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Whether the 1963 rules include checkmate remains an open question until someone gets ahold of the 1963 rules and checks. While the Zillions-of-Games file does treat capture of the King and not checkmate as the winning condition, it also says it is based on the original rules, and these would be the 1962 rules we can read on Abbot's website. Concerning Abbott's four Ultima puzzles, he describes the first one as "Mate in 1", and he describes the other three as "Mate in 2".

However, he says of these:

This follows the conventions of chess problems, even though in Ultima the object is to capture the king, not achieve check mate.

Notably, he is saying this within the context of having tested his puzzles with Zillions-of-Games, and this is the rule programmed into Zillions-of-Games. Zillions-of-Games claims to be based on the original rules. Since these rules say the game is won through capture of the King, and they do not explicitly mention checkmate, they could be interpreted to not include checkmate. It was presumably Mark Lefler or Jeff Mallett who programmed this interpretation of the original rules. However, the 1962 article on Ultima says "The same rules for declaring check apply as in chess." How are we to make sense of this if checkmate is not part of the game? Additionally, it ends with the sentence "We are especially interested in discovering the minimum number of pieces needed for checkmate." If checkmate was not part of the original rules, it would not make much sense to say this. So, it would have been valid to understand these rules as including checkmate, and Lefler or Mallett may have made a mistake, which Abbott repeated without accurately remembering his own rules.

Around the same time that Ultima appeared in Zillions-of-Games, David Pritchard published the rules in The Encyclopedia of Chess Variants and in Popular Chess Variants. In both of these, he claimed the object was checkmate.

In the 1968 rules, Abbott explicitly mentions that the game is won through checkmate, and this comes after a paragraph in which he said "The object of the game is to check the enemy king." Given the context, this sentence does not imply that the game is not won through checkmate, and accordingly, the similar sentence in the original rules, "The object of the game is to capture this king", may not have been intended to deny that the game is won by checkmate.

Given that checkmate may have been part of the original 1962 rules and was definitely a part of the revised 1968 rules, it's reasonable to expect that it was also part of the 1963 rules.

At this point, we probably have two traditions concerning the rules of Ultima. We have those who followed Pritchard in treating checkmate as a winning condition, and we have those who followed Zillions-of-Games in making capture of the King the winning condition.