Okay...I've received a lot of e-mail that describes some fantastic effort to solve this particular Hard Hat Challenge. Many of the messages tell me enough to confirm that a number of people are on the right track. Knowing all the brainpower that's being exerted (there are some development
teams that are working on the problem), I started to wonder if I was wasting the collective developer effort in the world with a faulty Challenge. So, I
manually solved the problem again tonight with my wife's help (I feel sorry for her

) to re-confirm its accuracy. I also re-referenced the foundational work that the puzzle depends on to ensure that I wasn't off base.
DON'T READ ANY FURTHER IF YOU DON'T WANT A FEW HINTS
Here are a few hints that might push some of you in the right direction:
- Yes, the left side of the image is a frequency chart.
- The primary technique is based on a paper from 1952.
- The verbal hint from the show is very important. If I avoid the silly rhyme, I might re-state it like this: if two structures are considered "equal," first pick the structure that contains the element with the lowest overall character value.
I hope this helps!