written by Alex Fisher, The Blake School
Hey folks! If you’ve been eagerly awaiting a solution to the flag problem, perhaps while stitching your own replacements and making plans to kick Texas out of the Union, wait no more!
The easiest way to think about this grid is to actually separate it into two overlapping grids:
From this picture, we can see two sub-grids: one in blue that has m rows and n columns (in this case, 5 and 6 respectively) and another in pink, that measures m-1 by n-1. Add them together and I have T stars total. It’s tempting to try to multiply (2m-1) by (2n-1), but that overcounts the total number of stars. However, what if we split these two grids up?
Now we have those same two rectangular grids, the blue m x n and the pink (m-1) x (n-1). Together they almost make a larger grid, but not quite: there’s a column missing. It’s tempting to just add those stars to the pink grid… However, if I add a column, I’m changing the total number of stars by m or n, making it harder to determine whether a given number of stars T can be arranged in this pattern. I want to find a way to make sure that the number of stars in my overall picture can be represented in terms of T and constants.
Now we’re getting somewhere! By doubling the previous arrangement, we actually end up with that previously mentioned (2n-1) x (2m-1) grid! Well, almost. There’s one star of overlap. Even so, this means that if I double T and take away that one overlapping star, I have a perfect grid. Thus, this equation: 2T-1 = (2m-1)(2n-1).
This seems like an odd conclusion to draw, but it is actually the key to answering all of our questions. Consider the situation where Puerto Rico, Guam, and Washington DC all are given statehood. We now have 53 states, so T = 53, and 2T-1 = 105. If I can factor 105, that means I can find numbers m and n to make that grid. In this case, since 105 factors into 3*5*7, I can actually do this in three different non-trivial ways: 5*21, 3*35, and 7*15. Solving for m and n in these scenarios gives us the following diagrams:
And while the last one is a little silly, the 4 x 8 case is relatively nice looking. Also, we can see that if only one state joined the Union, 2T-1 = 101, which isn’t factorable, so this arrangement wouldn’t be possible at all. Likewise, if we lost a state, we’d be left at 97, also not a number with nice factors. So, if you’re a big fan of this star arrangement, then bad news: you’re stuck with Texas. At least, unless you convinced them to take Florida on their way out…
Also, note that while I did this work geometrically, there is also an algebraic approach: in the original set-up, T = the blue rectangle + the pink rectangle, or T = m*n + (m-1)(n-1) = 2mn-m-n+1. I will leave the algebraic manipulation to get this to match our answer of 2T-1 = (2m-1)(2n-1) as an exercise for the reader.