Does it ring a bell?
Can a wind chime be viewed from an angle so that the tips of the chime rods fall in a single plane? (Non-trivial solutions only allowed)
posted by littlecow at 1/20/2006 12:38:00 AM
March, Find, Analyze, Siege, Attack and Conquer.
posted by littlecow at 1/20/2006 12:38:00 AM
6 Comments:
Firstly, the tip of a whindchime is itself a plane, so if you do NOT consider the tip to be a point, it's quite obvious that all the tips can never be in the same plane.
If you consider the tips to be a point, then they still cannot lie in a plane.
Reasoning: These are open pipes, with frequencies: first harmonic, second harmonic, third harmonic etc. I'm assuming that the tubes are not fine tuned to intermediate frequencies. If this is the case, the lengths will also increase in a linear fashion, since in the general equation L=nc/2f (for an open pipe), the n is the only one that is increased to achieve different lengths with different frequencies.
Now lay all of these lengths side by side, and with the tops aligned. The bottoms will all be in the collinear (due to the ratios being linear) and thus in same plane. In a wind chime, the tops are aligned, but the rods are not laid out in a line, they are placed with their axes on a circle. Thus, the tips now have an x,y and a z coordinate. A plane can pass through them only if the z and one of x and y coords have a linear relationship. The z does, but the x and y now have a sinusoidal relationship, and thus, they cannot line on a plane.
A clarification: the above reasoning is assuming that the lengths are of a linear ratio, not necessarily double, triple etc. In real life, wind chimes are often adjusted and fine-tuned to change the length to some other arbitrary number, which is very close to the original design length.
upon further thought, here is a better reasoning: Assumption: all wind chime pipe tips are points, not planes. z axis = parallel to the pipes of the windchime. origin = bottom tip of the longest pipe. x axis = parallel to line joining the tops of the longest and shortest pipes, which are always adjacent to each other in a windchime. y axis defined according to right hand rule.
Join the bottom tips of the longest and shortest pipes. This is a line AB in the xz plane. Now, define a unique plane such that it contains the line AB and the bottom tips of the second shortest pipe. This plane has a negative slope along the zy plane. But in order to make contact with the second longest rod, it has to have a negative slope, thus there is at least one tip which cannot lie on the plane defined, and thus the tips cannot be viewed such that they are on a single plane.
@balaks: you are correct, it's GP, not AP, the first analysis wil lstill hold, becos for points to be coplanar, the ratio of x to y and y to z and x to z should be the same across all points, but with the z in GP and the x and y sinusoidal, that can't happen.
i think the second analysis holds, and the assumptions are that it is some set of rods arranged in around the circumference of a circle in ascending order of length. i have used the fact that any 3 points define a plane. if you take a bunch of long rods and slide them, u will get a coplanar arrangement, but the longest and shortest rods will not be adjacent to each other as it always is in a windchime. restrictions on lengths and positions: as i described above. The key factor in windchime arrangement is that the longest and shortest rods will be adjacent.
@grenade: good point, i don't know in what random state of mind i was when i addressed that to balaks.
@lenscrafter: your analysis is generally right. and yes, i wanted you to make all the standard assumptions, which you have made. and yes, the lengths form GPs as grenade has pointed out. despite that, you will be awarded the points.
@grenade: when unspecified, make suitable assumptions. think back to your goldman sachs interviews! ;)
@all: lenscrafter's logic of exploiting the fact that the smallest and largest pipes are adjacent is right. infact, if you work through the equations (for the standard set of C,D,E,F,G,A,B notes), you will realize that the equations for the largest and smallest length rods (one of them starts from (0,0,L) - use cylindrical polar coordinates) are inconsistent... and hence the impossibility.
yes, this puzzle was rather simple - it was inspired by one of the previous photo puzzles on lenscrafter's website.
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