Circle Packing, Wine Rack Problem, Mathematical Diamonds

The present design helps make sense of the wine-rack problem, one of the mathematical diamonds described by Honsberger (2003, p. 1), who mentioned that the problem was first discovered by Charles Payan and later proven by Hung Dinh.

The problem goes like this. There is a rectangular frame or rack, which can hold three bottles in a row but is not wide enough to hold four bottles (circles). If bottles are placed on top of each other, with the leftmost and rightmost ones touching the walls, then the three bottles in the fifth row form a horizontal line (see pictures). The proof boils down to the properties of tangency and those of a bunch of rhombi. The position of the middle circle at the bottom makes no difference (see animation below)!

This is one of those problems that physical modeling is as interesting as dynamic modeling (using GeoGebra, for example). The 3D cylinders can be placed on either end of the rectangular frame to set up the problem. The pieces can be used for model other mathematical structures such as triangular numbers. Be careful with small pieces when working with young children! Have fun.

Among the Files

1. All in one: Frame plus 13 cylinders.
2. Single cylinder.
3. Rectangular frame only.

References

1. Honsberger, R. (2003). Mathematical diamonds. Washington, DC: Mathematical Association of America.
2. https://www.cut-the-knot.org/Curriculum/Geometry/NCircleRack.shtml

FlashForge

Finder

No

No

0.2mm

10%

Animation is created by LGBU using GeoGebra.