Is it possible in UNO to run out of cards to draw?

Question

The question Uno discard and draw pile empty? deals with what would happen according to the rules if a player has to draw one or more cards but is unable to do so because both the draw pile and the discard pile are empty. The author of the question assumed "I'm guessing this is impossible mathematically or they were missing cards".

But is it actually mathematically impossible for that to happen?

Assuming that:

• You are playing with a complete 108 card uno deck.
• You have between 2 and 10 players, as the rules say the game was designed for. Whatever number works best for achieving this game state.
• Players are playing according to the official rules and refrain from self-destructive behavior (like drawing cards even though they don't have to), but otherwise follow a strategy which provokes this situation.
• The cards happen to get shuffled in the ideal order necessary for this to happen.

Answer 1

Yes. Here's how:

10-player game. Designate players 5 and 10 "lucky", meaning that these players monopolize the number that initially flops, and also monopolize the wilds, ensuring that they always manage to play, and the other eight "unlucky" players are never able to play. Assign two of the four colors to players 1-5, and the other two colors to players 6-10, ensuring that the current color is always inconvenient when an unlucky player's turn comes around.

Let's say players 1-5 got the colors blue+green, and the red 1 initially flopped. The "lucky" players proceed to play out the other 7 1's, with the unlucky players drawing (8*4=32) cards in the process, leaving only 5 cards in the discard pile. Player 10 plays a Wild, calling one of his own colors, and then the endgame begins:

• Player 1 draws a junk card
• Player 2 draws a Wild Draw 4 and plays it, calling one of player 10's colors
• Player 3 draws the last 3 cards from the deck, including a Wild Draw 4, then reshuffles the discard into the draw and also draws a Wild.
• Player 4 draws a previously played blue or green 1
• Player 5 plays a Wild, calling one of his own colors
• Player 6 draws a previously played red or yellow 1
• Player 7 draws a previously played red or yellow 1
• Player 8 draws a previously played red or yellow 1
• Player 9 draws a previously played red or yellow 1
• Player 10 plays a Wild Draw 4 that he has held since the start
• Player 1 draws draws the last three cards from the deck (all 1s), then reshuffles and draws a Wild
• Player 2 draws and plays a Wild Draw 4, calling one of player 10's colors
• Player 3 reshuffles that Wild Draw 4 into the draw pile and draws it, -there are no cards left to draw-
• Player 4 has nothing to play and no cards to draw

Here's a sample deck order for producing this result: https://pastebin.com/hzBiFQ0B