1. ## arithmetic problem

Family Thomas Shawn David Sui Lisa

Number of
Consecutive 10 5 8 6 8
Nights

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 Sui family’s stay did not overlap with the Lisa family’s stay, which of the
14 nights could be a night on which only one of the five families stayed at the hotel?

(A) The 3rd (B) The 5th (C) The 6th (D) The 8th (E) The 10th

2. Hello, sri340!

The problem seemed to have no unique solution . . . a silly problem.

Then I had a thought (always a dangerous event):

If the given answer-choices are the only ones available,
. . then there is a solution . . . and the problem is even sillier!

$\displaystyle \begin{array}{c|ccccc} \hline \text{Family} & \text{Thomas} & \text{Shawn} & \text{David} & \text{Sui} & \text{Lisa} \\ \hline \text{Cons.nights} & 10 & 5 & 8 & 6 & 8 \\ \hline \end{array}$

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 Sui family’s stay did not overlap with the Lisa family’s stay,
which of the 14 nights could be a night on which
only one of the five families stayed at the hotel?

. . $\displaystyle (a)\;3^{rd}\qquad (b)\;5^{th}\qquad (c)\;6^{th} \qquad (d)\;8^{th}\qquad (e)\;10^{th}$

From the given clue, Sui's family and Lisa's family spanned the entire fortnight.

There are two possible scenarios.
Here's one of them:

. . . . . . . . . . . . . . . $\displaystyle \text{Nights}$
$\displaystyle \begin{array}{c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Family} & 1&2&3&4&5&6&7&8&9&10&11&12&13&14 \\ \hline \hline \text{Sui} & *&*&*&*&*&*& & & & & & & & \\ \hline \text{Lisa} &&&&&&& *&*&*&*&*&*&*&* \\ \hline \end{array}$

Thomas' family covered 10 consecutive nights:
. . ranging from [1,10] to [5,14]

If Thomas stayed [1,10], [2,11], or [3,12]
. . all 5 answer choices are eliminated.

Hence, Thomas must have stayed [4,13] or [5,14].

Therefore, the $\displaystyle {\color{red}3^{rd}}$ night could have had only one family.

. . [So why tells us about Shawn or David?]

3. Hello sri340
Originally Posted by sri340
Family Thomas Shawn David Sui Lisa

Number of
Consecutive 10 5 8 6 8
Nights

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 Sui family’s stay did not overlap with the Lisa family’s stay, which of the
14 nights could be a night on which only one of the five families stayed at the hotel?

(A) The 3rd (B) The 5th (C) The 6th (D) The 8th (E) The 10th
If you study the attached diagram, you'll see that the only possible nights are in the range 1 - 4 or 11 - 14. So, of the possibilities given in the question, the only available night is the 3rd.