So, I didn't enter the contest from long ago, but had I done so, I would've gone with the following map.
The first thing this map does is identify a number of whole-county groupings, each centered around a metro area and nearby rural counties, which are close enough to a multiple of the idea CD population. I have five such regions here: Cincinnati/Dayton (1, 2, 3,
; Toledo (4, 9); Columbus (7, 12, 15); NE Ohio (5, 10, 11, 13, 14, 16); and Zanesville/Ohio River (6). District 6 has the largest deviation, at 1,198, but it is also entirely whole counties; every other district is within 850 of the ideal, and could be brought to exact equality with microchops.
The second thing this map does is rigorously adhere to municipal boundaries. Only three cities are split: Cleveland and Akron are split to allow a 50% BVAP VRA district; and Columbus is split because municipal boundaries in Franklin County are beyond insane and I give up. Except for Hamilton and Cuyahoga, no county has more than 2 CDs; those two have three.
Obviously, with the high-level groupings as well as the within-group splits, I've attempted to keep metro areas as close together as possible. There are a couple portions which are not entirely satisfactory: putting Madison in with the southeast group rather than Columbus is the worst offender, but as far as I can tell the 4-CD math made it hard to avoid.
Onto the districts!
1: Entirely Hamilton; Cincinnati and close-in suburbs. 55.3% Obama, 52.2% Dem. Lean D.
2: Splits Hamilton with 1 and 8, and Greene with 3. Western suburbs of Cincy/Dayton and rural southern Ohio. 37.3% Obama, 41.4% Dem. Safe R.
3: Dayton/Springfield; takes Fairborn and Bellbrook from 2 to equalize population. The decision to pair Dayton with Springfield rather than Beaverbrook/Xenia is pretty much the only thing Repubicans have any reasonable cause to complain about with this map IMO; but having 2 hook over to take Springfield would be uglier, and Dayton/Springfield is reasonable. 51.0% Obama, 50.5% Dem. Tossup.
4: Lima, Findlay, and lots of farms in the northwest; splits Seneca with 9. 36.9% Obama, 39.8% Dem. Safe R.
5: Centered on Elyria/Lorain, this entirely new compact collection of exurbs and small cities in northern Ohio is mostly within Cincy's orbit, but on its outer reaches. Splits Lorain with 10. 50.6% Obama, 55.6% Dem. Tossup.
6: Zanesville and Portsmouth are the largest cities here, I guess, but really it's a rural and Appalachian-flavored Ohio River valley district. No splits whatsoever. Ancestrally Dem, but Bill Johnson would probably be pretty happy with it, since I pushed it further south and away from Youngstown. 46.2% Obama, 58.1% Dem. Lean R.
7: The southern Columbus-area district. Splits Licking with 12 and Franklin with 15; the city of Columbus itself is split, too, because it's impossible to get any sort of reasonable lines otherwise. Also the deviation is down to -182; splitting Columbus helps get the numbers quite low here. Most of the more urban and liberal areas are in 15, but a few bleed into this district, which should still be R for now but is likely to trend more competitive. 47.8% Obama, 49.4% Dem. Lean R.
8: Boehner's district is still mostly the northern suburbs of Cincy with some rural areas to fill out population; it sensibly retreats from Dayton and the only split is in Hamilton with 1 and 2. 34.6% Obama, 36.7% Dem. Safe R.
9: Toledo and environs; splits Seneca with 4. 59.2% Obama, 61.2% Dem. Safe D.
10: I would have loved to keep this district entirely within Cuyahoga, but that wasn't compatible with a 50% BVAP district on Cleveland's East Side. Pushing one township deep into Lorain is the next best thing, though. Splits Lorain with 5 and the city of Cleveland with 11. 56.3% Obama, 61.6% Dem. Lean D.
11: The VRA district does not quite maximize the black percentage, but it gets it over 50 with a minimum of ugliness and a maximum of useful ripple effects to neighboring districts (esp. 13). 53.3% black, 50.6% BVAP, splits Cuyahoga with 10 and 14 and Akron (both city and county) with 13. 80.3% Obama, 80.0% Dem. Safe D.
12: North of Columbus, exurbs and rural areas. Splits Licking with 7, deviation 189. 40.0% Obama, 43.0% Dem. Safe R.
13: Akron (minus the parts taken for Fudge's district) and Canton! These two counties seem to be a natural fit in my eyes, though maybe that's because they're linked in my mind due to being the birthplace of the National Football League, with its first champion (Akron Pros) and first dynasty (Canton Bulldogs). 52.4% Obama, 56.9% Dem. Lean D.
14: LaTourette's district has no choice but to expand out to the borders of Cincy and Youngstown; it now splits Cuyahoga with 10/11 and Trumbull with 16 (though it takes nearly all of Trumbull). 53.5% Obama, 58.5% Dem. I'll call this one Lean R, despite having better D numbers than 13, due to having such a popular incumbent.
15: The Columbus district. Entirely within Franklin, takes in a couple northern suburbs and chops off the southern edge of the city to 7 in the name of sane lines and a -92 deviation. 63.2% Obama, 60.6% Dem. Safe D.
16: Youngstown and similar areas to the south, such as New Philadephia and Steubenville. Would've been nice to connect Warren, too, but 14 is already painted into a corner, so it eats into the old 6 instead. Splits Trumbull with 14. 54.2% Obama, 66.0% Dem. Safe D.
Final tally:
4 Safe R (2, 4, 8, 12)
3 Lean R (6, 7, 14)
2 Tossup (3, 5)
3 Lean D (1, 10, 13)
4 Safe D (9, 11, 15, 16)
Seems perfectly balanced to me.