This solution explains how to determine the day of the week for a specific date, April 17, 2014, using a standard calendar calculation method.

Understanding Calendar Calculations

Determining the day of the week for any given date involves understanding how calendar days progress and accounting for leap years. A common method uses a mathematical formula that considers the day, month, year, century, and leap year rules.

Choosing a Calculation Formula

We will use a well-known algorithm to calculate the day of the week. The formula requires specific inputs derived from the date:

• The day of the month ($q$).
• A code for the month ($m$).
• The year, split into the last two digits ($K$) and the century ($J$).

The formula is:

$$ h = (q + \lfloor \frac{13(m+1)}{5} \rfloor + K + \lfloor \frac{K}{4} \rfloor + \lfloor \frac{J}{4} \rfloor - 2J) \mod 7 $$

Where:

• $q$ is the day of the month.
• $m$ is the month's code: March=3, April=4, May=5, ..., December=12. Importantly, January and February are treated as months 13 and 14 of the *previous* year for this formula's calculation sequence.
• $Y$ is the year.
• $J$ is the century ($\lfloor Y/100 \rfloor$).
• $K$ is the year of the century ($Y \mod 100$).
• $\lfloor x \rfloor$ denotes the floor function (the greatest integer less than or equal to $x$).
• $\mod 7$ means the remainder after dividing by 7.

The result ($h$) maps to the day of the week as follows: 0 = Saturday, 1 = Sunday, 2 = Monday, 3 = Tuesday, 4 = Wednesday, 5 = Thursday, 6 = Friday.

Step-by-Step Calculation for April 17, 2014

Let's apply the formula to the date April 17, 2014:

1. Identify the values:
   • Day ($q$): $17$
   • Month ($m$): April corresponds to $4$.
   • Year ($Y$): $2014$.
2. Determine Century ($J$) and Year ($K$):
   • Century $J = \lfloor 2014 / 100 \rfloor = 20$.
   • Year $K = 2014 \mod 100 = 14$.
3. Check for Month Adjustment: Since the month is April ($m=4$), it is not January or February, so we do not need to adjust the year $Y$.
4. Calculate the terms in the formula:
   • Term 1: $q = 17$.
   • Term 2: $\lfloor \frac{13(m+1)}{5} \rfloor = \lfloor \frac{13(4+1)}{5} \rfloor = \lfloor \frac{13 \times 5}{5} \rfloor = \lfloor 13 \rfloor = 13$.
   • Term 3: $K = 14$.
   • Term 4: $\lfloor \frac{K}{4} \rfloor = \lfloor \frac{14}{4} \rfloor = \lfloor 3.5 \rfloor = 3$.
   • Term 5: $\lfloor \frac{J}{4} \rfloor = \lfloor \frac{20}{4} \rfloor = \lfloor 5 \rfloor = 5$.
   • Term 6: $-2J = -2 \times 20 = -40$.
5. Substitute and calculate $h$: $$ h = (17 + 13 + 14 + 3 + 5 - 40) \mod 7 $$ $$ h = (52 - 40) \mod 7 $$ $$ h = 12 \mod 7 $$ $$ h = 5 $$

Final Day of the Week Determination

The calculated value is $h=5$. Using the formula's mapping (0=Saturday, 1=Sunday, ..., 5=Thursday, 6=Friday), the value 5 corresponds to Thursday.

Therefore, the day of the week on April 17, 2014, was Thursday.