Est. 1998 — 27 years of woodworking knowledge
Tueday - late entry
Bill Earl
5 fraternity brothers go into a math test with a plan. Each has studied one of the 5 topics to be covered and plans to telegraph his answer to the others by blinking his eyes in Morse code. The desks in the test room are arranged in a regular grid. It is a full class, so all seats will be occupied.
Can you find a seating arrangement that allows each of the 5 to clearly see each of the others? Can you prove it?
Re: Tueday - late entry *LINK*
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Morse translator
Tuesday Hint
Bill Earl
Alex knows the answer but he cant prove it. 
Considering the grid intersections as integer coordinates, how many ways can you seat someone on the grid?
A start
Larry Barrett
If A is at grid location 0,0, then anyone at immediately adjacent grid locations (i.e at 0,1 1,1 1,0) will be visible.
Also, anyone at a grid location that can be connected by a straight line to A that does not intersect an intermediate grid location. For example, someone at grid location 2,1 would be visible to A, but someone at grid location 2,2 would not because a straight line to 0,0 would intersect 1,1 which is occupied (doesn't matter if one of the five, or not).
So we could have A, B, C, D located at 0,0 0,1 1,1 1,0, all visible to each other. E would then have to be at a location not on the same rows or columns, and satisfying the restriction above with respect to each of the other four. For example, if E was at 3,2 he would be visible to A (at 0,0) B (at 0,1) C (at 1,1), but not to D (at 1,0) because with respect to D, grid location 2,1 is on the same line.
A,B,C,D could be more scattered and still be visible to each other; for example, at 0,0 1,3 2,1 3,4.
So far I can not find an acceptable location for E for either example or prove there is none for these cases (or the more general case).
Correct line of reasoning...
Re: A start
Alex Y
That's about where I am. I also would note that if A and B are at positions (ra, ca) and (rb, cb), then A and B can see each other iff rb-ra and cb-ca are relatively prime. But I can't get any further. ["yet", he says hopefully!]
Good puzzle, Bill!
More hints
Bill Earl
Can you make any generalizations about the coordinates of the first 4 seats?
Answer?
Alex Y
Following up on my relatively prime idea, and consistent with Bill's second hint:
There are only four values that the points can have mod 2: (E,E), (E,O), (O,E), and (O,O). Since there are five points, at least two have the same values mod 2. There will be someone sitting exactly half-way between these two, blocking their view of each other.
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Bill Earl
Considering the seats as integer coordinates in a two-dimensional grid, the five brothers sit at positions (x1,y1) - (x5,y5).
As Alex correctly observed, each brother's x,y coordinates will be in of four categories:
(Even,Even)
(Even,Odd)
(Odd,Even)
(Odd,Odd)
Therefore two of the five brothers' coordinates must be in the same category. Let's call those two brothers A and B.
Consider the midpoint (xm,ym) between the positions of A and B: ([xA+xB]/2,[yA+yB]/2)
Note that: (xA+xB) and (yA+yB) must both be Even. (because Even + Even = Even and Odd+ Odd = Even). This means that the midpoint of the line must also have integer coordinates meaning that (xm,ym) is a grid position between (xA,yA) and (xB,yB) with someone sitting there.
Therefore brothers A nd B cannot see each other.