In algebra, dealing with parabolas involves graphing quadratics or finding the max/min points, also known as vertices. In the context of conics, a parabola is formed by a line, called the directrix, and a point, called the focus. The parabola is the curve formed by all points equidistant from the directrix and focus. The line perpendicular to the directrix, passing through the focus, is the axis of symmetry. The point on this axis, midway between the focus and directrix, is the vertex, where the parabola changes direction.

The name "parabola" refers to the fact that the distance from the parabola to the focus is always equal to the distance from the parabola to the directrix. The vertex is exactly midway between the directrix and focus. Parabolas can be "right side up", "upside down", or "sideways", with axes of symmetry parallel to the x-axis, opening to the right or left.

A key property of parabolas is that any light or sound ray entering parallel to the axis of symmetry and hitting the inner surface will be reflected back to the focus. This property is utilized in "parabolic dishes" such as radio telescopes. The focus is always inside the parabola, the vertex is on the parabola, and the directrix is outside.

The general form of a parabola's equation is y = ax^2 + bx + c, unless it's "sideways", in which case the equation is x = ay^2 + by + c. The important difference is which variable is squared: for regular parabolas, x is squared, while for sideways parabolas, y is squared.

The vertex form of a parabola with its vertex at (h, k) is y = a(x - h)^2 + k. The conics form of the parabola equation is 4p(y - k) = (x - h)^2. The relationship between the vertex form and conics form is a rearrangement, where 4p is equal to 1/a.

The variable p represents the distance between the vertex and focus, and also the distance between the vertex and directrix. The distance between the focus and directrix is 2p. By comparing the equation with the conics form, the center can be determined as (h, k). The coefficient of the unsquared part is 4p, which can be used to find p.

For example, given the equation, the center can be found at (h, k) = (5, 3), with p = 2. Since the y part is squared and p is positive, the parabola opens to the right. The focus is two units to the right of the vertex, at (7, 3).

In another example, the conics form of the equation has subtraction inside the parentheses, indicating the vertex is at (-3, 1). The coefficient of the unsquared part is -20, which is also the value of 4p, so p = -5. Since the x part is squared and p is negative, the parabola opens downward, with the directrix five units above the vertex, at y = 6.


## JavaScript Rounding

JavaScript provides several methods for rounding numbers, including `Math.round()`, `Math.floor()`, `Math.ceil()`, and `Math.trunc()`. These methods can be used to round numbers to the nearest integer or to a specified number of decimal places.

### Rounding to the Nearest Integer

* `Math.round()` rounds a number to the nearest integer. If the decimal part is exactly halfway between two integers, it rounds up.
* `Math.floor()` rounds a number down to the nearest integer.
* `Math.ceil()` rounds a number up to the nearest integer.
* `Math.trunc()` removes the decimal part of a number, leaving only the integer part.

Examples:
```
Math.round(2.3)  // 2
Math.round(2.5)  // 3
Math.floor(2.3)  // 2
Math.floor(2.5)  // 2
Math.ceil(2.3)   // 3
Math.ceil(2.5)   // 3
Math.trunc(2.3)  // 2
Math.trunc(2.5)  // 2
```

### Rounding to Decimal Places

* `Number.toFixed()` rounds a number to a specified number of decimal places.
* `Number.toPrecision()` rounds a number to a specified number of significant figures.

Examples:
```
2.4387587.toFixed(2)  // "2.44"
2.4387587.toPrecision(2)  // "2.4"
```

Note that `Number.toFixed()` and `Number.toPrecision()` return strings, so you may need to use the `Number()` function to convert the result back to a number.

### Problems with Rounding Numbers

JavaScript stores numbers as 32-bit single precision binary values, which can lead to rounding errors in some cases. For example:
```
0.1 + 0.2 === 0.3  // false
```
This is because 0.1 and 0.2 cannot be represented exactly in binary, resulting in a small error when adding them together.

### Choosing the Right Rounding Method

The choice of rounding method depends on the specific use case. If you need to round a number to the nearest integer, `Math.round()` is usually the best choice. If you need to round to a specified number of decimal places, `Number.toFixed()` or `Number.toPrecision()` may be more suitable.

### Example Use Cases

* Rounding a number to the nearest integer: `Math.round(2.3)` => 2
* Rounding a number to two decimal places: `2.4387587.toFixed(2)` => "2.44"
* Rounding a number to two significant figures: `2.4387587.toPrecision(2)` => "2.4"

By understanding the different rounding methods available in JavaScript, you can choose the best method for your specific use case and avoid common pitfalls related to rounding errors.


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#### Question

The odometer of a car reads 57321.0 km at 08:30 AM and 57336.0 km at 08:50 AM. Calculate the distance moved by the car and its speed in km/min and km/h.

#### Solution

Initial odometer reading: 57321.0 km
Final odometer reading: 57336.0 km

Distance covered = Final reading - Initial reading = 57336.0 - 57321.0 = 15 km

Time taken: 08:50 AM - 08:30 AM = 20 min

Speed in km/min = Distance / Time = 15 km / 20 min = 0.75 km/min

To convert time to hours: 20 min = (1/60) * 20 = 1/3 h

Speed in km/h = Distance / Time = 15 km / (1/3 h) = 45 km/h


**RRB Arithmetic Practice Questions**

1. A clock shows 6 a.m. and gains 2 minutes every hour. How many minutes will it gain by 9 p.m.?
(a) 30 minutes
(b) 25 minutes
(c) 28 minutes
(d) 34 minutes

2. Find the next number in the series: 2, 4, 8, 32, 256, ?
(a) 4096
(b) 8192
(c) 512
(d) 1024

3. Find the missing number: 10, 11, 23, 39, 64, ?, 149
(a) 100
(b) 103
(c) 78
(d) 128

4. A bus travels from Trivandrum to Attingal in 45 minutes at 40 km/hr. If the speed increases by 10 km/hr, how much time will it take?
(a) 34 minutes
(b) 36 minutes
(c) 38 minutes
(d) 40 minutes

5. The difference between 6 times and 8 times a figure is 14. What is the figure?
(a) 12
(b) 9
(c) 7
(d) 6

6. If 92y = 36, what is 9y?
(a) 4
(b) 6
(c) 9
(d) 18

7. One-fourth percent of 180 is:
(a) 4.5
(b) 0.45
(c) 0.045
(d) 45

8. A candidate needs 40% marks to pass. He scored 40 marks and failed by 20 marks. What are the maximum marks?
(a) 100
(b) 200
(c) 180
(d) 150

9. Find the missing number: 32, 52, 74, 112, 135, ?
(a) 16
(b) 15
(c) 17
(d) 14

10. If 250 is increased to 300, what is the percentage increase?
(a) 16.67
(b) 20
(c) 23
(d) 17

11. The ratio of 9 seconds to 10 hours is:
(a) 1:40
(b) 1:4000
(c) 9:10
(d) 1:400

12. A person sold goods at Rs.153 and lost 10%. At what price should he sell to gain 20%?
(a) 204
(b) 250
(c) 240
(d) 210

13. What will be xy if 7862xy is divisible by 125?
(a) 25
(b) 00
(c) 75
(d) 50

14. A 100-meter train runs at 36 km/hr. How long will it take to pass an 80-meter bridge?
(a) 30 seconds
(b) 36 seconds
(c) 20 seconds
(d) 18 seconds

15. If two-thirds of a bucket is filled in one minute, how long will it take to fill the bucket?
(a) 90 seconds
(b) 70 seconds
(c) 60 seconds
(d) 100 seconds

16. If a quarter kilogram costs Rs.60, how much will 150 grams cost?
(a) Rs.30
(b) Rs.24
(c) Rs.36
(d) Rs.40

17. If 3 men or 6 boys can do a job in 20 days, how many days will 6 men and 8 boys take?
(a) 5
(b) 8
(c) 10
(d) 6

18. What is the sum of the first 100 natural numbers?
(a) 5050
(b) 5005
(c) 9900
(d) 9050

19. Two poles of height 6 meters and 11 meters stand 12 meters apart. What is the difference in distance between their tops?
(a) 12m
(b) 5m
(c) 13m
(d) 11m

20. How many balls of radius 4 cm can be made from a solid sphere of radius 8 cm?
(a) 4
(b) 8
(c) 16
(d) 2

21. Solve the equation x2 + 6x + 9 = 0.
(a) x1 = +3, x2 = -3
(b) x1 = 3, x2 = 3
(c) x1 = -3, x2 = -3
(d) No solution

22. What is the probability of getting a 2 or 4 on a die?
(a) 2/3
(b) 1/6
(c) 1/3
(d) 1/2

23. At what rate will an amount double in 10 years?
(a) 10%
(b) 7.5%
(c) 16%
(d) 15%

24. Five times an unknown number is 5 less than 50. What is the unknown number?
(a) 10
(b) 11
(c) 9
(d) 5

25. What is the angle between the hour and minute hands at 4 PM?
(a) 90°
(b) 120°
(c) 150°
(d) 240°

26. Water fills a vessel, doubling every 5 minutes. If it takes 30 minutes to fill, when will it be one-fourth full?
(a) 20 minutes
(b) 25 minutes
(c) 7 minutes 30 seconds
(d) 10 minutes

27. If 10 cats kill 10 rats in 10 minutes, how long will 100 cats take to kill 100 rats?
(a) 1 minute
(b) 10 minutes
(c) 100 minutes
(d) 10000 minutes

28. If 75% of a number added to 75 equals the number, what is the number?
(a) 250
(b) 750
(c) 400
(d) 300

29. Food for 400 children lasts 12 days. How long will it last if 80 more children join?
(a) 6 days
(b) 7 days
(c) 10 days
(d) 8 days

30. The sum of two consecutive numbers is 55. What is the larger number?
(a) 25
(b) 28
(c) 26
(d) 27

31. A shopkeeper sells 2/3 of an item at the cost price of the whole lot. What is his profit percentage?
(a) 33 1/8%
(b) 66 2/3%
(c) 25%
(d) 50%

32. A rectangular field has a perimeter of 480 meters and a length-to-breadth ratio of 5:3. What is its area?
(a) 7200m²
(b) 15000m²
(c) 54000m²
(d) 13500m²

33. If adding 100 to a number results in more than multiplying it by 1000, what is the number?
(a) 1.5
(b) 1.0
(c) 2.5
(d) 2.0

34. A student needs 40% marks to pass. He scored 150 and failed by 30. What are the maximum marks?
(a) 400
(b) 500
(c) 475
(d) 450

35. Where is the circumcenter of an obtuse triangle?
(a) Interior of the triangle
(b) Midpoint of the diameter
(c) Exterior of the triangle
(d) Midpoint of the side of the triangle

36. What is the degree measure of a semicircle?
(a) 360
(b) 90
(c) 270
(d) 180

37. What is the point of intersection of a triangle's medians?
(a) Circumcenter
(b) Centroid
(c) Orthocenter
(d) Incenter

38. A cone's height and base diameter are equal. If the base radius is 'r', what is its slant height?
(a) 3r
(b) 4r
(c) √5r
(d) √3r

39. The radii of two spheres are in the ratio 2:3. What is the ratio of their surface areas?
(a) 4:9
(b) 2:3
(c) 8:27
(d) 4:6

40. What is the common ratio of the progression 3√2, 6, 6√2?
(a) 3
(b) 2
(c) √2
(d) √3

41. In a class of 100 students, 50 passed math and 70 passed English. 5 students failed both. How many passed both?
(a) 25
(b) 30
(c) 50
(d) 45

42. A boat's speed in still water is 9 km/hr. It travels 12 km downstream and back in 3 hours. What is the speed of the water?
(a) 3 km/hr
(b) 4 km/hr
(c) 4.5 km/hr
(d) 5 km/hr

43. A student added 10 and subtracted 16 from a number instead of adding 16 and subtracting 10. The result was 14. What is the correct answer?
(a) 20
(b) 26
(c) 30
(d) 32

44. Find the area of a right triangle with a hypotenuse of 10 cm and a base of 8 cm.
(a) 48 sq.cm
(b) 34 sq.cm
(c) 24 sq.cm
(d) 42 sq.cm

45. Find the next term in the series: 3, 6, 9, 18, 27, 54, ?
(a) 81
(b) 69
(c) 63
(d) 57

46. A number is 20 plus 20% of its value. What is the number?
(a) 20
(b) 22
(c) 25
(d) 30

47. Calculate 20% of 5 + 5% of 20.
(a) 5
(b) 2
(c) 6
(d) 21

48. What is the angle between the minute and hour hands at 8:30?
(a) 80°
(b) 60°
(c) 105°
(d) 75°

49. Rs.1581 is divided among A, B, and C in the ratio 10:15:6. What is B's share?
(a) 306
(b) 765
(c) 700
(d) 510

50. The sum of four consecutive numbers is 154. What is the smallest number?
(a) 36
(b) 37
(c) 38
(d) 31

Answers:
1. A
2. B
3. A
4. B
5. C
6. B
7. C
8. D
9. C
10. A
11. B
12. A
13. D
14. D
15. A
16. C
17. D
18. A
19. C
20. C
21. C
22. B
23. A
24. C
25. B
26. A
27. B
28. D
29. C
30. C
31. A
32. D
33. B
34. D
35. C
36. D
37. B
38. C
39. A
40. C
41. A
42. A
43. B
44. C
45. A
46. C
47. B
48. D
49. B
50. C


To perform an Analysis of Variance (ANOVA) to contrast differences between two groups, researchers first need to collect and summarize their data. Although statistical software can easily compute ANOVA, calculating it by hand helps understand the individual steps involved in showing the differences between groups.

### Step 1: Summarize Statistics
List the summary statistics for easy reference, including individual data points for the first group ("x"), the second group ("y"), and the number of data points for each group ("n").

### Step 2: Sum Data Points
Add all data points for each group, resulting in "SX" for the first group and "SY" for the second group.

### Step 3: Calculate Mean Correction
Use the formula C = (SX + SY)^2 / (2n) to calculate the mean correction.

### Step 4: Calculate Sum of Squares Between Groups
Calculate the sum of squares between groups (SSB) using the formula SSB = [(SX^2 + SY^2) / n] - C.

### Step 5: Square Data Points
Square all data points and organize them for easy reference.

### Step 6: Sum Squared Data Points
Sum all squared data points into a single sum, "D", without separating the groups.

### Step 7: Calculate Sum of Squares Total
Calculate the sum of squares total (SST) using the formula SST = D - C.

### Step 8: Calculate Sum of Squares Within Groups
Calculate the sum of squares within groups (SSW) using the formula SSW = SST - SSB.

### Step 9: Compute Degrees of Freedom
Compute the degrees of freedom for between groups (dfb = 1) and within groups (dfw = 2n - 2).

### Step 10: Compute Mean Square Between Groups
Calculate the mean square for between groups (MSB) using the formula MSB = SSB / dfb.

### Step 11: Compute Mean Square Within Groups
Calculate the mean square for within groups (MSW) using the formula MSW = SSW / dfw.

### Step 12: Calculate F Statistic
Calculate the final F statistic using the formula F = MSB / MSW.

#### Required Materials
- A data set


## The Most Asked Coding Interview - #5

**Problem Statement:**
Transform an array such that each element contains the greatest element on its right side. Given an array A of size N, modify A in place such that A[i] = M, where M is the maximum value among all elements A[j], j > i. If such an element does not exist, set A[i] = -1.

**Importance:**
This problem has been asked at Amazon coding interviews and is considered a must-do for coding interview preparation.

**Example Solution:**
The solution to this problem is straightforward and is a good example of an easy coding interview question.


Wien's Law Calculator
This calculator estimates the temperature of an object based on the peak wavelength or frequency of its thermal emission spectrum. Wien's displacement law describes the relation between the emission spectrum of a black body and its temperature, stating that the higher the temperature, the lower the wavelength λ_{max} for which the radiation curve reaches its maximum.

Wien's Displacement Law
The law states that λ_{max} (peak wavelength) is inversely proportional to temperature. The shift to shorter wavelengths corresponds to photons of higher energies. The Stefan-Boltzmann law takes into account the total power radiated from a body at any temperature. Together with Wien's law, they originate from Planck's distribution.

Wien's Law Formula
The equation describing Wien's law is:
λ_{max} = b / T,
where:
λ_{max} – Peak wavelength of light;
T – Absolute temperature of a black body; and
b = 2.8977719 mm·K – Wien's displacement constant.
The formula for the peak frequency is:
f_{max} = k × T,
where k = 5.8789232 × 10¹⁰ Hz/K is a numerical constant.

Estimating Temperature
To estimate the temperature of an object, apply Wien's displacement law. For example, to calculate the temperature of the Sun's surface:
1. Find the peak wavelength of the solar spectrum (approximately λ_{max} = 501.7 nm).
2. Transform the Wien's law formula to obtain the temperature: T = b / λ_{max} = 2.8977719 mm·K / 501.7 nm = 5,776 K.

FAQ
1. How do I find the temperature using Wien's law?
To find the temperature, determine the peak wavelength, take the Wien's displacement constant (b = 2.8977719 mm·K), and divide this constant by the estimated peak wavelength.
2. What does Wien's law reveal about stars in the sky?
Wien's law tells us that different star types have surface temperatures reflected by their apparent colors, with the coldest visible stars appearing red and the hottest ones appearing blue.
3. Can you use Wien's law to tell star temperature?
Yes, but it's usually a rough estimate. Wien's law determines the ideal black body temperature, and not all stars' spectra match this model.
4. How do I find an A-type star temperature?
Given the peak wavelength (λ_{max} = 340 nm), convert it to meters, divide the Wien's displacement constant by this value, and obtain the temperature (approximately 8,523 K). A-type stars' surface temperature spans 7,400 to 10,000 K.


## Making Change using Partners of One Hundred

### Objective:
Students will be able to make change from one dollar using dimes by utilizing the relationship between addition and subtraction to solve problems.

### Warm Up (10 minutes)
To activate prior knowledge, students respond to problems in their math journals:
- How much is ten less than 45, 88, and 61?
- What is changing in each problem? (The tens place decreases by 1 group of 10.)
- What would 20 less than 45 be? Which place is changing? (The tens place.)
- Are we adding or subtracting? How do you know? (Answers are getting smaller.)

### Teaching the Lesson (15 minutes)
Using a Smart Board or classroom objects, the teacher:
- Displays a dollar bill, objects, and dimes.
- Explains that we are shopping with a one-dollar limit and no change.
- Asks students for suggestions on how to buy an item costing less than a dollar without spending the whole dollar.
- Reminds students of a previous lesson using a balance scale to pay with a quarter and add pennies or nickels to make it equal.
- Models making change with a dollar and dimes using a toy car priced at 80 cents.
- Asks students to find the difference between the dollar and the object's price using dimes.

### Partner Practice (15 minutes)
- Students are given a card and find their partner with the same number.
- Each student receives one dollar, and each pair receives five dimes.
- Students draw objects priced between 50 and 90 cents and take turns buying the object with a dollar and asking their partner for change.
- The teacher observes and adjusts the prices to 45-95 cents, adding nickels for more complex change calculations.

### Closing (10 minutes)
- Students return materials and sit down.
- The teacher draws a balloon priced at 60 cents and asks how much change they would get if they paid with a dollar.
- After students solve the problem, a volunteer shares their answer, and the class discusses.
- A second problem with a book priced at 30 cents is presented, and students repeat the process.

### Resources and Similar Lessons
- Cover Up
- Domino Addition: Understanding the Part/Part/Whole Relationship
- It's As Basic As That
- Unit and lesson listings for further context and related topics.


All digital computers, from supercomputers to smartphones, are electronic devices that can detect whether an electrical signal is on or off. This basic information is called a bit (binary digit) and has two values: 1 (true) when the signal is on and 0 (false) when the signal is off.

Larger values can be stored by a group of bits. For example, 2 bits can store 4 different values (00, 01, 10, and 11), 3 bits can store 8 values (000, 001, 010, 011, 100, 101, 110, and 111), and 4 bits can store 16 values (0000, 0001, ..., 1111). In general, n bits can store 2^n different values. However, large numbers using only 0s and 1s are unwieldy for humans. For instance, a computer would need 19 bits to store numbers up to 500,000.

Fortunately, different number systems can be used to work with large binary strings that represent numbers within computers.

## Different Number Systems

A number system is the way we name and represent numbers. The decimal number system, or base 10, is used in daily life and is based on 10 different digits: 0, 1, 2, ..., 8, and 9. The base of a number is written as a subscript. For example, the decimal number "twelve thousand three hundred and forty-five" is written as 12345_10. Without a base, it's assumed that a number is in base 10.

In computer science, three additional number systems are used: binary (base-2), octal (base-8), and hexadecimal (base-16). The table below compares these number systems:

| Number System | Base | Digits Used | Sample Numbers |
| --- | --- | --- | --- |
| Binary | 2 | 0, 1 | 10110_2, 10110011_2 |
| Octal | 8 | 0, 1, 2, 3, 4, 5, 6, 7 | 75021_8, 231_8, 60012_8 |
| Decimal | 10 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | 97425_10 or 97425 |
| Hexadecimal | 16 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F | 54A2DD0F_16 |

Octal and hexadecimal are used to represent binary numbers in a user-friendly way. Each octal symbol represents 3 binary bits, and each hex digit represents 4 binary bits. For example, the number 53201, stored as the binary number 10000001111000101100, is represented by the octal number 2017054 and the hex number 81E2C.

In general, N bits have 2^N different values. Computers with 8-bit words of memory can store 256 different values, displayed using 2 hex characters.

The following table shows the first 20 numbers in decimal, binary, octal, and hexadecimal:

| Decimal | Binary | Octal | Hexadecimal |
| --- | --- | --- | --- |
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 2 | 10 | 2 | 2 |
| 3 | 11 | 3 | 3 |
| 4 | 100 | 4 | 4 |
| 5 | 101 | 5 | 5 |
| 6 | 110 | 6 | 6 |
| 7 | 111 | 7 | 7 |
| 8 | 1000 | 10 | 8 |
| 9 | 1001 | 11 | 9 |
| 10 | 1010 | 12 | A |
| 11 | 1011 | 13 | B |
| 12 | 1100 | 14 | C |
| 13 | 1101 | 15 | D |
| 14 | 1110 | 16 | E |
| 15 | 1111 | 17 | F |
| 16 | 10000 | 20 | 10 |
| 17 | 10001 | 21 | 11 |
| 18 | 10010 | 22 | 12 |
| 19 | 10011 | 23 | 13 |
| 20 | 10100 | 24 | 14 |

## Formation of Numbers

The formation of any number involves multiplying each digit by its place value and adding them together. For example, the number 12345 can be broken down as follows:
12345 = 1 × 10000 + 2 × 1000 + 3 × 100 + 4 × 10 + 5 × 1 = 1 × 10^4 + 2 × 10^3 + 3 × 10^2 + 4 × 10^1 + 5 × 10^0.

All other bases work similarly:
1101_2 = 1 × 2^3 + 1 × 2^2 + 0 × 2^1 + 1 × 2^0 = 8 + 4 + 0 + 1 = 13_10
175_8 = 1 × 8^2 + 7 × 8^1 + 5 × 8^0 = 1 × 64 + 7 × 8 + 5 × 1 = 64 + 56 + 5 = 125_10
A5E_16 = 10 × 16^2 + 5 × 16^1 + 14 × 16^0 = 10 × 256 + 5 × 16 + 14 × 1 = 2560 + 80 + 14 = 2654_10

## Grouping Binary Digits for Octal and Hexadecimal Numbers

To convert from octal to base 2, replace each octal digit with its corresponding 3 binary bits. For example:
375_8 = 011 111 101_2

To convert from hex to base 2, replace each hex digit with its corresponding 4 binary bits. For example:
FD_16 = 1111 1101_2

To convert from binary to octal or hex, group the bits by 3s or 4s (starting from the right) and convert each group:
10000001111000101100 = 10 000 001 111 000 101 100 = 2017054_8
10000001111000101100 = 1000 0001 1110 0010 1100 = 81E2C_16

It's easy to convert between base 8 and 16 by expressing the number in base 2 and then converting that number from base 2.

## Using Hexadecimal Numbers to Represent Colors

Computers use hexadecimal numbers to represent colors in computer graphics. Each color is represented by a combination of red, green, and blue (RGB) light. Two digits are used for each color, resulting in 256 possible shades for each. The hash tag or number sign denotes a hexadecimal number. For example, the color red is #FF0000, green is #00FF00, and blue is #0000FF. Black is #000000, and white is #FFFFFF.

## Resources

Ryan's Tutorials cover this topic in detail, including number systems, binary conversions, binary arithmetic, and fractions in other bases.

The CoolConversion.com online calculator is a useful tool for practicing conversions between decimal, hexadecimal, octal, and binary.

## Format of ACSL Problems

ACSL problems will focus on converting between binary, octal, decimal, and hexadecimal, as well as basic arithmetic operations in these bases.

To be successful, you should know the following facts:
- The binary value of each octal digit 0, 1, ..., 7
- The binary value of each hex digit 0, 1, ..., 9, A, B, C, D, E, F
- The decimal value of each hex digit 0, 1, ..., F
- Powers of 2, up to 4096
- Powers of 8, up to 4096
- Powers of 16, up to 65,536

## Sample Problems

### Sample Problem 1

Solve for x where x_16 = 3676_8.

Solution: Convert 3676_8 to base 10 and then to base 16.
3676_8 = 011 110 111 110_2
= 0111 1011 1110_2
= 7 B E_16

### Sample Problem 2

Solve for x in the hexadecimal equation: x = F5AD_16 - 69EB_16.

Solution: Perform the subtraction directly in base 16.
x = F5AD_16 - 69EB_16
= 8BC2_16

### Sample Problem 3

How many numbers from 100 to 200 in base 10 consist of distinct ascending digits and also have distinct ascending hex digits when converted to base 16?

Solution: There are 13 numbers that meet the criteria:
123 (7B), 124, 125, 126, 127 (7F), 137 (89), 138, 139 (8B), 156 (9C), 157, 158, 159 (9F), 189 (BD)

## Video Resources

Several YouTube videos cover computer number systems, including:
- Number Systems - Converting Decimal, Binary and Hexadecimal (Joe James)
- Hexadecimal Tutorial (Carl Herold)
- Hexes and the Magic of Base 16 (Vaidehi Joshi)
- Collins Lab: Binary & Hex (Adafruit Industries)


# Problem H: Star Wars Order

## Definition
The *starwars-order* operation rearranges a list of numbers by moving the first third to the center, the center third to the front, and keeping the last third in place.

## Problem Statement
Given a list of numbers, determine their *starwars-order*.

## Example
For the Star Wars movies, the *starwars-order* of $1, 2, 3, 4, 5, 6, 7, 8, 9$ is $4, 5, 6, 1, 2, 3, 7, 8, 9$.

## Input
- The first line contains an integer $n$, the number of numbers, which is a multiple of $3$.
- The second line contains $n$ distinct integers separated by spaces, each ranging from $1$ to $10^9$.

## Output
The *starwars-order* of the input values, on a single line, separated by spaces.

## Scoring
| Group | Points | Constraints |
| --- | --- | --- |
| 1 | 20 | $n = 9$ |
| 2 | 80 | $3 \leq n \leq 3*10^5$ |

## Sample Inputs and Outputs
| Sample Input | Sample Output |
| --- | --- |
| 9
 1 2 3 4 5 6 7 8 9 | 4 5 6 1 2 3 7 8 9 |
| 3
 5 9 10 | 9 5 10 |
| 6
 6 55 1 3 8 34 | 6 8 1 3 34 55 |


The wave motion is characterized by the oscillation of water molecules at any given point. During propagation, a water molecule completes one complete oscillation, taking time $T$ to do so. This time $T$ is equal to the time taken for one complete wave propagation.

The displacement-time plot for oscillation shows that the distance $\lambda$ (one wavelength) takes $T$ amount of time. The speed of wave propagation can be calculated using the formula:

$v = \frac{distance}{time} = \frac{\lambda}{T} = \lambda f$

where $v$ is the speed, $\lambda$ is the wavelength, $T$ is the time period, and $f$ is the frequency.

For example, given a sound wave with a frequency of $400Hz$ and a wavelength of $0.8m$, the speed of the propagating sound wave can be calculated as:

$v = \lambda f = 0.8m \times 400Hz = 320ms^{-1}$

This demonstrates the relationship between wavelength, frequency, and speed of wave propagation.


The behavior of functions as x approaches infinity varies. For example, the function f(x) = 1/x approaches 0 as x approaches infinity. In contrast, the function f(x) = x^2 + x approaches infinity as x approaches infinity.

In polynomial fractions, such as f(x) = (ax^2 + x) / (bx^2 + 1), the function approaches a/b as x approaches infinity.

A polynomial is an expression of finite length formed by variables and constants using addition, subtraction, multiplication, and non-negative whole number exponents. Examples of polynomials include:

* p(x) = a0 + a1x + a2x^2 + ... + an*x^n, a polynomial with a single variable and n terms
* p(x) = x, a polynomial with a single term (also called a monomial)
* p(x) = a + bx, a polynomial with two terms (also called a binomial)

The following are not examples of polynomials:
* p(x) = 5/x, because 1/x has a negative exponent
* p(x) = x^2 + x^3/2, because the second term has a non-integer exponent

To determine whether a polynomial increases or decreases as x approaches infinity, we examine the leading term (the term with the highest exponent) and its coefficient. If the coefficient is positive, the polynomial increases; otherwise, it decreases.

A polynomial approaches infinity as x approaches infinity if the leading term has a positive coefficient. It approaches negative infinity as x approaches infinity if the leading term has a negative coefficient. A polynomial cannot have a finite limit as x approaches infinity (or negative infinity), unless it is a polynomial with only a constant term.

Examples:
* p(x) = x approaches infinity as x approaches infinity
* p(x) = x^2 - x approaches infinity as x approaches positive or negative infinity, because the term x^2 has a positive coefficient.

Question: Do the values of a polynomial go to infinity?
Answer: A polynomial's behavior as x approaches infinity depends on the leading term and its coefficient. If the coefficient is positive, the polynomial approaches infinity; if negative, it approaches negative infinity. Only a polynomial with a constant term has a finite limit as x approaches infinity.