Logic+ 11: UNIZOR.COM - Math+ & Problems - Logic

Notes to a video lecture on http://www.unizor.com

Logic+ 11

Problem A

During an epidemic people tried to sit as far from each other as possible.
Consider a row of 11 seats.
Maximum number of people to accommodate in these seats, so that none sits next to another and there is only one empty seat between people is 6:
I-I-I-I-I-I,
where 'I' signifies a seat taken by a person and '-' is an empty seat.
Considering every new person takes a seat as far as possible from those who already sit, where would you put the first person to be able to sit the maximum number of people?

Hint
If you put the first person at the left or the right edge of a row, or right in the middle, you will not be able to accommodate 6 people.
Here is an example:
I----------,
I---------I,
I----I----I,
and the next person would inevitably leave 2 empty seats in a row, which would prohibit the 6th person to sit without another person sitting next to him.

Solution A
To reach the maximum occupancy of 6 people without anyone sitting next to another, that is to reach configuration
I-I-I-I-I-I,
the configuration after 3 people have taken their seat should be
I---I---I-- or --I---I---I,
because that would force the next 3 people to take seats in-between the already taken with one empty seat between them.
Then the previous configuration after 2 people have taken their seat should be
I-------I-- or --I-------I,
because the 3rd person, taking the seat as far from those taken, would bring the configuration to the one above.
Therefore, the first person should take either the seat #9 or #3 to get the configuration
--------I-- or --I--------,
which would lead to all the above configurations step-by-step.

Answer
To reach the maximum number of people to sit on 11 seats without them being next to another, start from pointing to seats #3 or #9.
Then the sequence of fillings of the seats would be
--------I-- or --I--------,
I-------I-- or --I-------I,
I---I---I-- or --I---I---I,
which leaves 3 empty seats for the next 3 people exactly in-between already taken ones, leaving an empty seat between people.