Minimal Underpromotions (Fairy Chess)

Minimal
_Underpromotions

The simple mate-in-two chess problems presented here illustrate various underpromotions. For each case I have sought to find a setting with the minimum number of total pieces. The resulting problems are not challenging to solve, and deliberately have no variations. The problem is finding the minimal settings. The diagrams are also intended as instructional exercises in the use of certain "fairy" chess pieces.

The first group of problems uses orthodox pieces only, but for the second and larger group we also allow promotion to a Cannon (Pao), Vao, Leo or Dummy Pawn. The moves of the Cannon, Vao and Leo, called "Chinese" pieces, are described at the beginning of Chess with Chinese Pieces. A "Dummy Pawn" is simply a Pawn that has moved to the eighth rank without promoting. It can never again move, capture or promote. Such a move to the eighth rank never need occur in a two move direct mate problem; but Sam Loyd composed a direct mate in three for which promotion to a Dummy Pawn is the unique key (see EMPLOY THE DEAD!). So we allow Pawns to appear on their eighth rank below, and also on the first rank, as if the problem setting arose from a chess variant that allowed "free placement" of units before commencing play. These conventions for chess compositions are called the "CVL" (Cannon, Vao, Leo) rules in Chess with Chinese Pieces. Whenever nonorthodox Pawns appear below, a simple alteration (usually replacing a Pawn with a Vao) will preserve the problem.

Define an "underpromotion" as promotion by a Pawn to any piece other than a Queen. For problems, if White's first move is a promotion we require it to be unique. So, for example, a key move can not be both a promotion to a Queen or to a Rook. However every mating promotion can be effected by promotion to either a Queen, Knight or Leo. Mating promotions to a Rook, Bishop, Cannon or Vao can always be effected by promotion to a Queen or Leo instead, and it is these options that we use to classify the problem. As an example, suppose we seek a minimal mate-in-two problem in which both the first and second move

must be promotion to a Leo. In the diagram here, the Black unit on h6 is a Vao, represented by the symbol for a Bishop turned by 90 degrees to the left. Other Chinese pieces are represented analogously. This setting, employing seven pieces, is a possible solution to the task. The key is 1. d8 (Leo) K-g8 or h8 2. e8 (Leo) checkmate. Note that 1. e8 (Cannon) does not solve the problem for then Black can play 1. ... K-f6. Nor does 1. d8 (Vao) mate in two since then 2. e8 (Leo) is not a check. And note that White could also mate by 2. e8 (Cannon), but we do not count this as a dual, and characterize the problem by the promotions Leo-Leo. It is possible that some setting with only six pieces could also be a mate-in-two with solution some Leo-Leo promotions. We would also find preferable a setting again with seven pieces, but with less total "force" (such as using a Cannon instead of a Rook).

All possible mate-in-two problems characterized by either one or two underpromotions are listed in the table below. The designation "Any" means any move (not any promotion). Except for a few cases, the types link internally to a mate-in-two diagram that is my best effort at a minimal setting. The purely orthodox tasks are grouped first in the table, and the settings for these employ only orthodox pieces. In none of these cases was I able to find a CVL setting with fewer pieces. Tasks involving at least one Chinese piece are grouped second in the table.

ALL POSSIBLE MATE-IN-TWO PROBLEMS CHARACTERIZED BY WHITE UNDERPROMOTIONS

Underpromotion Type	Number of Pieces	Notes
ORTHODOX	********	****** All pieces and promotions are orthodox. ******
Rook - Any	3	Promoting to a Queen must stalemate Black. Easy - two fundamentally different settings. No diagram below.
Bishop - Any	4	Is there a fundamentally different setting still with but four pieces?
Knight - Any	4	Minimal.
Any - Knight	4	Minimal number. Can the force be reduced?
Rook - Knight	6	It seems the use of Chinese line pieces can not reduce this number, nor for the next two tasks.
Bishop - Knight	6	Has this been anticipated?
Knight - Knight	5	Anticipated?

CVL	********	****** At least one of the promotions is to a Pao, Vao or Leo. ******
Rook - Leo	7	Perhaps the number can be reduced. The White Dummy Pawn could be replaced by a White Vao.
Bishop - Leo	6	Generally, if promoting to a Leo is mate, then promoting to either a Cannon or to a Vao will also mate.
Knight - Leo	5	Surely minimal.
Pao - Any	5	Surely minimal. Choosing a Leo initially instead of a Pao must stalemate Black.
Pao - Knight	7
Pao - Leo	10	If you can reduce this number please send your setting and I will present it here under your name.
Vao - Any	5	Five must be the minimum number of pieces, but can the force be reduced?
Vao - Knight	7
Vao - Leo	7
Leo - Any	4	The key must be promotion to a Leo only. Easy - move the Leo diagonally and check orthogonally. No diagram.
Any - Leo	4	Easy - no link.
Leo - Knight	5	Surely minimal.
Leo - Leo	7	Given in the example above. No diagram below.

.Following are all the diagrams linked to in the above table, presented in random order. Orthodox and CVL problems are mixed together, but if the reader only wishes to consider orthodox problems the ones designated "CVL" should be skipped. No solutions are given. The reader can solve these, or attempt to construct a simpler setting for the task

Diagram A.