Question

Carl Friedrich, Joachim, and Jürgen met at a Sylvester party on Thursday, 31 December \(1998 .\) They agreed to play Skat (a German card game) together some day as soon as all of them find the time to do so. But they got into the usual troubles: Carl Friedrich was busy except on Fridays, Joachim had time on 7 January and then again every 9 th day, and Jürgen was free on 6 January and then again every 11th day. Which date did they agree upon?

Short answer

The agreed date was February 5, 1999.

Step-by-step solution

Determine Joachim's Available Days

Joachim is initially free on January 7 and then every 9th day after that. To find Joachim's availability, we follow this pattern and add 9 to 7 iteratively to get the sequence of dates: January 7, 16, 25; February 3, 12, 21; etc.

Determine Jürgen's Available Days

Jürgen is initially free on January 6 and then every 11th day. By following this pattern, we add 11 to 6 iteratively to get the sequence of dates: January 6, 17, 28; February 8, 19; etc.

Identify Common Dates with Joachim and Jürgen

From Steps 1 and 2, we list the dates each is free and find the common dates between them. The available dates are 17 January, 28 January, and so forth.

Determine Availability on Fridays

Since Carl Friedrich is only available on Fridays, check the common dates from Step 3 to see which fall on a Friday. January 28, 1999 was a Thursday, so it'll be February 5, 1999. Verify the first joint availability on a Friday.

Key concepts

Linear Algebra

Linear Algebra is a branch of mathematics that deals with vectors, matrices, and systems of linear equations. It provides the mathematical framework for discussing the nature of linear relationships and transformations. This topic is incredibly relevant in a wide variety of fields, such as engineering, physics, and computer science where modeling and solving systems are essential.

In our exercise, while not directly involving vectors or matrices, there is a conceptual relation to Linear Algebra in terms of patterns and periodic solutions. By understanding sequential patterns (like 9-day and 11-day intervals), you are essentially creating a matrix-like sequence to solve for common availability, much like finding a solution for simultaneous equations.

• Vectors and matrices define transformations in space.
• Systems of equations can be represented and solved through matrix operations.
• Understanding intervals and sequences aids in problem-solving strategies similar to linear transformations.

Linear Algebra aids in simplifying complex scheduling problems by using systematic approaches.

Number Theory

Number Theory is the study of integers and integer-valued functions. It involves chains of thought that are often theoretical but have practical applications, such as cryptography. In our exercise, we use concepts from Number Theory—specifically related to cycles and modular arithmetic—to determine the availability pattern of workers.

The task involves finding the least common multiple of Jürgen's 11-day cycle and Joachim's 9-day cycle. The method includes iterating through potential dates to find when both become available, akin to solving congruences:

\[ x \equiv 7 \pmod{9} \]
\[ x \equiv 6 \pmod{11} \]
This approach helps identify common days through numerical patterns.

• Patterns in numbers can determine cyclic events.
• Finding the least common multiple aligns two numerical cycles.
• Solving congruences helps align different periodic availabilities.

Number Theory enriches our problem-solving toolkit by allowing us to handle such scheduling puzzles efficiently.

Calendrical Calculations

Calendrical Calculations involve understanding days, dates, and how they work together over given periods of time. These calculations emerge in planning and scheduling, where it is crucial to coordinate events across different timescales. The exercise demonstrates how calendrical understanding aids in collective scheduling.

For Carl Friedrich, Joachim, and Jürgen, determining the common available date requires interpreting the monthly calendar. By calculating the specific weekdays associated with periodic patterns, we can hone in on the exact Fridays that work. Methods include:

• Understanding the day's progression through the week.
• Adding intervals correctly across weekly boundaries.
• Checking against specific weekdays, like Fridays, to match individual availability.

Using such calculations ensures precise and accurate timing for future commitments. Calendrical Calculations provide structure to daily, weekly, and monthly planning, emphasizing the importance of detail when solving logistical problems.