Water level problem

This is the solution to the water level problem discussed in my blog ankit-mathur.blogspot.in (alternatively ankitmathur.in).

The box is accelerating in the direction BA, creating a pressure differential between two ends of the box. Since the acceleration is constant, the ratio of pressure at the front and rear end of the box is constant, and so is the ratio between the heights in the tubes.

In simpler words, if the box is accelerating in direction BA then there is a pseudo force acting on the water in the direction AB, which is pushing water in B tube higher.

The 8 Ants Problem - The Time Dimension

This one's about the 8 ants problem i discussed in my earlier post. It has just grown more interesting! You had to prove that each ant meets every other ant at least once. A friend of mine just got another witty solution to it. Probably the 'elegant' solution that the friend who first put this puzzle to me talked about.

Consider a three dimensional space with time as the third dimension. The plane in which the ants are moving would be the x-y plane. Now, let us imagine, how the path of the ants looks like - many skew straight lines in the 3-D space. The z coordinate of these lines represents time. Now, if two of the ants meet anywhere at any point in time, their corresponding lines will intersect somewhere: somewhere in the 3-D space, and the z-coordinate of the intersection point would represent the time when the meeting occurred. Now, let lines A and B represent the ants which meet all other ants. These two lines together define a plane (statement 1). Any other line which intersects these two lines has to be coplanar with A and B (statement 2). It means all lines are coplanar. And since all coplanar lines intersect (unless they are parallel) (statement 3), we can say that all ants meet at some point of time (statement 4). Hence proved.

Give some thought to this before you read further.

Being unarguably a more interesting solution than the one I suggested, this one has some terrible flaws. If you can visualize quickly in 3-D, the first flaw would manifest itself quite clearly - what if one of these lines is not coplanar, but meets lines A and B at their point of intersection (i.e. the three lines intersect at the same point)? Now, this partcular line meets A and B and does not necessarily meet other lines. Hence, our solution breaks at this point. However, we can make minor alterations in the problem statement itself, to prevent thee ants from meeting at the same point, or something of that sort.

What really breaks this solution down is this: some of the lines intersect below the t=0 (or z=0) plane. So these meetings, in a sense, are virtual. Since these lines never met in t >= 0 space, the ants never actually met. Any meeting before t=0 cannot be considered as a real meeting.

Hence, the solution breaks down here. A nice thought, though!