EXACT ECHOES Exact Echoes by G. P. Jelliss Arrangement of Contents Introduction.....................................1 4. 90° Rotations.............................25 1. Identity........................................5 5. 180° Rotations...........................28 2. Unit Translations.........................9 6. Lateral Reflections.....................29 2a. Vertical with pawns........9 7. Diagonal Reflections..................36 2b. Horizontal with pawns...18 8. Transflections.............................41 2c. Pawnless....................... .19 9. Multiple Echoes..........................42 3. Longer Translations...................21 10. Fairy King Compositions..........48 Introduction Repetition of the same pattern of pieces or moves at a different position on the board or at a different stage in the play, known as an echo, is one of the principal themes to be encountered in chess problems. It is a theme with both artistic and scientific appeal. The first published collection of problems showing echoes, in direct-mate chess problems, was Echo by František Dedrle (published in A. C. White's ‘Christmas Series’ in 1927). An extensive essay on echoes in help-play problems was ‘L'Echo dans le Mat Aidé’ by W. Roese (published in L'Echiquier September-October 1931, pages 1469-1487). One of the artistic and scientific aims with any theme is to show it in its purest, most economical, form. An exact echo occurs when all the pieces in the two formations concerned are in the same positions relative to each other, but possibly in a different position relative to the board. It is the pattern formed by the pieces that is echoed, not the actual pieces. For example, in a problem involving two white knights the two similar men may be interchanged between one formation and its echo. Similarly, in a pattern with pawns, their direction of movement may be different from one formation to the other. An echo is an overall visual or geometrical effect, and is independent of the underlying dynamics of move-power by which it may be accomplished. This document contains a collection of chess compositions that show exact echoes. The problems are classified according to the type of echo, and listed in each section according to the problem stipulation and the forces used, which may include fairy pieces. The idea for this study originated in the successful ‘Exact Echoes Tourney’ which was announced in Chessics issue 15 in 1983, entries being published in issues 17 to 21 and the award in issue 23 in 1985. Composers were asked to send in not only their original compositions but also outstanding examples of their previous work, or the work of others that they considered should not be missed, with a view to publishing a booklet on the subject. Thanks are due to Eugene Albert, Erich Bartel, Andrew Kalotay, Russell E.Rice, and Michael McDowell for providing examples, the latter especially of direct mates. However, this project was never completed, partly because of Eugene Albert’s more ambitious Encyclopedia of Ideal Mate Chess Problems, which was proposed around the same time, and would contain much of the same material. Twenty-five years on I can now publish the collection in electronic form (as a web page and as a PDF), but I have not so far attempted to bring the collection up to date to cover work done in the intervening years. Echoed Play The emphasis here will be on exactly echoed formations (by which we mean an arrangement of pieces fixed relative to each other but not relative to the board). It is also possible to show echoed moves or play, but this is not studied here. 1 EXACT ECHOES The following curious example, which will serve to show exactly echoed play, was provided by Michael McDowell. I give diagrams of both parts of the twin for clarity. --------------------------------------------------------------------------------------------------------------------------- A. Benedek ¶H542 The Problemist v/1977 w________ww________w [wdwdwdwd][wdwGwdwd] [dwdw0wgw][Iwdw0pdw] [w0pdwdwd][wdpdw0wd] [dpdwdwdw][drdkdwdw] [BdwiwdrG][wdwdwdpd] [dw0wdwdw][dwdwdwdw] [wdw4wdbd][wgw4wdbd] [dKdwdwdw][dwdBdwdw] w--------ww--------w Helpmate in 2, (b) turn 90° clockwise. * (a) 1.Kc5 Bb3 2.Bd4 B×e7‡ (b) 1.Kc5 Bb3 2.Bd4 B×e7‡ --------------------------------------------------------------------------------------------------------------------------- It may be noted that turning the formation 90° is not quite the same as turning the position (consisting of board and men) 90°, since the latter reverses the chequering pattern, while the former moves the pieces relative to the board, leaving a1 as a dark square. Types of Exact Echo In geometrical terminology, the echoed formation is related to the original by an isometry (also termed a ‘congruence’), that is a transformation in which all the relative distances and angles remain the same. Geometrical theory shows that these transformations are of five possible types: ‘identity’, ‘translation’, ‘rotation’, ‘reflection’ and ‘glide’ (also termed ‘glide reflection’). See for example Introduction to Geometry by H. S. M. Coxeter (John Wiley & Sons, 2nd edition 1969, page 46). Many works on geometry make no mention of the fifth type of isometry. In place of the terms ‘glide’ or ‘glide reflection’ I coined the more systematic name ‘transflection’, however the shorter term is often convenient, as is ‘shift’ for translation or ‘turn’ for rotation, where space is limited. In an identity the two formations are the same and similarly placed on the board. As mentioned earlier in the introduction, if there are two or more pieces alike it may be that they interchange places, since it is the visual pattern that is echoed. In a translation or ‘shift’ the pieces are in the same relative position to each other and the pattern is in the same orientation with respect to the board. All the pieces, in effect, move the same distance in the same direction. Translations can be of various lengths and directions, and can be specified by the leap {m, n} made by each piece (apparently) in making the transition direct from one formation to its echo. On the 8 by 8 board 35 different patterns of leap are possible, ranging from the {0, 1}, which is a single rook step, to the {7, 7}, which is a bishop move from corner to corner. The maximum area that can be occupied by the pattern echoed by an {m, n} translation on the 8 by 8 board is (8 – m) by (8 – n). Thus a {7, 7} translation could echo only a single cell. The shorter translations, with m and n taking the values 0, 1 or 2, are by far the most common types shown. Any combination of translations is equivalent to a translation or to the identity, which can be regarded as a null translation, of pattern {0, 0}. In a rotation all the pieces move through the same angle about a fixed point. A piece at the fixed centre remains unmoved. The angle of rotation, bearing in mind the chessboard pattern, can only be 90°, 180° or 270° (in other words rotations through one, two or three right angles). Following the usual convention in geometry, rotations anticlockwise are counted positive. A rotation of 270° anticlockwise is of course equivalent to a rotation of 90° clockwise. Any combination of rotations is 2 EXACT ECHOES equivalent to a rotation or to the identity, which can be regarded as a null rotation, of 0° (or any multiple of 360°). The centre of rotation in the 90° or 270° cases can be at either the centre or corner of a cell. In the case of a 180° rotation, which is also known as a ‘half turn’, the centre may also be at the middle of the edge of a cell. It may also be noted that a 180° rotation can be regarded as a ‘reflection in a point’, namely the centre of rotation. However this is a different kind of transformation to reflection in a line which is what we now consider. In a reflection there is a line that acts like a sort of ‘mirror’, although it should be noted that a geometrical ‘mirror’ reflects on both sides. Pieces on the mirror line stay there while other pieces pass perpendicularly across the mirror line to a cell the same distance beyond it as they were before it. Because of the chessboard pattern of square cells reflections that always leave the pieces at the centres of the squares can only be ‘lateral’ (parallel to the sides of the board) or ‘diagonal’ (at 45° to the sides). A diagonal mirror must pass through cell-centres, but a lateral mirror either passes through cell centres or along the edges of cells, that is between two successive ranks or files. A transflection or ‘glide’ is equivalent to a reflection combined with a translation along the line of the mirror, or equivalently a translation combined with a reflection in a line parallel to the direction of motion. The simplest illustration of this complicated type of isometry is that of a pair of successive footprints, one of the right foot and one of the left. Transflections like reflections can be lateral or diagonal. In lateral transflections, as for reflections, the lateral mirror may be either a rank or file, or the line between two successive ranks or files. However, in diagonal transflections the mirror, may be a diagonal line through cell centres, but can also be a line midway between two successive diagonal rows. This type of transflection is difficult to visualise since the reflection places the pieces at the corners of cells instead of at the cell-centres. The translation component must then be of length (n+½) diagonal steps to ensure the pieces end up at cell centres, where the integer n is usually 0 or 1. Notations In the source-details, given above the diagrams, dates are expressed with small roman numerals i, ii, ..., xii for the months, ¶ indicates problem number, # indicates issue number, these details are given, where known, to assist those who wish to check the original publication.