Mate Problem in Chess: Mate in N, Helpmates and Selfmates

A mate problem is a chess position composed with a task: the solver must find how checkmate can be forced within a stated number of moves. It is a puzzle between a composer and a solver, not a game between two players, and the position must still be reachable by legal play from the starting array.

The main types

In a directmate White moves first and must give mate within the stated number of moves against any defence. A mate in two is a two-mover, a mate in no more than three is a three-mover, and a mate in a larger fixed number is a more-mover. In a helpmate Black moves first and cooperates with White so that Black’s own king is mated in the set number of moves. In a selfmate White moves first and forces Black to checkmate White. Studies are a related class in which White must win or draw with no fixed move limit.

Notation and conventions

Counts refer to moves by the side that carries out the goal. The hash sign stands for checkmate, so a mate in three is written #3. A sound problem has a unique first move, called the key. A problem with two keys is cooked, and ideally White also has only one correct reply to each Black move, since a second one is a dual. Castling is assumed to be allowed unless the position shows otherwise, and en passant is assumed not to be allowed unless the last move must have been a two-square pawn move.

Length records

Some compositions need more than 100 moves, and the many-mover record is over 200. Lutz Neweklowsky composed a mate in 267 in 2001, building on work by Thompson and Karrer in 2000, which the source describes as the longest more-mover without obtrusive units.

Where it appears

The article on the mate in 64 examines a long forced mate that engines can solve, marked as cooked in a problem database. Engines can solve long forced mates by search, a tablebase covers the endings with few pieces, and a position is shared as a FEN string. The story of composed chess belongs to the broader chess history of the game.

A first look at a two-mover

The 1905 to 1906 first-prize problem by Godfrey Heathcote, printed in the Hampstead and Highgate Express, is a standard teaching example of a mate in two. The key is 1.Rcc7!, which threatens 2.Nc3. Black has many defences, and each one meets a different mate: after 1…Nxb3 White plays 2.Qd3#, after 1…Nb5 it is 2.Rc5#, and after 1…Rc5 it is 2.Rxc5#. The key does not give check and does not capture anything, which is deliberate. A solver who tries checks and captures first, as a player would, goes wrong, and composers treat a key that is a check, a capture or a mere restriction of the black king as a weak one.

Stipulations and abbreviations

The task printed under a diagram is the stipulation. The abbreviations are short: the hash sign stands for checkmate, an equals sign for stalemate, h for helpmate, s for selfmate, r for reflexmate and ser- for a series-mover. A mate in three is #3 and a helpmate in two is h#2. Helpmates in which White moves first carry a half move in the stipulation, so a helpmate in two and a half moves is h#2½, and these are very popular today.

Helpmates and selfmates in detail

The first helpmate was composed by Max Lange and printed in the Deutsche Schachzeitung in December 1854, with White to move and several duals. Sam Loyd published the first helpmate with Black to move in The Chess Monthly in November 1860, and the first completely sound one came from A. Barbe of Leipzig in 1861. The name help-mate appeared in T. B. and F. F. Rowland’s The Problem Art of 1897. A helpmate may have more than one solution, but the composer must say so; if the stipulation is silent, there is exactly one.

Selfmates were once called sui-mates. A well-known short example is Wolfgang Pauly’s selfmate in two from 1912. The only first move is 1.c8=N!, a promotion to a knight, because any other piece could interpose and spoil the forced mate. Black then has three replies, and each one ends with …Bxg2# on White’s king. A reflexmate is a selfmate variant in which either side must give mate if it can.

Cooks, duals and the idea of soundness

A problem is sound when its solution is unique. A second key makes it cooked, which is judged unsound or defective. Records show how hard it is to avoid. Ottó Titusz Bláthy’s 342-move selfmate of 1922 was later found cooked, and so was Andriy Stetsenko’s 359-move problem from 2016, which held the record until December 2021. The current selfmate record, 203 moves, is by Karlheinz Bachmann and Christopher Jeremy Morse in 2006, based on the cooked Bláthy composition. In helpmates a set of thematically related solutions is acceptable, which is why the cook rule works differently there.

Construction tasks and the composing community

Not every problem asks for mate. In 1866 Sam Loyd set a construction task in Le Sphinx: build a game that ends with Black delivering a discovered checkmate on move four. The answer is 1.f3 e5 2.Kf2 h5 3.Kg3 h4+ 4.Kg4 d5#. Composing has its own institutions. FIDE awards composition titles: the International Master title was established in 1959 and the Grandmaster title was first awarded in 1972. A problem earns one point in the FIDE Album and a study one and two-thirds, and the Master titles need 12, 25 and 70 points for FIDE Master, International Master and Grandmaster. The World Chess Solving Championship is organised by the FIDE composition commission, known as the PCCC. Solving titles for Grandmaster and International Master were first awarded in 1982.

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