Can an unequipped Chump really lose to the Rubber Ducky?
Solution 1:
18% Chance of Death by Ducky
You are correct in thinking that the Chump cannot lose this fight. It takes the Rubber Ducky 3 turns to cycle through its deck. In those 3 turns, it will deal 3 damage to you and 2 damage to itself. In the meantime, you also have to play 3 cards.
If you play "Cower" twice and "Eyes Closed Punch" once, you will deal one damage to the Ducky and block 1 (or 2) damage to yourself, leaving you with at least 3 health and the Ducky with 1. There is no card the Ducky can play on turn 4 that can hurt you enough to kill you.
If you play any other card combination on the first 3 cards, you will have done at least 2 damage and the duck will be dead by the end of turn 3.
To complete this achievement you need to use a different dungeoneer and purchase one of the blessing items (spoilers at bottom). In order to do this, grind your way up to 100 gold through map exploration and treasure, then lose a fight or quit the dungeon, and make your purchase. When you re-enter the dungeon, you will start the stage over, and therefore will have a chance to fight the Ducky again.
Even after this, there is some luck involved; you may have to retry the fight a few times since the duck has less than a 20% chance to win.
You must purchase the Apprentice and the Talisman of the Crone. The Apprentice has a 7-card starting deck but also the card "Intuition" which draws two cards but does not attack or block. Spam this ability at every opportunity. On your first two turns, play intuition and a 1-damage spell. You will have drawn your entire deck at this point (4 card starting hand + 1 card drawn on turn 2 + 2 cards drawn from Intuition). If you draw Intuition again on turn 3, continue to cast it until you lose.
Solution 2:
Fun answers! (creator of the game here)
We specifically created the rubber ducky's moveset so that it was impossible to lose to them. They only turn up in that very first tutorial fight in the first quest.
A few days after release someone turned up on our forums saying the game was too hard and that they had even lost to the rubber ducky in the tutorial. We called them out saying it was impossible to lose to the ducky, when someone else posted a screenshot (see this thread) showing them dying to the duck!
We were so impressed by their ingenuity that we added an achievement for it.
Personally I did it with:
the cat burglar (they have no blocks and have two attacks that queue up damage for later turns) and no shrine, so only needed to grind 50 gold losing the tutorial quest.
Solution 3:
There is also a third solution:
As before, grind out 100 gold with the Chump and fail the quest. Purchase the Bruiser and the Talisman of the Crone. On the Ducky's first cycle through the deck, you must play "You tryin' something" against "Uhhh...." You must play "Nice try, chump" and "Oi!" or "How do you like this?" against his other two cards. On the second cycle, as long as the Ducky plays Headbutt before Anger you can succeed.
The odds are more complex for this case so I didn't calculate them yet.
Solution 4:
41% Chance of Death by Ducky
For completeness sake I offer a 2nd solution: (PS: I had no idea it would turn out better!)
Again, grind out 100 Gold with a Chump & fail the quest. You might as well get the "Look At The Size Of This Thing" achievement while you're at it. Purchase the Talisman of the Crone & the Hidden Den (Cat Burglar). When you face the Ducky, your 1st turn must be a 1 damage attack (Pounce or Scratch), followed by a succession of Catnip plays. Ducky will kill himself on turns 4 or 5, so you won't need more than 3 Catnips in a row. (Feel free to smack him into oblivion on that last turn!)
The smaller 6 card starting deck dramatically improves the odds that you have a good card for the 1st turn. The 4 card hand size and 2 cards that don't directly deal damage make turns 2 & 3 a guarantee. Giving you a 93% chance of keeping him alive to his reshuffle. A larger chance of failure appears on turns 4 & 5 due to the possibility of Ducky suicide (Anger) or your inability to hold off killing him (which you will do in style) because you drew your damage card when he draws "Uhh...".
18% Chance of Death by Ducky - 50 Gold Edition.
This character can die to the ducky without you spending 50 Gold on "the blessing", but you've only got a 54% chance of keeping Ducky alive until turn 4, and, as you will be killing him on turn 4, Ducky must deliver the killing blow on turn 4.