Count of balanced non-terminating starting deals

Determine the total number of balanced non-terminating starting deals—starting configurations with 26 cards in each player's hand—that lead to periodic cycles of play in Beggar-My-Neighbor with a standard 52-card deck under the 1911 Encyclopaedia Britannica rules.

Background

The paper presents a constructive method that yields a balanced non-terminating game and identifies 30 starting deals that reach the same 62-trick cycle. Achieving balance required careful backward play to select pieces that shift many cards to the second player during reconstruction.

Earlier searches often produced unbalanced non-terminating configurations, highlighting the rarity of balanced non-terminating games. The authors' success opens the question of how many such balanced infinite games exist across all possible starting deals.