Airlines Frequency puzzle

Question

Source: It is from a book of logical reasoning questions, the name of which I am not able to verify.

Five airline companies P, Q, R, S and T applied to government permission to run their flights in a certain city. The government gave them permission to have certain number of trips every year but with uniform frequency based on the fee they paid. A regular traveler observed the trips of different flights on different days. P on 2nd July, Q on 3rd July, R on 4th July and so on. After some days he also observed the below trips: T on July 30th, S on July 31st and R on August 1st and so on. No flight has a trip every day. Every flight has a trip at least once in every week.

Question 1: Frequency of which flights among P, Q, R, S can be uniquely found out ?

Question 2: Assume that each flight has a distinct frequency. For what all flights can their frequency be definitely inferred?

---

My thoughts on the puzzle:

A flight's frequency should lie between 1 flight/week to 3 flights/week.

I focussed on R first as two data points have been provided for it which are 4th July and 1st August.

Considering the initial week from 2nd July to 8th July:

• Case 1: R's frequency = 1 flight/week

  Dates on which R can fly will be July : 4, 11, 18, 25 and 1st August

  This satisfies our conditions given for R

• Case 2(a): R's frequency = 2 flights/week

  Pair of dates on which R can fly in a week will be:

  July: (4,5), (11,12), (18,19), (25,26)

  August: (1,2)

  This also satisfies the conditions for R i.e. 4th July and August 1

• Case 2(b): R's frequency = 2 flights/week

  July: (4,6), (11,13), (18,20), (25,27)

  August: (1,3)

• Case 2(c): R's frequency = 2 flights/week

  July: (4,7), (11,14), (18,21), (25,28)

  August: (1,4)

• Case 2(d): R's frequency = 2 flights/week

  July: (4,8), (11,15), (18,22), (25,29)

  August: (1,5)

  All cases for R = 2 flights/week are valid

• Case 3: R = 3 flights/week

  which will provide me 6 more subcases in which the dates for in the initial week can be as follows:

  July: (4,5,6)

  July: (4,5,7)

  July: (4,5,8)

  July: (4,6,7)

  July: (4,6,8)

  July (4,7,8)

This would involve a lot of checking, am I doing it correctly or have misunderstood the problem completely?

Answer 1

My initial response to this is that the question is poorly worded.
We need to make some slightly dodgy assumptions to be able to get answers that are valid options to the original multiple choice questions.

So let's assume that:

"P on 2nd July, Q on 3rd July, R on 4th July and so on." means that there was an S flight on 5th July and a T flight on 6th July.

"T on July 30th, S on July 31st and R on August 1st and so on." means that there was a Q flight on 2nd Aug and a P flight on 3rd Aug.

Now, we can work out the intervals between known flights for each airline, and then look at possible "frequencies" (between once every two days and once per week) that would work.

 Airline   Interval between    Possible
           known flights       intervals
   P         32 days             2, 4
   Q         30 days             2, 3, 5, 6
   R         28 days             2, 4, 7
   S         26 days             2
   T         24 days             2, 3, 4, 6 

(Q9) From that, the only airline whose frequency is uniquely determined is S with flights every two days.

(Q10, Q11) If we are now told that each airline has a unique frequency:
We already know the S has flights every two days.
P cannot have flights every two days, so must be every four days.
And then R must have flights every seven days.

We cannot determine the frequencies for Q (3, 5 or 6) and T (3 or 6).

(Q12) Finally, if we are told that Q flies every three days, we know T must fly every six days.
None of the other options work.