Self-Neatening Wine

I’m not a big wine-drinker but I tend to make sure there are always a certain number of bottles in the house for, er, entertaining (interestingly the number of wine bottles kept on standby seems positively correlated with the number of children I have, but I’ll leave the statistical inference for another day).

I generally just order a box of wine online and then push all of the bottles into a cupboard. Sounds messy, right? Well then, get this: one day I just went ahead and shoved 13 bottles onto a shelf and the front row of bottles lined up perfectly. It was ridiculous.

It took me a lifetime and a half to find this, but it turns out that this has been stumbled upon before! It is in Section 1 of Ross Honsberger’s book Mathematical Diamonds.

Suppose you have a rectangular space that fits three bottles across but not four. If you push three bottles in and make sure the outer two are in the corners, and then you place two bottles to sit on top of those three, and then three bottles to sit on top of those two and so on, then you will find that the fifth row will always line up perfectly horizontally.

This is madness. Indeed, no matter where circle B ends up, the centres of K, L and M will be on the same horizontal line.

Apparently this property was discovered by Charles Payan, one of the creators of the drawing package CABRI. He was experimenting with the program and discovered this property.

Feel free to go ahead and prove it. I’ll get you started by drawing in some lines:

Given that we know that FK and HM are vertical, we just need to show that angles FKL and HML are right angles. And probably proving one of those is enough to prove the other too.

Good luck!