Getting Lost

I feel that a couple points need to be clarified here. First, how do the sensors work? Are the binary, in that they sense the presence of at least one person in a row or column? Or do they actually count the number of people in the row and/or column? Do we know if a person sensed is in a row or in a column? Without knowing these things, it's hard to know a precise answer.

Also, must the squares be tested simultaneously, or can we use a sequential approach?

In any case, I reduced the problem to that of searching a 3x3 grid, with all three people searching simultaneously, and with the sensors not functioning in a binary fashion, but rather being able to differentiate between multiple signals.

If this is the case, I put Aaron at position (1,1), Bobby at position (1,2), and Cathy at position (3,3). If Danny is in any of these three squares, they see him immediately.
If Danny is in position (1,3), all three detect him. If he's in (2,1) or (2,2) or (2,3) only Aaron (or, respectively, Bobby or Cathy) will detect him. If Danny is in position (3,1) Aaron and Cathy detect him, but Bobby does not. If Danny is in position (3,2), Bobby and Cathy detect him, but Aaron does not.

First, draw a 6x6 grid made out of 36 unit squares.

Choose any 3 points (To represent Aaron, Bobby, and Cathy) that are not on the same row or column. Cross out all the squares in the row or column they

are in.

Then count the remaining number of squares. Danny has to be in one of those sections, and the worst case scenario is that Aaron, Bobby and Cathy check

the squares Danny is not in.

Since there are 9 squares left after step 2, they need to check 8 squares in order to guarantee that they find Danny. (Since the last square has to be the one Danny is in)

Why is it true for the general case?

Suppose we first give Aaron a random section. If we set him in any of the 36 squares, then there will be a total of 11 squares in the same row or column as

Aaron. Thus, we only have 36-11=25 ways to put the next person.

We then put Bobby on another available square. After we set him on a square, there will only be 25-9=16 squares left.

We then put Cathy on another available square. After we set him on a square, there will only be 16-7=9 squares left.

So Danny has to be in one of the last 9 squares.

Now we consider the last few squares. There are 9 possible squares, and they are arranged to that there are 3 rows and 3 columns so that each row and

column has 3 squares they didn't check. So when they check a square, it rules out 5 squares, then when they check a second square they didn't check

before, it rules out 3 more. So in the worst case scenario, the last one is the square Danny is in. So, in this case, they need to check $\boxed{3}$ squares to

guarantee they find Danny.