Sequential Problem Solving and Program Design

Fundamental Methodology of Problem-Solving in Programming

• Solving programming problems is analogous to solving high school mathematics word and story problems.
• The process begins with reading the problem to identify the specific requirements and logical steps needed for a solution.
• A standard approach involves isolating three core sets of information to ensure the resulting program is accurate: inputs, processes, and outputs.

Step 1: Identifying and Categorizing Inputs

• The first step in any problem is to understand the inputs. This requires distinguishing critical data from non-essential information included in the problem description.
• For a computer program to process these inputs, the programmer must define specific characteristics about the data:
  • Data Types:
  • Numeric Inputs: These can be classified as whole numbers (counting numbers or integers) which may be positive or negative. Alternatively, they may be floating-point numbers or real numbers, which include decimals.
  • Character Inputs: Individual letters or symbols. It is essential to determine if casing matters (uppercase vs. lowercase).
  • Strings: These are defined as collections of characters or full sequences of words.
  • Restrictions and Limitations:
  • Programs must define if a number needs to be positive, negative, or a zero.
  • Specific data types, such as those used for money, have their own unique logic and restriction requirements.
• It is necessary to determine the method by which the program will receive these inputs from various devices.

Step 2: Determining Calculations and Unit Conversions

• The second step in problem-solving is identifying the calculations or "process" required to reach the solution.
• Formulas and Functions:
  • Basic geometric formulas include the circumference of a circle or the volume of a cube ($v^3$).
  • More complex formulas involve conversions, such as converting Temperatures to Celsius.
  • Advanced mathematical operations may utilize trigonometric functions, exponentials, or logarithms ($\ln$).
• Handling Units and Percentages:
  • Unit conversions are a vital part of the process identifying phase.
  • For instance, if a problem provides a percentage of $10\%$, the calculation within the program must use its decimal equivalent, which is $0.1$. This transition is achieved by moving the decimal point relative to the percent sign.

Practical Application: The Chicken Wood Fence Problem

• To illustrate the Input-Process-Output (IPO) model, consider a story problem where a circular fence is being built around "chicken wood."
• Identifying Inputs:
  • In this descriptive problem, the specified input is the "rating" or radius ($r$).
• The Process and Specific Calculations:
  • To find the required amount of fencing, one must calculate the circumference ($C$).
  • Circumferential Formula: $C = 2 \times \pi \times r$
  • Resource Allocation Logic:
  • Suppose the calculated circumference is $55\text{ feet}$.
  • If the fence rolls are sold in units of $20\text{ foot rolls}$, you cannot purchase a partial roll (e.g., half a roll).
  • Simple division ($\frac{55}{20}$) results in $2.75$. However, the practical requirement is three full rolls ($3 \times 20 = 60\text{ feet}$).
• The Ceiling Function:
  • In programming terminology, rounding up to the next largest whole number is performed by the "ceiling" function.
  • Calculation Rule: $\text{Number of rolls} = \text{ceiling}(\frac{C}{\text{fence roll length}})$

Step 3: Defining the Output

• The final step is determining what the program should display or produce.
• In the fencing example, the output is the total number of fence rolls.
• Data Integrity of Output:
  • The resulting value must be an integer, which is a whole, counting number.
  • The output for this counting process should be zero at its base level, representing a non-negative whole value.