I found a group of 3 piece armies that are like rock paper scissors
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Here are 3 positions showing the 3 armies matched up against each other
The black army in the first position and the white army in position 2 are equivalent. With best play the army with pawns beats the army with rooks no matter who’s turn it is, the army with rooks beats the army with neither pawns nor rooks no matter who’s turn it is, and the army with neither pawns nor rooks beats the army with pawns no matter who’s turn it is.