Best Way to Sum Values in Variable Positions Across Multiple Sheets?

Basically, I'm trying to tabulate stats that would be manually entered on a grid, where the values would be in a different column for each sheet. Think of it like a box score in baseball, where the lineup is different each game. I want the main team page to automatically calculate a player's total hits, but I can't just do a 3D sum to the same cell on each page, because the individual player won't always be on the same cell (but the row will always be constant). On a discussion I found about a similar problem, one suggestion was to do SUMIF on each sheet, with each player's total for that sheet being in a fixed position, so then you can just do a SUM for that cell across every sheet. It seems like that will work, but I'm just not sure if it's the most expeditious way (also, I'm slightly OCD about having the sheets as "clean" as possible, even with cells that aren't immediately visible). Another suggestion said something about =SUMPRODUCT(SUMIF(INDIRECT... It seemed like that would be a lot more straightforward, but it's a bit too far above my current understanding to figure out exactly how that string of nested formulae works.

sum_variable_sample.xls

So, in the attached example, each page after the first has a record of "stuff" and "things" for Bob, John, Steve and Ted, and I'm looking for the best way to have the first page automatically tabulate each person's total of Stuff and Things, even though their placement changes on each page.