title="prime\_optimized.c" lineNumbers="true" fullWidth="false" %}

```c
#include "cs1010.h"

bool is_prime(long num)
{
    if (num == 2)
    {
        return true;
    }
    if (num % 2 == 0)
    {
        return false;
    }
    for (long i = 3; i * i <= n; i += 1)
    {
        if (num % i == 0)
        {
            return false;
        }
    }
    return true;
}
```

{% endcode %}

### 4. Digits\*

The most important idea in this probelm is to define two variables (so that we won't use `arrays` here) to store:

1. The **digit(0-9)** of the longest sequence
2. The **length** of the longest sequence

This idea comes from the exercise we have done, which is in [/pages/jq4cqBADR1aadQWFcM1I#problem-2.1](https://wenbo-notes.gitbook.io/cs1010-notes/past-year-exam/midterm-pe/pages/jq4cqBADR1aadQWFcM1I#problem-2.1 "mention") about finding the maximum number in a list. Here we just use two variables to store the properties of the "largest" structure we want to find.

{% hint style="warning" %}
There is a trivial case that may be ignored in this probelm if you use `while` loop. That is when you input `num` as $$0$$. To deal with this case correctly, you should initialize the variables correctly and use `do-while` loop.
{% endhint %}

{% code title="goldbach.c" lineNumbers="true" %}

```c
long longest_consecutive_digits(long n)
{
    long longest_count = ­-1;
    long longest_digit;
    long current_count = 0;
    long current_digit = n % 10;
    do
    {
        // Increase the counter if we see the same digit.
        // Otherwise reset counter to 1.
        if (n % 10 == current_digit)
        {
            current_count += 1;
        }
        else
        {
            current_count = 1;
        }
        
        // Checks if we find a longer (or equally long)
        // consecutive sequence. Update longest_digit
        // and longest_count if so.
        if (current_count > longest_count)
        {
            longest_digit = current_digit;
            longest_count = current_count;
        }
        else if (current_count == longest_count)
        {
            if (current_digit < longest_digit)
            {
                longest_digit = current_digit;
            }
        }
        
        // Update the current digit to the last digit of n
        // and shorten n by one digit.
        current_digit = n % 10;
        n = n / 10;
    } while (n > 0);
    return longest_digit;
}
```

{% endcode %}

In this program, the use of `current_digit` is awesome! Actually, it serves the purpose of `previous_digit` and `n % 10` at the first of the loop acts as the true `current_digit`. Initializing the `current_digit` to be `n % 10` deals with the problem of there is no `previous_digit` at the start in an elegant way.

The use of `do-while` loop also guarantees that we will enter the loop **at least** once, which deals with the trivial case of $$0$$ as well.

### 5. Square

The "heart" of this problem is **pattern recognition**. Below are the patterns that are crucial:

1. The first and last rows `(row == 1 || row == width)` are where we have to draw `##...#`
2. The second and penultimate rows `(row == 2 || row == width - 1)` are where we have to draw `#...#`
3. The rest `(row == 3 && row <= width - 2`, where we have to draw `#...#`, but with inner squares with the recursion `print_square(row - 2, width -4)`.

## Tips

1. Always regard **real numbers** as `double` type variable!
2. Include the function `prime` in your cheatsheet!
3. **When to pay attention to the&#x20;*****Big Number*****&#x20;input in the question involving `prime`?**\
   The rule of thumb is **to pass all the test cases**, these cases should be **strick** enough! Also, the `is_prime(num)` function provided here [#id-3.-goldbach](#id-3.-goldbach "mention") should be **quick enough!!!**