# Bottles in a Slanted Rack

The applet bellow is a dynamic illustration of the extension (due to Nathan Bowler) of the Bottles in a Wine Rack problem: for a slanted rack, i.e., for a rack with both walls tilted to the same angle, there is, under certain conditions, a layer of the bottles with centers collinear on a horizontal line. Nathan's derivation shows that this should happen in layer

A note about the applet: an important point about the problem is that the bottles are assumed to be in "layers", with alternating quantities of N and N-1 bottles. With a slanted rack, this condition may be easily violated: the order of bottle placement may affect the configuration and mix the layers. The applet does not make a serious attempt to resolve the configuration when layer mixing takes place. Judging the configuration unrelated to the problem as stated, it simply stops drawing subsequent layers. Do not be surprised or alarmed by this behavior.

What are the exact conditions under which the problem's statement is correct? It looks like, both the original problem and Nathan's extension need layer separation: bottles in layer K may only touch the bottles in layers K - 1, K, and K + 1. But I'll be ready to amend this assertion if warranted by a good argument.

What if applet does not run? |

Suppose there are N bottles in the bottom layer. We will label the bottles in even layers (those containing ^{th} bottle from the left in layer 2m (because it will be useful later, we also include an imaginary layer 0 beneath layer 1).

for k < N-1 by considering a couple of rhombi | |

for k > 1 similarly | |

by considering a rhombus and an isosceles triangle | |

similarly | |

since the bottom layer is horizontal. |

We then deduce:

l(2N - 2, k) | = l(2N - k - 1, 1) |

= 2a - r(2N - k - 2, 1) | |

= 2a - r(N - k, N - 1) | |

= l(0, k) |

Similarly, r(2N - 2, k) = r(0, k). In particular, then,

Note that a similar argument gives the original result in the case that the sides are vertical, since then

= 2a - r(0, N - k) + 2a - l(0, N - k) | |

= 4a - 0 = 0. |

### References

- 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

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