First Street: every Thursday Second Street: every fourth Tuesday Third Street: every Monday Central Street: every third Wednesday Grand Street:...

OWNING THE DATASET

Visual Representation of Cleaning Schedule

Immediate Inferences

• Streets cleaned weekly (First, Third) will be cleaned more frequently than others
• No street is cleaned on Sunday
• For \(\mathrm{X} \rightarrow \mathrm{Y}\) to work, Y must be cleaned on the day after X's cleaning day

SEEKING THE CRITICAL INSIGHT

Key Pattern: For "every cleaning of X to be followed by cleaning of Y":

• Y must occur on the day immediately after X
• Y's frequency must be equal to or more frequent than X's frequency
• If X occurs every nth period, Y must occur at least every nth period on the following day

UNDERSTANDING THE QUESTION

Task Analysis

• Must find exactly one street for X and one for Y
• Constraint: EVERY cleaning of X must be followed by cleaning of Y the next day
• This is a strict requirement - no exceptions allowed

PROCESSING THE SOLUTION

Identify Valid Day Sequences

Possible day transitions:

• \(\mathrm{Monday} \rightarrow \mathrm{Tuesday}\)
• \(\mathrm{Tuesday} \rightarrow \mathrm{Wednesday}\)
• \(\mathrm{Wednesday} \rightarrow \mathrm{Thursday}\)
• \(\mathrm{Thursday} \rightarrow \mathrm{Friday}\)
• \(\mathrm{Friday} \rightarrow \mathrm{Saturday}\)
• \(\mathrm{Saturday} \rightarrow \mathrm{Sunday}\) (impossible - no Sunday cleaning)

Systematic Analysis

\(\mathrm{Wednesday} \rightarrow \mathrm{Thursday}\)

Central Street (every 3rd Wednesday) \(\rightarrow\) First Street (every Thursday)

• Central Street cleaned on weeks: 3, 6, 9, 12...
• First Street cleaned on weeks: 1, 2, 3, 4, 5, 6, 7, 8, 9...
• Since First Street is cleaned EVERY Thursday, it covers all Thursdays following Central Street's Wednesdays
• This works! ✓

\(\mathrm{Thursday} \rightarrow \mathrm{Friday}\)

First Street (every Thursday) \(\rightarrow\) Main Street (every 2nd Friday)

• First Street cleaned on weeks: 1, 2, 3, 4...
• Main Street cleaned on weeks: 2, 4, 6, 8...
• Week 1: First Street cleaned Thursday, but NO Main Street Friday
• Not every First Street cleaning is followed by Main Street
• This fails ✗

Other Combinations

• \(\mathrm{Monday} \rightarrow \mathrm{Tuesday}\): Third Street (weekly) \(\rightarrow\) Second Street (every 4th) - frequencies don't align ✗
• \(\mathrm{Tuesday} \rightarrow \mathrm{Wednesday}\): Second Street (every 4th) \(\rightarrow\) Central Street (every 3rd) - frequencies don't align ✗
• \(\mathrm{Friday} \rightarrow \mathrm{Saturday}\): Main Street (every 2nd) \(\rightarrow\) Grand Street (every 4th) - frequencies don't align ✗

FINAL SOLUTION SYNTHESIS

Solution Path Recap

• Identified that Y must occur the day after X with compatible frequency
• Recognized that Y's frequency must be at least as frequent as X's
• Found that Central Street \(\rightarrow\) First Street satisfies all conditions
• Verified no other combination works

Final Answer

• X = Central Street
• Y = First Street

Key Insight

The solution leverages frequency compatibility: a weekly cleaning (First Street) encompasses all instances of a less frequent cleaning (Central Street), ensuring the "every" condition is satisfied.

Exam Strategy

When solving TPA scheduling problems:

• First identify compatible day sequences
• Then check frequency alignment
• Remember that "every" means no exceptions - the following event must occur at least as frequently as the preceding one