Which one of the 14 nights could be one on which only one of the families stayed at the hotel?

Question

Family      Number of consecutive nights

 

Jacksons                     10

Callan                          5

Epstein                        8

Liu                              6

Benton                        8

 

The table above shows the number of consecutive nights that each of five families stayed at a certain hotel during a 14-night period. If the Liu family’s stay did not overlap with the Benton family’s stay, which of the 14 nights could be a night on which only one of the families stayed at the hotel?

 

A) The 3rd

B) The 5th

C) The 6th

D) The 8th

E) The 10th

 

I’ve read a few solution methods online but I still don’t understand the thinking and reasoning behind them, I’d appreciate the help.

Answer 1

  1   2   3   4   5   6   7   8   9  10  11  12  13  14

 

   L   L   L   L   L   L   B   B   B   B    B    B    B    B

   B   B  B  B   B   B   B   B   L   L    L     L    L     L       [one or the other; we don't know which]

 

            3  4   5    6       8       10                             [only one (see above), so others have to fit in blanks]

 

After selecting one of these, the only places where there are 10 consecutive blanks (for Jacksons) is 4-13 or  5-14.  That makes at least two families for those nights.

 

Of the choices given, only (A) 3rd night could possibly have only one family staying.

 

 

Answer 2

The Jackons spent 10 nights. They could have stayed  nights 1-10, 2-11, 3-12, 4-13, or 5-14.

 

If they started on the 4th night or later, the Lius or Bentons would be the only family on nights 1-3, so your answer is A.