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Yeah, I was doing some napkin math trying to figure out how likely it is to get the 5 round run properly and gave up lol. Nice job on the 6 round run!

Some probabilities:

Round 1 damage:

5 damage: 2.5% chance

4 damage: 36.4% chance

Round 2 would be similar, but slightly better because you have stuff from round 1.

Getting 4 or more summon spells in rounds 3-5 is ~3.7% chance

Getting 7 or 8 or 9 rats/skeletons/summon spells in rounds 3-5 is ~7.5% chance.

These aren't all independent probabilities and I wasn't too thorough checking them, but you're looking at a 1 in 5000 chance to get what you need and even after that you need to play well to actually win. This does seem in the realm of doable legitimately though.

For the record, I haven't gotten a 5 round run without twiddling probabilities, so the challenge of being the first legit 5 round run is still open ;)

The probability of getting the market selections you need for the 4 round run is 1 in 40343846 alone. It's probably just as rare to get the enemy spawns you need so that's never happening.

Update:

4 turns is actually possible too but it's probably never going to happen in an actual run. You need the best luck in the market and the enemy spawns:

This is definitely the minimum you can do.

Turns out winning in 5 turns is possible. 

Strat: 

Rounds 1&2: Add 2 level 2 skeletons and anything else OR 1 level 2 skeleton and 1 level 2 skull. Best is 2 level 2 skeletons and archer/stone thrower. A level 2 archer is good for rounds 2-3 when you don't spy since it can target any lane. Always get skeletons first if they're there so you can afford everything. You can afford having some inefficiency, with a 4 castle damage combo instead of the ideal 5 damage for one or both of these rounds if you get lucky later.

Round 1:

Round 2:

Rounds 3-5: You want summon spells, rats and more level 2 skeletons (with the aim of getting at least one level 3 summon book by round 4/5). Bones are good if you already have skulls, but you probably don't want to make a new skull unless it's the best option available. You can probably afford 1/2 other buys if you must and they can protect the castle damage dealers. Use the summon spells as fodder to protect the level two skeletons/skulls as they do more damage to the castle. Rats should be protected the most as they do the most damage to the castle vs their ability to take out units. Level two rats are better than two level 1 rats, but level 3 isn't useful until fighting the king. When you get the level 3 summon book, potentially buy the tier 3 unit produced (ideally it's a summoner). For rounds 4-5, use the spy as you may need to sacrifice a lane to deal more damage. Don't merge summoners into level 2.

Round 3:

Round 4:

Round 5:

King Fight: 3 of any: summoner, bard, rocker, rats, bat spell to remove shield. Use spy to make sure you don't die and pick the best options. Don't bother with princes or knights in other lanes.

(Note that I altered the market probabilities to reduce the number of retries I needed when I was testing out the strategy, but it is still doable if you get extremely lucky. Using this strat across 6 rounds is much more likely and gives some leeway in terms of luck.)

I think five rounds is the best possible as the only way (I think) to do 60 castle damage in 4 rounds even if no units die is:

Round 1: 2 level 2 skeletons + 1 level 1 skull = 5 damage

Round 2 : 4 level 2 skeletons + 1 level 2 skull = 11 damage

Round 3 : 1 level 2 rat + 1 level 1 rat + 4 level 2 skeletons + 1 level 2 skull = 18 damage

Round 4 : 3 level 2 rats + 4 level 2 skeletons + 1 level 2 skull = 26 damage

Total : 5 + 11 + 18 + 26 = 60 exactly, but I can't think of anyway for all units to survive every round without losing more than 9 health.