# A Logician's Game of UNO Flip

Two players are in the middle of an ongoing one-on-one game of UNO Flip, with all its standard rules in effect.

Play is currently Light Side Up. It is Player B's turn. Player A has 3 cards remaining, and Player B has 2 cards remaining. Below are the players' hands, the top card of the draw pile, and the top two and bottom two cards of the discard pile.

[Light]{Dark}

• A's Hand: [Yellow 3, Blue 1, Blue 1]{?, ?, ?}

• B's Hand: [Green 9, Blue Flip]{Orange +5, Purple 7}

• Draw Pile: {Orange 4, ...(no more than 103 other unknown cards)...}

• Discard Pile: [Red Flip, Red 4, ...(103 or fewer other unknown cards)... Yellow 4, Yellow 4]

In order to maximize their chances of winning, how should Player B play out the remainder of this game, and why?

(space for clarifications, hints, and additional info)

• The Red Flip is the card that B may play on.

• I included the [no-computers] tag because I think it's way more fun without them, but if someone has software that can solve this I'd love to know about it.

• Explain your logic, there's work to show. If you got the answer, you'll see why it's really the optimal strategy.

• Umm.. I know UNO well but.. How to play UNO Flip? – athin Jan 9 '20 at 5:33
• – ZanyG Jan 9 '20 at 7:32
• Are we to assume that the information listed is exactly what both players know, and that each of them knows that the other knows exactly that, etc., and that each knows that, etc.? (I'd have thought that in a real game each player would know the backs of their opponents' cards. I have no idea whether you're supposed to know how the dark and light sides are paired.) – Gareth McCaughan Jan 9 '20 at 9:29
• Given the rules say that when light side is in play, the dark side of the cards are visible to opponents, the first thing that player B would do is look at the dark side of A's cards (shown as {?, ?, ?} in the problem description), which would allow them to determine a more precise strategy. Knowing what could be on the back of the two Yellow 4 cards also seems key to determining the optimal strategy - as one of them will become the top of discard pile for A to play on (or to be unable to play on). Any owner of the actual game would be able to trivially check this... – Steve Jan 9 '20 at 9:39