Most probably the kind of solitaire most common among Carpatho-Rusyn immigrants was the game known as "Russian" or "Strict Yukon" solitaire.
The game starts with seven piles of cards. There is one card in the first pile, two in the second pile, and so on. The top card in each pile is face up, the remainder are face down. Then four additional face-up cards are placed on piles two through seven. The picture below shows such an arrangement.
The object of the game is to group all the cards into 4 sets of 13 cards in sequence down in suit from King to Ace (i.e. King on top).
The game is played by moving groups of face-up cards regardless of sequence. The card heading any such sequence can only be placed upon another card immediately down in suit. Suits are considered cyclic in value - that is, an ace it considered less a two in value and greater than a king in value. When a face-down card is uncovered, it is automatically turned face up and is available for play. Spaces or empty piles can be filled with any sequence of face-up cards.
This game is much less easily won than the standard American solitaire game which specialists refer to as "Klondike".
There is another version of "Russian" or "(ordinary) Yukon" in which the object of the game is to group all the cards into 4 sets of 13 cards in sequence down from King to Ace in alternating colors. It is not definitely clear which of these versions was most common among Carpatho-Rusyn immigrants in the early years of the 20th century.