Rechtman's Theorem of the Food Coop Boxes

Dear Gazette,

Enclosed please find some mathematical thoughts regarding the box storage method we use at the coop.

Rechtman's Theorem of the Food Coop Boxes

the·o·rem Pronunciation Key (thr-m, thîrm)

An idea that has been demonstrated as true or is assumed to be so demonstrable.
Mathematics. A proposition that has been or is to be proved on the basis of explicit assumptions.

As anyone who is worth his membership card in the Park Slope Food Coop knows, there is always something else to do... Take for example, stacking up the boxes near the front door. The coop, being environmentally conscious, recycles its boxes by making them available for shopping members. The boxes are packed on a shelf and shoppers use them to stock groceries on their way home.

But wait a minute...! The boxes are stacked into each other; that is to say, some boxes are inserted inside other boxes, to save space. As I discovered, if you consider any set of two boxes that are not of the same dimension, there is at least one way in which one box can be "inserted" into another.

Say, ABC is a "4 x "5 x "6 box and the XYZ is a "3 x "5 x "7 box. One of the sides of the XYZ box is a "5 x "7 which will allow the ABC box to be inserted into the XYZ box on the "4 x "5 side. What I am trying to formulate here is that for every two boxes, one will always 'go into' the other. But is it always the case? And if it is always so, then why?

In general, two hypothetical boxes are ABC and XYZ. ABC's dimensions (HxWxD) are Height of "A" inches, Width of "B" inches and Depth of "C" inches. Now take a second box, call it XYZ where the Height is "X" inches, Width is "Y" inches and the Depth is "Z" inches. I assume here that whatever the wall width is as slim as we want it to be. Further, I observed that boxes of the same dimension (for example, two boxes of 2 x 3 x 4) will not be able to be inserted one into the other.

Why is it the case that one box will always fit into another? Can there be a counter example, other than 2 boxes of the same dimension? At the moment I am seeking comments, counter examples or solutions to this observation (I am guessing that the proof will be recursive or by contradiction, but any proof will do...) I can be reached at Rechtman@aol.com.