Game over, we win! (alternative version)

“We made it through the worst,” says Haw. “I was worried that if we drew one more bad card it’s an instant loss”

“Three suits removed, one empty column, this is a shoo-in” says the Dumb Bunny.

“I was wondering,” says the Eagle “Can we prove we are mathematically guaranteed to win?

Ninja Monkey immediately grabs two decks of cards and lays them according to the diagram above, and implements his patented look-ahead algorithm. He has 10,000 wins from 10,000 tries.

“Guaranteed win, no probs” coos Ninja Monkey.

“Not so fast,” says the Eagle. “No matter how many times you win, you can’t prove the game is a mathematical lock with Monte Carlo simulation. Besides, your face down cards were arranged the same way in all 10,000 iterations. We need to prove a guaranteed win regardless of how the face-down cards are arranged”

Ninja Monkey pulls a frowny face.

“I think it is a win,” says the Wise Snail.

“How do you prove it?” asks the Elephant.

“I recommended we do the following”, says the Wise Snail

• Clear the Spade suit
• Exchange the 6-5-4-3-2-A of Hearts in Column 5 with the 6-5-4-3 of Clubs in Column 6.
• Dump the 9-8 of Clubs in Column 3 into the empty column
• Clear the Heart suit, winning back the empty column
• Shift the Qh-Jd onto the Kh in Column 1, turning over a face-down card in Column 6 (and keeping an empty column)

“Note that I went to the extra effort to clear a card in Column 6 rather than Column 5. This is because clearing cards in Column 6 is harder than Column 5 (especially since the Q-J are offsuit). As a general principle it is often wise to save an easy task for later and get the “difficult task” over and done with whenever possible – this helps avoid the embarrassing situation of “One Hole No Card” as alluded to in a previous post.”

All the animals are paying their utmost attention. For once, the Wise Snail has been given a chance to truly shine.

“The resulting position is shown below, with the newly-exposed card redacted,” continues the Snail.

“Let us consider all possible face-down cards (which we identified from last week):”

• Queen of Clubs: this can go “underneath” the Jack of Clubs (Jack onto Queen, winning an empty column, Q-J to Column 8, losing an empty column)
• Queen of Diamonds: this goes onto the King of Clubs
• Ten of Clubs: this goes onto the Jack of Clubs
• Ten of Diamonds: this goes onto the Jack of diamonds
• Ten of Hearts: this goes onto the Jack of Hearts
• Nine of Diamonds: this goes underneath the 8-7 of Diamonds
• Seven of Clubs: this goes onto the Eight of Clubs
• Six of Hearts: we will count this as a “bad card” since the 7 of Diamonds is offsuit (and will counterfeit the Nine of Diamonds). This goes into the empty column
• Five of Hearts: This is a bad card and goes into the empty column.

“Note that the first seven cards are good, and we don’t even require an empty column to achieve the corresponding action. The only possible snag is there are two bad cards and only one empty column. But wait! If we draw both the Five and Six of Hearts then we can immediately place the Five on top of the Six. The net effect is to condense two bad cards into one – hence there is no snag after all.

Finally we also check that there is no issue with one-hole-no-card. Assuming we turnover all cards in Column Six first we will eventually get an empty column in Column Six and then we can choose randomly between shifting the Jh in Column 2 or the 9-8-7 of Hearts in Column 5 into the new empty column. Essentially we are pretending that all nine face-down cards are in Column 6.”

At last, the Wise Snail had finished his discourse and everyone was convinced the game was mathematically won. Quod Erat Demonstrandum. All that remained was the formality of executing the final moves to remove all eight suits and win the game.

“Remarkable,” says the Eagle.

“Elementary,” replies the Wise Snail.

“But most of the credit goes to Haw,” says the Spider GM. “He used his fantastic analytical skills to find the right moves when all seemed lost.” Spider GM is especially pleased to see his students are helping each other improve with little supervision required.

“And none of it goes to Hem,” sneers the Smart 65,83,83. “He ran away as soon as we were forced to give up our empty column. Nobody has seen him since.”

There’s always one in every group, sighs Spider GM.

Then Haw heard what he thought was the sound of movement. As the noise grew louder, he realized something was coming. Could it be that Hem had turned the corner? Was he about to find out they had managed to win the game?