Liam is asleep, but I am not sure how long that will last. That means time is of the essence here, and so I am going to forego any pretense of an introduction and jump right into some math.
Definition: Two natural numbers are said to be an isoperimetric pair if the number of bricks needed to enclose units of area (arranged as an rectangle) is as small as possible. That is, if
They may not word it exactly like this, but the REU students have been working hard (and playing hard!) on answering this question: Given a fixed number for what values of is an isoperimetric pair? (In my previous post, I called the product a hydropronic number.)
A brute force approach will confirm that when the only possible values for are or . Similarly the only pairs for which are and . Of course these students have been busy pattern-hunting, which allowed them/us to make a number of conjectures. I list some them below, labeling all of them as claims, and placing a check-mark next to one to indicate that it has been proved. Unchecked claims are open conjectures that might be fun to spend a friday afternoon exploring.
- Claim 1: () All pairs of the form and are isoperimetric pairs.
- Claim 2: All pairs of the form are isoperimetric pairs.
- Claim 3: If is an isoperimetric pair, then so is .
- Claim 4: Fix and let Then is an isoperimetric pair for all natural numbers where .
- Claim 5: Using the same notation as in Claim 4,
- Claim 6: Given the number where is … not yet known. In other words, this conjecture is complete; the pattern for is not yet clear ….
What do you think? Interestingly, pictures can be very helpful in deciding how best to approach or verify any of these claims. Also, Liam’s still asleep!
Update (11:39 am): Someone really should write a program that finds and plots these pairs. Something like this: IF and THEN print and you need to repeat this, increasing until it fails. THEN this entire thing needs to be repeated, changing to — some sort of nested if-then thingy, I suspect.
Update (12:57 pm): I am not too certain why this only now occurred to me, but if we take our condition for being an isoperimetric pair and remove all of the ceilings from the expression (I know, but stick with me for just a second) one obtains which of course simplifies to
If you divide both sides by the left hand side becomes the mean of and while the right hand side becomes what is called the geometric mean. So (very) roughly speaking, our pairs consist of numbers whose arithmetic and geometric means coincide. This is a well studied situation; in fact, an inequality between the two means exists, with equality holding if and only if .
Again, this is a very rough approximation to our situation, since our equation holds for various choices of and . Still, I’m curious that there may be some more geometry lurking in the background here. Any thoughts?