in 30 games. The inability of theoreticians to precisely calculate these odds has been referred to by mathematician Persi Diaconis as "one of the embarrassments of applied probability". The best information about the winnability of Klondike concerns a modified version of the game called "Thoughtful Solitaire" or "Thoughtful Klondike", in which the location of all 52 cards is known. The probability of winning Thoughtful Klondike (with draw three rules) has been calculated as being approximately 82%, more precisely as having a confidence interval of 81.956% ± 0.096%. Thoughtful Klondike is not quite the same as simply playing with all cards face up, as this would allow an impossible movement of a pile if the top downturned card happened to be in sequence with the upturned card underneath it. Using physical cards, Thoughtful Klondike can be played by peeking at the face-down cards or by dealing the downturned cards face-up but sideways to differentiate them from the tableau piles. With electronic programs Thoughtful Klondike can be played by allowing unlimited use of undos to return to the start if a choice turns out to be unfavorable. Because the only difference between the two games (regular and thoughtful) is the knowledge of card location, all winnable Thoughtful Klondike games will also be winnable in regular Klondike. The win percentage of 82% for Thoughtful Klondike gives an upper bound of 82% on the win percentage of regular Klondike when the location of all cards is unknown. The true probability with best play might be much smaller, because the difference between a right and wrong move cannot be known in advance whenever more than one move is possible and some cards are still hidden. Ultimately, very little is known about the winnability of regular Klondike. A Klondike-playing AI using Monte Carlo tree search was able to solve up to 35% of randomly generated regular Klondike games, placing a lower bound on the winnability percentage. One experiment found a skilled player could win 189 out of 442 games (43%), but this gives a discrepancy of almost 40% between regular and Thoughtful Klondike. Variants Single deck Below are some single-pack variants of Klondike: Agnes (US variant): the stock is dealt in batches of seven on reserve piles and every one is available. Furthermore, the bases of the foundations depends on the twenty-ninth card, which is dealt on the foundations. Easthaven (less commonly Aces Up,) twenty-one cards