In chess, it is possible to double check whilst giving promotion. An example of this would be a pawn promoting to a queen while a discovered attack from a rook behind the pawn checks the opponent’s king. Both the new queen and rook check at the same time, thus making it a double check.
Construct a sequence of promotions with double check with maximal possible length. Assume that white is to move first and black is to help. You must also prove your answer to be optimal via any reasonable method.
A double check can occur either by a classic discovery, or by an en-passant capture. The latter is impossible here so it has to be a discovery. There are three cases:
Type A: Black King in the 8th rank right in front of the pawn to be promoted, promotion = Q or R, with another Q or R in the same column as the King.
Type B: Black King in the 7th rank, checked by Rook or Queen also on the 7th rank; all four types of promotions can occur.
Type C: Black King in the 6th rank, attacking-close to the pawn to be promoted. Promotion = Q or R or N, while the other checking piece can be anything.
Now here is why there can't be too many moves:
Fact 1: there can be only one type A in the sequence. Because (long story short) the King has to leave the 8th rank and it can never come back.
Fact 2: there can be only one type B in the sequence. Because (long story short) the King has to leave the 7th rank and it can never come back.
Therefore the only possible sequences are (possibly truncated) ABCCC... and ...CCCBA.