More Bottles in a Wine Rack

This is an extension of the Bottles in a Wine Rack configuration, which was problem #44 in the delightful problem book Which Way Did the Bicycle Go?. At the the end of the solution section, the authors ask whether the solution is extendable to more than the original total of 13 bottles. The applet below allows for experimentation in the more general case.

With N bottles at the bottom, we have to consider 2N-1 layers of bottles. It's not hard to see that the situation repeats itself. Much of the time the bottles in the top layer are all horizontal. However, when the radius is so small as to almost allow an (N + 1)^st bottle in the bottom layer, the top layer may fail to be level. If this happens, the very top bottles (it appears there are always two of them) are still horizontal. Nathan Bowler has given a necessary and sufficient condition for the horizontal top layer. He has also proved the two-bottle conjecture and, in addition, showed that the bottles in the top layer to the left and right of the aforementioned pair are also horizontal, so that, in general, the bottles in the top layer are situated on three horizontal levels.

References

1. J. Konhauser, D. Velleman, S. Wagon, Which Way Did the Bicycle Go?, MAA, 1996

• A Circle-Stacking Theorem
• A Property of Rhombi
• Bottles in a Slanted Rack
• Bottles in a Wine Rack
• More Bottles in a Wine Rack
• Proofs and Generalizations

Copyright © 1996-2009 Alexander Bogomolny