Fun math fact

I'm prepared to believe there are some interesting decisions when playing with Bunnies, though I'm not sure I've can remember any. Maybe choosing the swing die on the first round, before seeing one's opponents swing die might sometimes be interesting.

But I do think this sort of thing is mathematically sort of cool, if not a deep decision problem: in the picture below,  J. is forced to take my p20 and then I need to sweep his remaining dice. I can afford to lose either one of my remaining dice, but not both. I could either start with the d12 and then switch to the d6 if the d12 is captured, or start with the d6 and switch to the d12 if the d6 is captured. Does it matter which one I start with?

Can you generalize the result? Can you give a good intuition to back up the computation?

Game #8952  •  ElihuRoot (Coil) vs. Jota (Bunnies)  •  Round #2

UBFC 73

Opponent's turn to attack

p(4) (12) p(20) (20) (V) Button: Coil Player: ElihuRoot

W/L/T: 1/0/0 (3)  •  Score: 9 (-7 sides) Dice captured: (X=19), (1) (12) 7 p(20) 3 (V=6) 5 1 (1) 1 (1) 1 (1)  1  (1) Dice captured: (20), p(4) W/L/T: 0/1/0 (3)  •  Score: 19.5 (+7 sides)

Player: Jota Button: Bunnies (1) (1) (1) (1) (X)

Have I mentined how much I like Crab?

Crab is one of my favorite buttonmen ever. Not just because she often wins, but the decisions and gambits those f20 lead to are just delicious -- they are what buttonmen is all about!

I even like some of the really tough situations she brings -- like this one. I can't keep mcculloch from getting initiative, is it even worth trying? I'd like to keep my f20 showing 19 safe,
but that's not possible, if I only use by f20 with 11 to get initiative, he can win it back with his f6 and I'm dead... even if I could try to use both dice to get initiative, he'd still take it from me and I'd just have two d20s with value 1. It's hopeless, right?