Good job everyone, especially those with the dedication to get 22 or all 23 of the gnomes!
Looking at the scoring system, you should actually go for a gnome only if the "expected time" to get the gnome is less than 4 days, as a gnome is worth 1,000 and each day costs you 250 points. I essentially split the gnomes into 2 categories: set-up (those where you need to prepare conditions under which the gnome can spawn, and wait) and activity (those where you need to do some direct action to get the gnome). I tried to set up the conditions for the set-up gnomes as early as possible (Vampire, Blingaboo, Cranstan Boonitz and the pet gnomes, in particular) and then took care of the activity gnomes (the rest, basically).
I used a Fairy, but used a Genie elixir part-way through. Also, I set the season length to 4 days - this is critical for my strategy of getting the seasonal gnomes from the festivals. Traits: Ambitious, Dog Person, Flirty, Hopeless Romantic, Eccentric.
I played in Bridgeport. Right off the bat, I used the Midas Touch and Wish Enhancing Serum elixirs to quickly satisfy my Sim's LTW of Swimming in Cash. I bought the Motive Mobile, the Festival Tickets LTR (can't remember what it's called), and went on vacation to France for a day, to get the WA gnomes and also to read up on Gardening. Returning back, I got my gardening to 7, started the Charisma skill for the celebrity challenge, and also said hi to several high-level celebrities, promptly throwing Jars of Potent Friendship on them to get my own celebrity level up to 5.
Romantic Interests
Jar of Potent Friendship, plus me being a Hopeless Romantic Flirty Sim was what I used, mainly. I also worked up to Level 5 Celebrity and put in some work on the charisma skill; this also gave me a good set of best friends to work with when doing the RI kick-out job. Not too hard, actually.
Gnomes obtained
Mysterious Mr. Gnome - I got him by drilling with the Miner. You can also get Mysterious Mr. Gnome by transfiguration of gems/metals that are worth a little over 1,000 I think; was doing the miner more for the evil gnome, though.
World Adventures:
Sultan Sam, Little Leon and Master Manchu - You can take these from the art museum in France.
Ambitions:
Magical Gnome of Invention - There's a 1.5% chance of getting one of these when completing an invention. Eccentric Trait (reduces time to 85%), and mass-producing cheap (cuts time by half) Dog Toys was what I used. The RNG wasn't too friendly, though; I made about 130 of these before getting the gnome - given that my Sim had 3 discoveries and 130 toys, that would place him in the bottom 13% in terms of luck...
Blingaboo - Came as a celebrity gift. I made sure to get up to level 5 celebrity ASAP (played in Bridgeport), though wasn't counting on this, actually.
Abracadabra the Laundry Gnome - Use the expensive laundry machine and dryer; it should come after a few cycles.
Vampire Gnome - Grow 3 Plasma Fruit bushes on your home lot. There will be a 15% chance that it will spawn every night (I think). I'm not sure if they only count when matured or not, but anyway since I had a Fairy, I Bloomed the bushes to maturity.
Pets:
Felix Felinenimus - Buy the pet toy-box until you can get a cat-nip toy. Then, get the cats to play with it; they will get the Whacked Out moodlet. There's a small chance (0.7%) that the gnome spawns when this happens. I had 5 cats and micromanaged their time on this until I got it.
Zenyatta - Adopt a horse, have it lick a salt-lick... it should spawn soon enough.
Carter Caninenimus - 0.7% chance of spawning, per Dog best friend. Once I had the first 2 gnomes, I removed all of the cats and the horse, and moved in dogs. I had the attractive lifetime reward, and the Celebrity charisma skill-challenge, so that plus Dog-Person helped me make the Dog BFs quickly and smoothly.
Showtime:
Lounge Lizard - You don't have to be a singer. I changed my Sim to a Genie to cast summon food - summon a fried peanut butter and banana sandwich. 15% chance of spawning every night. As for why this does this, I have no idea.
Cranstan Boonitz - I joined the Magician Career and did some performances... it spawned after a while. I've heard you can get this from summoning food (genie) but seemed to have no luck in testing.
Supernatural:
Gnome of the Darned - Fairies can cheat at the claw machine as Ricalynn has said, making this quite quick.
Seasons:
4 Seasons Gnomes - I got the LTR that doubles carnival tickets (can't remember what it's called offhand). I then queued up and immediately cancelled 8 Skate cues on the carnival grounds; repeat a few times. This is best done when you can't/shouldn't do other stuff (e.g. I try not to use the miner at night, in case my BFs / future RIs get offended). Your Sim will skate one lap around the rink and then get out, and score ten tickets. Repeat ad nauseum (until you have 2000 tickets). During the appropriate season, you can buy the seasonal gnome for 500 tickets.
Evil Mr. Gnome - I used the Miner. You can also use trick-or-treating, or have some luck with submitting a lot of gems and metals to be cut/smelted.
Gnomes not obtained
Sculpting Gnome - Forgot about the MoveObjects On trick, so I wrongly concluded that it would take too long and not be worth it!
Troglodyte - Time Machine travel to the past, or using the Miner. My test results were less than positive for this, in terms of expected value...
Freezer Bunny - 0.1% chance of spawning every time a cat freaks out at the fridge, or a dog barks! Didn't think of getting this (too random) - I figured that if I got it, I got it, and if I didn't, I wouldn't pursue it.
University Gnome - Don't have University, so automatically out.
Thanks for a solid (and pretty tricky) challenge - I like these decision-making ones, where you can choose to stop or keep going, honestly. Off to Memory-ville now...
