HOME | MAGAZINE: SUBSCRIBE • RENEW VIEW CART View Cart   
THE BRIDGE WORLD

Mystery #3

Heaton's System

It is not easy to find an item in bridge literature that was brought forth even though the publisher did not understand it. Perhaps the most outstanding of this rare breed is the packet of material that falls under the general characterization of Heaton's System, a proposal for one or more kinds of movements for pair games. Below, we present the late Albert H. Morehead's introduction to the subject, and an article by the system's inventor. To resolve the mystery, one must (a) explain Heaton's proposals in everyday language or using standard terminology, (b) describe their relationship, if any, with currently-used methods, and (c) evaluate their practical applications.

Introduction to Heaton's System

by Albert H. Morehead

For about five years, the editors of The Bridge World, and the tournament directors of the American Contract Bridge League (ACBL), and perhaps others who occupy some publicly recognized position in contract bridge, have been receiving frequent letters and printed material from a Dr. D. R. Perry Heaton of Lake Worth, FL. Dr. Heaton excoriates many essential features of the duplicate tournament movements that are in general use, and offers a method of his own to cure the defects to which he calls attention.

But his letters go farther than that. They accuse the editors of this magazine and the tournament directors and committees of the ACBL of willfully suppressing the Heaton method so that certain individuals may profit from the sale of the movements that are generally used. Dr. Heaton's letters usually direct such accusations against a list of peopIe excluding the person to which the particular letter was addressed: Thus, in a letter to Moyse he would accuse Morehead, Mc- Kenney, Baldwin, Sobel, Mott-Smith, Ach, Kennedy and others, but not Moyse; in a letter to Morehead the list would be substantially the same, except that this time Moyse would be included among those accused, and Morehead would not.

Dr. Heaton, in other words, believes that the Howell system cards, the McKenney-Baldwin guides, and the Ach-Kennedy cards, are profit-making properties; and that those who profit from them are so anxious to avoid competition from a superior product that they condemn duplicate players to inferior games rather than permit an improvement to be publicized. As evidence, Dr. Heaton adduces the refusal of The Bridge World and of the ACBL's Bulletin to publish the details of his method.

Many of our readers have accused us of being apologists for the present duplicate methods, but the charge is not true. We have refused to print articles criticizing those methods, but only because the contributors of the articles offered criticism without remedy. No such statement can be made of Dr. Heaton. He does offer a remedy. Furthermore, I believe that his method has merit.

But, in my opinion, Dr. Heaton's writing, while it is English, has a rare and unfortunate quality that makes it practically unintelligible to most readers. His tables and his explanations leave me bewildered; they also bewilder Sobel, Baldwin, Ach, Mott-Smith, Moyse, and others with whom I have often discussed them.

Despite Dr. Heaton's accusations, no one makes any money out of the sale of duplicate movements. The ACBL uses McKenney-Baldwin movements in most tournaments; neither McKenney nor Baldwin receives a royalty, and the sale of cards to others does not net the ACBL ten dollars a year.

The ACBL has no paid officers, not even MeKenney. Employees of the ACBL are paid so little that they would have to resign if they did not make some money in outside jobs. Bridge, unlike golf, tennis, track and other amateur sports, is not a spectator game; the ACBL must exist on a gross annual income of some $30,000. [circa 1944--Ed.], paid wholly by its members in dues and entrance fees. Figure out what sort of salaries it can pay to nine employees, plus rent and overhead, out of that. It cannot hire anyone to study the Heaton method.

Dr. Heaton came into my office in 1940. He had a set of blocks with which he explained his duplicate movement. He made it very clear to me then; but after he had left I could not recapture that understanding, and recourse to his printed explanation left me once more bewildered. I suggested to Dr. Heaton that he employ someone to rewrite his explanation for publication; he chose to take offense at that suggestion. He returned to Florida and resumed his letter writing.

When I became publisher of The Bridge World, I resolved that I would give Dr. Heaton the publicity he wanted. In the meantime, he has produced in addition to his original duplicate method, a "guideless How-ell," which, he says, is not so effective as the Heaton method but which is somewhat simpler, and which is far superior to regular Howell or Mitchell.

The following article was written by Dr. Heaton; I publish it exactly as he submitted it. I have not edited it; I have not even submitted it to the usual "copyreading." [We tried to correct obvious errors in typography and spelling.--Ed.] I was forced, by space limitations, to omit some of Dr. Heaton's closing remarks on the faults of Howell movements.

I still cannot understand Dr. Heaton's method; but I still believe, on the strength of that one brief glimmer of light in 1940, that it has merit. Perhaps readers of The Bridge World will understand it; perhaps some reader will wish to obtain the full booklet on the Heaton method from Dr. Heaton, and clarify it for me and for readers who, like me, cannot make it out at present.

The Inept Mechanics of Duplicate

by D. R. Perry Heaton

Without mechanics, to breed experts and pioneers, we would still have only Whist, a game for a few highbrows. Contract, for the 17 million, its authors, teachers, etc., therefore owe existence to mechanics.

But after 40 years, sportsmanship and efficiency for the one million devotees of duplicate, is wholly subservient to the ease and profit of the ignorant "operator" (the term "director" usually implies attainment).

Mitchell, requiring only boards as equipment, is: half sit; the rest go circularly one way; boards circularly the other way. How discrimination, mild to raw, is dished out 90% of the time, is not in scope of this article since the fault lies with the human element, not mechanics. Where the profit lies in fees of latecomers, it is most difficult to remedy.

The Howell non-division method does so, but as now used, is a classic of doing things the wrong, the stupid way; hence hara kiri, popularly. Inanimate boards stutter ringwise (mix-ups often) twice the Mitchell number dart each round on frenzied course, to peck at boards as they pass.

As four special faults in Howell,

*1) Slavery to guides; mental effort to con (maximum case, 5340 times)

*2) Cost of full printed cardboard sets, up to $32.50

*3) Walk-collide beyond next tables; maximum aggregate, 17 miles.

*4) Time loss, extra at changes; maximum aggregate 40 hours.

As tables add, horrors multiply. Common to both Mitchell and Howell:

#5) Cafeteria stuff: customers, not vendor, do the work-except in some Howells, operator mistrusts woolgatherers and does it himself.

#*6) Time loss in starting (to nail the profit): average aggregate 10 hours.

#7) Gyp on length, when pure accident of how many attend may cut to as few as 17 boards: your 50 cents buys 12 eggs one day; 6, unpredictably, the next.

*8) Horse racing beats records while duplicates fold; teachers can't make a living. The vital factor is: frequent handicaps to give all a chance.

With the 1900 model ox-carts, brains went on ice, except for pseudo-scientific drivel on comparisons. Yet the "guideless Howell" below can cure faults with *; the Heaton method cures these and those marked with #.

Print on 16 cards: "EW go to next higher-number table, EW NS' go to next lower-number table, NS Use TOP board if more than one at table. Take boards to T-? (One of Nos. 1-16). If others there, put them at BOTTOM. Handicap form: point arrow to pair with the higher number."

Special cards (supersede others): At T-I: NS becomes EW; go to highers. Last T; EW pivot to NS; go to lowers. Pair-number cards, serially: "sit anywhere."

Place a card, NOT by table number, but by figure given under "to", at tables:

[beginning of figure--Ed.]

T   totototototototototototoT
4&6&7&8&9&10&11&12&13&14&15&16&
  1424-4-5-8-72722-3-11*12-12*  1
  23-5-7*8-7-10-4*1234*727-1*  2
  31*632-1-928*10-10312-6*8-11-  3
  42-3*6323126*8-8-13-3*549-  4
  5--1*1-6*6-9-3-11-8212-14315-  5
  6--2-324*221-1-3-1-1-1*7*  6
  7----5-323*5-9-5-11*8-11-162  7
  8------7-5*2-11-1-1055515-4-  8
  9--------4-32276-5-4-3-14-  9
10----------436-7*6323632610
11------------5-442-13-13*8*11
12--------------9-7-14-2-5-12
13----------------9-10-426-13
14------------------9-9-10214
15--------------------10-13-15
16----------------------3*16

(Within the above table, horizontal rules appear after the lines labeled 4 and 8 in the final T column.)

5 Tables

A   B   C   D   E
F   B   C   G   I
H   E   C   D   B
G   I    C   E   A
D   AxA   I    F
B   F   H   E   G
C   F   H   I    D
E   G   H   A   F
I   D   H   G   B

Since the system fails at 5 tables, the boards are placed at tables (using Heaton lettering method) by the schedule given--there being an exchange of As on Rd 5, tables 2-3. *, instead of Fig. 1, means: one board here.

[end of figure--Ed.]

As the upper part of Fig. 2 [None of the figures are labelled. They are reproduced here in order and in correct location relative to the text, except that the material headed "5 tables" appeared set Southwest to the previous information's Northeast.--Ed.] shows, players move circularly in same direction until reversed by special cards at first and last (pivot) table. The boards have absorbed the "dizzy element" T-10 example: with "take to" card 4#7 at T-1, North takes board used to T-7, putting it under the two already there; then joins partner at table he moved to. While twice the needed number move and twice as often, each move is the shortest possible and accomplished in a standard 10 seconds, as in Mitchell. Costs nil.

If the gross loss in popularity is in lack of practical, individual handicapping, a single basis is: the weaker you are, the fewer the hands played the same way as tops. So seed #1, as weakest, up, at same tables to pivot table; thence backwards to T-1; if 6 Ts:

[beginning of figure--Ed.]

  1  2  3  4  5  6
121110  9  8  7

[end of figure--Ed.]

Then arrows to point always to the higher-number pair. If "class nights" (without quarantining the weak or muscling-in on their prizes), have strong, stranger (insufficient data) average and weak quarters: graded prize per quarter and a bonus for tops-wherever appearing.

PART TWO [I know, I know. There is no labeled PART ONE.--Ed.]

A simpler way for Man to go ringwise, while inanimate, stackable boards adapt is: move Mitchellwise. But a given EW alternately refuses to move-making NS become EW mover; the arriving, so far EW, pair fills NS vacancy. Alternately moving, "Key" shifts 2 pairs at each new table- the each-meet-each resulting. Boards are stacked and dealt each round to adapt.

   Details are beyond space here available and properly commercial. Only these applications of "Heaton" to the faults will be given in theory:
   5) Cafeteria: director handles 50 to 100% of boards; those liable to hold him up are passed by players, using automatic "dials".
   7) Gyp length: an added board, per table, moves Mitchellwise until total play is 26 or more except at 9 Ts; here too if some players exchange seats.
   6) On-time start: implies a standard preplacement of boards, with option to continue with any system after the prompt started, at named hour. Latecomers start as more tables fill; but they are prevented from completing boards that would hold up the prompt (gaining nothing) by a call of: "start no new board now"-made when the prompt start their last board. Boards so missed may be staggered to scoring session for play off, without penalty: another normal move restores round 1 couplings: place boards (best marked as missed) to adapt.

[beginning of figure--Ed.]

T               B      BX      G      R      B      G      T
  1A  1/  44/31P16
  2B  3/  22/29O15
  3C  5/  66/30OR15*
  4D  7/  88/32Q16*
  5E  9/12&/27N14
  6F11/10Z/25M13
  7G13/14Y/28MS13*
  8H15/16X/26NT14*
  9I19/18*W/23L12
10J21/22*V/24LU12*
11K17/20*  /

(The words "1-side vul" appear under the consecutive columns B and G on the final table row.)

5 Tables

A   B   C   D   E
F   B   C   G   I
H   E   C   D   B
G   I    C   E   A
D   AxA   I    F
B   F   H   E   G
C   F   H   I    D
E   G   H   A   F
I   D   H   G   B

x: swap
of 2nd set.
[end of figure--Ed.]

To permit 1-4 boards per table, letter set thus: that at far left on three boards in line; in BLACK if under B, in RED if under R. Given GREEN letters under G, on board at left; in last column, also the black letter under B. Those under X receive a legal type red SEAL. Now the 1-4 boards at a table travel under a single-digit letter or symbol.

Always place black pairs by the number-alphabet series given. If Mitchell, none switch from filled tables; if under 12 tables, add reds to mates; if under 9, also unmarked spares having the missing vul. (P-Q a pair if 16 tables). In "G.H." (guideless Howell), if over 8Ts, remove the even-number (*) boards; add reds if 5-4; unmarked spares if 4Ts. (Data for Heaton with sets). The "enterlater" groups (under "&" in list) continue alphabet by black pairs, if under 9 tables; by black-green series if 9 up.
   Should a pair, arriving after the call and before the "deadline" (change for Round 2) compel one less board per round, remove and cancel the one last added--or the even-number one. If even-tables at Mitchell, kill set at last table; reshuffle; place a chair between halves (usual swaps). Missed boards may be averaged as mild penalty; or zero-ed, as severe.

   If routine seems impossible at 5 tables, place boards by schedule given, on each round: on one round those at table 2 and 3 exchange boards.
   The strong-weak at Mitchell never gives the weakest a chance; only a match-point bonus can serve-sure to involve rigid bookkeeping if to squelch squawks. Here basis is inexact and mysterious: if not knowing whether to kick or not, least bookkeeping is involved.
   The Howell type is eternally damned, since exactly at common and paying numbers, the gyp on length is extreme; only Heaton can cure this defect.

PART THREE

"Readers are not interested in mechanics"-from the editor. But perhaps gentlemen are not aware of unsporting division and those proud about efficiency elsewhere, of the horsing here. The director-and the motorman-works handles; but without slightest knowledge of what makes the wheels spin (if excepting a handful related to printed Howell matter sales). A whole book was devoted to subject--chiefly about the "idiot's delight" way; only a few pages needed to review, show faults and remedy them.
   As customers-and judges; experts, who are keen to spike inside tracks; who huddle ad lib and get to bed at 2-seldom having other jobs. Also a humble million who must be abed by midnight; ready for job tomorrow. They may wish most card-play, least clowning, in the time available. If experts are allergic to efficiency and relish pauses and promenades, blow a whistle each round for 2-minute rests/walks; suit both classes. Other vendors blurb "service"; "customer is always right": here the parallel would be for vendor to handle maximum boards and be responsible for all mixups. But all now must suit the one serving a year's stretch, in rotation: the playing director; to give customers the benefit of guaranteed length, on-time start, handicaps-absolutely new additions to the routine-are not to be considered if vendor's work is increased.
   The subject of wooden custom, cranks, pseudoscientific "better comparisons" and "2-board play," to sell brands to the gullible, is relegated to author's discussions, with sets.
   The natives restored to Libya will resume with ox-carts, as for thousands of years, despite engineers trying to tell them how. Bridgers must use ox-carts-the atrocious mechanics-eternally, since there are no engineers other than brand sellers. That is to say: unless the reader, with no informed, intelligent and unbiased authority in existence, chooses to test bn the road and master details (perhaps improve) and prove the case. (For information on equipment, inquire of 217 No. 14 St., Ft. Lauderdale, Fla.)

Solutions and Comments

1. From N. Scott Cardell, Pullman, WA:

I. Introduction

   The first movement described by Perry Heaton in "The Inept Mechanics of Duplicate" is apparently a movement he developed and named the "Guideless Howell." The description of this movement begins with "Print on 16 cards:" and ends with "North takes board used to T-7, putting it under the two already there; then joins partner at table he moved to." In this section, the author refers to but does not explain an earlier development of his, which he refers to as "the Heaton method." Apparently, the Heaton method is described in a pamphlet that was available from the author (at 217 No. 14 St., Ft. Lauderdale, Fla) in the 1940's. In Part Two, the author also describes a second movement (which appears to be the Heaton method, or part of it). In addition, he describes a new way of numbering boards and makes several suggestions on handicapping and on accommodating late-arriving pairs.

II. Dr. Heaton's Guideless Howell

   The pairs take a number and (in the standard form) are allowed to sit anywhere, (the movement being "guideless"). If Dr. Heaton's handicapping method is to be used, than the pairs sit initially as follows:

   E-W pair number is equal to the table number.
   N-S pair number at table 1 is the highest pair number (equal to twice the number of tables) and descending in number with increasing table number.

   Thus, the N-S pair number is 2 times (number of tables) - table number + 1.

   For example, the numbering for 4 tables (by Table-EW number-NS number) is: 1-1-8; 2-2-7; 3-3-6; 4-4-5.

   For expositional purposes, I will use the above numbering at all times.

   The movement of the players for any number of tables (details for the movement of the boards are given for numbers from 4 to 16 tables) is as follows:

   E-W at table 1 (pair #1): Stationary.
   E-W at the highest table move to N-S, stay at the same table.
   All other E-W pairs move to E-W at the next higher table.
   N-S at table 1 move to E-W at table 2.
   N-S at all other tables move to N-S at the next lower table.

   The placement and movement of boards for 4, and for 6 through 16 tables is this:
   The boards are placed on the tables according to the "&" column in the first figure.
   "-" = place one set of boards on this table.
   "*" = place two sets of boards on this table.
   "2" = place three sets of boards on this table.
   "3" = place 4 sets of boards on this table, etc. ("n" = place n+1 sets of boards on this table.)

   The sets of boards are placed on top of each other. In each round, only the top set of boards is played. (He refers to the other sets as the "enterlater" groups.) The boards just played are then moved to the table listed under the "to" column in the first figure. The boards moved to a table are placed at the bottom of the stack. (Dr. Heaton recommends his unique color coded numbering, described below, to avoid possible confusion.)

III. Example: A Four-Table Guideless Howell

   Following Dr. Heaton's preferred method, I use capital letters to distinguish the sets of boards.

   Thus, the complete movement for a four-table Guideless Howell is as follows:

   Cards on the tables:

Table 1:
N-S go to E-W table 2; E-W stay stationary
Take boards to table 4

Table 2:
E-W go to table 3 E-W; N-S go to table 1 N-S
[or, instead of the above: E-W move to E-W at the next higher table, N-S move to N-S at the next lower table.]
Take boards to table 3

Table 3:
E-W go to table 4 E-W; N-S go to table 2 N-S
[or instead of above: E-W move to E-W at the next higher table, N-S move to N-S at the next lower table.]
Take boards to table 1

Table 4:
E-W stay at table 4 move to N-S; N-S go to table 3 N-S
Take boards to table 2

(for each round shown below, the four items separated by hyphens stand for Table-EW-NS-Board set-"enterlater sets"; tables are separated by semicolons)

Round 1: 1-1-8-A-EF; 2-2-7-B; 3-3-6-C-G; 4-4-5-D.

Round 2: 1-1-7-E-FC; 2-8-6-D; 3-2-5-G-B; 4-3-4-A.

Round 3: 1-1-6-F-CG; 2-7-5-A; 3-8-4-B-D; 4-2-3-E.

Round 4: 1-1-5-C-GB; 2-6-4-E; 3-7-3-D-A; 4-8-2-F.

Round 5: 1-1-4-G-BD; 2-5-3-F; 3-6-2-A-E; 4-7-8-C.

Round 6: 1-1-3-B-DA; 2-4-2-C; 3-5-8-E-F; 4-6-7-G.

Round 7: 1-1-2-D-AE; 2-3-8-G; 3-4-7-F-C; 4-5-6-B.

   (Dr. Heaton points out that one more move will restore the round one situation for a multi-session event.)

IV. The Five-Table Special Case

   The five-table guideless Howell uses the same pair movements. However, there apparently being no solution of the same type for the board movements, the boards are moved as in Heaton's second figure, where the AxA in round 5 indicates that tables 2 and 3 share the "A" set of boards on that round.

V. Dr. Heaton's Handicapping Method For Howell-type Movements

   Assign the pair numbers in order of weakest to strongest (pair 1 = weakest pair). At each table, at each round, point the arrow on the board to the higher numbered pair. Thus North-South on that board will always be the stronger pair regardless of the seating relative to the directions on the table. For example, in the four-table movement given above:

   Board set A will be played with:
in round 1: pair 8 playing the N-S cards and pair 1 the E-W cards;
in round 2: pair 4 playing the N-S cards and pair 3 the E-W cards;
in round 3: pair 7 playing the N-S cards and pair 5 the E-W cards;
in round 5: pair 6 playing the N-S cards and pair 2 the E-W cards.

   Note that, in rounds 3 and 5, board set A is not aligned with the directions on the table.

   The effect of this handicapping is that the weakest pair will always compare scores with the weaker of the two pairs in the other plays of the same cards. In any round, the weaker of the two pairs playing a given set of boards will compare scores with the weaker of the two pairs in all other plays of the same boards.

   When used with the guideless Howell, Dr. Heaton considers it important that the initial seating be as described above. Although no explanation is provided, it is clear that the initial seating does impact the degree of handicapping for all but the weakest and strongest pairs.

   Dr. Heaton makes the additional suggestion that the field be divided into quarters, by level, with a graded number of points awarded to best in quarter and with extra points to best overall.

VI. The Heaton Method: Dr. Heaton's Second Howell-Type Movement

   At the beginning of Part Two, Dr. Heaton introduces a second Howell-type movement, after which he makes the following statement: "Details are beyond space here available and properly commercial. Only these applications of 'Heaton' to the faults will be given in theory." This statement suggests that (1) this movement is part of the Heaton method, and (2) that further details such as the movement of the boards are not included in this article.

   In this movement, pairs are again assigned unique numbers, which (for simplicity) I will take to be in the same order as in the Guideless Howell. The moves after the first, third, and all odd-numbered rounds is straight Mitchell. N-S pairs stay put and E-W pairs move to the next higher table in cyclic order (E-W at the highest table moving to E-W at table 1). A single pair sitting E-W is denoted the "key pair" and assigned a special role in the movement. After each even-numbered round, the key E-W pair remains fixed while the N-S pair at that table moves to E-W at the next higher table and the E-W pair from the next lower table moves to N-S at the key E-W pair's current table. All other moves are E-W to E-W at the next higher table. Obviously, which pair is chosen to be the key pair does not matter because of the symmetry of a Mitchell movement.

   Using again the example of four tables, let pair 1 be the key pair. The pairs move as follows (within each round, a triple of numbers indicates table-EW-NS; tables are separated by semicolons):

Round 1: 1-1-8; 2-2-7; 3-3-6; 4-4-5.

Round 2: 1-4-8; 2-1-7; 3-2-6; 4-3-5.

Round 3: 1-3-8; 2-1-4; 3-7-6; 4-2-5.

Round 4: 1-2-8; 2-3-4; 3-1-6; 4-7-5.

Round 5: 1-7-8; 2-2-4; 3-1-3; 4-6-5.

Round 6: 1-6-8; 2-7-4; 3-2-3; 4-1-5.

Round 7: 1-5-8; 2-6-4; 3-7-3; 4-1-2.

   Apparently, there is a figure in Dr. Heaton's pamphlet on the Heaton method (similar in form to the first figure in the Bridge World article) that describes the movement of the boards. In any case, his assertion is that a similar movement of boards is possible. Assuming that this is correct, it should not be difficult to devise the appropriate figure.

   He also suggests that rather than playing fewer boards, one should add one board per set and play an appropriate number of rounds. Apparently, Dr. Heaton took great offense at paying his card fees and then playing 17 boards in a nine-table Howell. From the number of references to this problem (his term is "gyp length" or "gyp on length") and to nine tables or playing 17 boards, this must have happened to him a number of times. For reasons that are not given, Dr. Heaton suggests that only with the Heaton method is it appropriate to play fewer than the full number of rounds in a Howell-type movement. However, details, and perhaps justification, appear to be left to "Data for Heaton with sets."

VII. Dr. Heaton's Scheme for Marking Boards

   This scheme is only partially detailed, with the remainder being available with "sets" for the Heaton method available from the author. It is implied that these sets are (or include) sets of boards designed to be easily stackable. However, it may be that the sets were instructions on how to letter boards and how to implement the Heaton method. As best as I can determine from the article, the basic scheme is this:

   For each letter A, B, ... , place the letter on four boards. On one board per set, the letter is black, with no other mark; on one board per set it is in green; on another red; and on the final board, the letter is in black and "legal type red SEAL" is added. If sets of two boards are being used, then use the two black-lettered boards. If sets of three boards are used, use the black-lettered plus the green-lettered. If four-board sets, then add the red-lettered as well. Apparently, at least for the first 5 sets (A-E), each set of four boards has three distinct vulnerabilities. If five boards per set are needed, he suggests adding "unmarked spares having the missing vul."

VIII. Dr. Heaton's Suggestions for Dealing with Late-Arriving Pairs

   His primary suggestion is that a table or tables be added if pairs arrive late but before the call to move to the second round. If the number of boards to be played per round is to be decreased (say eight tables goes to nine and the decision is made to switch from a six-round Mitchell with four boards per round to an eight- or nine-round Mitchell with three boards per round), then the "last one added" can be canceled. Dr. Heaton suggests this procedure for Mitchell movements as well as for his Guideless Howell and Heaton Method movements. In the latter cases, the "enterlater" boards must be rearranged and additional sets added. So, if you go from four to three boards per set, cancel the red-lettered boards, keep the black- and green-lettered boards, and (for the new table) add the next set of black- and green-lettered boards. Missed boards on round one would be marked and, after the last round, the affected players and boards would move again to restore the initial pairings and board placements. The missed boards would then be played.

IX. Relationship of Dr. Heaton's Howell-Type Movements to the Standard Movements

   Like the conventional Howell movements, Dr. Heaton's Guideless Howell and Heaton Method movements allow 2 times T pairs to be seated at T tables, with 2 times T - 1 sets of boards to play 2 times T - 1 rounds playing one round against each other pair, playing each set of boards once. His movements are worked out for from four to sixteen tables. The primary advantage of Dr. Heaton's movements is that the movement of the pairs is minimal and simple. Each round, a pair either stays at the same table or moves to an adjacent table. Furthermore, the movements are easy to understand and can be guided by simple cards, such as, "E-W move to E-W at the next higher table, N-S move to N-S at the next lower table." In the Guideless Howell, all but the stationary pair move every round. In the Heaton Method, one half the pairs move each round, one pair is stationary, and with T tables one pair moves once, one twice, ..., two T-1 times, one ..., 2T-2 times. Because at least one pair must move at each table, the Heaton method gives the minimum possible number of moves. His method of moving the boards appears to be simpler, given boards designed to stack well. His lettering scheme for the boards might well reduce errors (everyone at the table would know that this round they are playing set "D", for example). However, in spite of his polemics about guide cards, one would certainly want a set that would allow checking each round to ensure that people were at the correct table playing the appropriate pair and using the right boards.

   For the unhandicapped movements, the largest disadvantage of Dr. Heaton's movements is that the comparisons are unbalanced. This applies to both the Guideless Howell and the Heaton Method. In the Guideless Howell, the stationary pair has balanced comparisons with all pairs, but some combinations of pairs sit the same way only once and other combinations of pairs sit the opposite way only once (in addition to playing against each other). In the Heaton Method, the stationary pair and the "key" pair always play in the opposite directions. In addition, another pair always plays in the same direction as the "key" pair (and in the opposite direction of the stationary pair) except when that pair plays against the key pair. Dr. Heaton views this disadvantage as minor: "The subject of wooden custom, cranks, pseudoscientific 'better comparisons' and '2-board play,' to sell brands to the gullible, is relegated to author's discussions, with sets."

   Dr. Heaton also suggests adding another board to the set and playing a partial movement to get a near optimal number of boards played. As he appears to think that 24 to 26 boards is optimal, he would with nine tables play the first 12 or 13 rounds of the 17-round Heaton method movement. He gives no reason in this paper but implies that this solution only works well with the Heaton method, not with a Howell or his "Guideless Howell." I do not see any reason why that should be true, but perhaps if some reader has a "Heaton set" and the supporting material that came with it, I could be enlightened. The three-quarter movement (described in the "Encyclopedia of Bridge") allows 26 boards with eight to twelve tables and has reasonably balanced comparisons. As with the standard Howell, the three-quarters movement requires more complex moves by the pairs.

   The handicapped Heaton method and Heaton's handicapped Guideless Howell are, to my knowledge, a unique approach. Heaton's handicapping approach could be applied to a standard Howell movement. However, the purpose of the handicapping scheme is to unbalance the comparisons in a way that will favor the weaker pairs at the expense of the stronger pairs, while a primary purpose of the Howell movement is to produce balanced comparisons. I applied the handicapping method to some example cases and the results seemed to lead to a reasonable stratification of the degree of handicapping. One result that can be easily computed, and that applies to any number of pairs, is: weakest pair of N pairs will be below their opponents by N/2 in rank on average; the pairs that they are competing against in the scoring will be N/3; similarly, the strongest pair will be on average N/2 higher in rank than their opponents and will be scored versus pairs that are on average N/3 higher than their opponents. Thus, if the ranking is accurate, the handicapping will be partial, giving the weaker players a much better chance, but not equalizing the chances.

X. Practical Applications of Dr. Heaton's Suggestions.

   Dr. Heaton's Guideless Howell and Heaton Method movements (without handicapping) provide an easier and probably faster movement for a Howell-type game. However, they sacrifice balanced comparisons. In some situations (for example a seven-table Howell-type event) a small club might well consider the trade-off worthwhile. Also, with awkward numbers of tables like 8, 9 or 10, a club might well want to try a simple-movement 13-round partial Guideless Howell or Heaton movement. If experience showed that the easier movements of the Guideless Howell and Heaton Method saved enough time to allow one extra round of bridge, then their popularity would no doubt increase. However, the competitive nature of many bridge players might make the widespread use of a method with very unbalanced comparisons unlikely. Instead, clubs could stick to the Howell and three-quarter movements, which offer reasonably balanced comparisons. However, many clubs use Mitchells without making any effort to balance the field between the N-S pairs and the E-W pairs. Usually, the earliest arrivers sit N-S, choosing to do so so they won't have to move. It is probable that a scheme for rotating the directions could be worked out so that Dr. Heaton's simple movement could be combined with balanced comparisons. (In a six-table Guideless Howell, simply switching the direction of play at table four gives fairly balanced comparisons.) However, that goes beyond the scope of what he proposed. Unfamiliarity with the movement and using a stack of sets of boards at some tables also would be a barrier to adopting either the Guideless Howell or the Heaton Method.

   Dr. Heaton's handicapping method is intriguing. It seems to me that it is well worth trying. It in effect allow experts and novices to play together in one event while players at each level compare scores primarily with other players of similar skill. The modified scoring based on winners within quarters of the field and overall winners allows the rewards to be commensurate with accomplishment. This seems to be a better way to get novice players involved in duplicate than having them play against each other in a novice game. The conventions allowed could be highly limited, as they are in novice games, and yet if first overall and first in the top 1/4 are reasonably rewarded, novice players would get a chance to compete against experts in a situation where they had a reasonable chance of winning overall and an excellent chance to win the "novice quarter." It would also work well in a club that wants to have regular tournaments, with one's rank in the next tournament depending on one's past performances in club events. The handicapping method would work best with the Guideless Howell or perhaps with the Heaton Method. While the handicapping method could be applied to a standard Howell, the complex movements of a Howell would not combine well with switching the direction of the boards.

   Heaton's approach to marking sets of boards certainly has the possibility of reducing errors. It combines well with his other suggestions. For example, a change of the letter marked should be a good cue that you have played all the boards in the set. However, a large part of his motivation seems to be to avoid producing a large number of guide cards that list the boards to be played each round at each table. Unfortunately, the large table at the beginning of "Part Two" remains largely a mystery, at least to me. I suspect that in order to understand exactly how he selected the boards to be marked with each letter, one needs to have the "Data for Heaton with sets." There would seem to be no great difficulty in creating guide cards that would allow the use of the Guideless Howell or the Heaton Method with a standard set of boards.

   Dr. Heaton's preference for partial movements and playing more boards seems quite mundane today. At most clubs, at least 22 boards will be played no matter how many contestants show up. In effect, that his suggestions in this area have already been widely adopted, though not in combination with his movements. My own choice with eight or twelve tables would be a Mitchell-type movement, playing twenty-four boards. To play eight rounds with eight tables or twelve rounds with twelve tables, let odd numbered E-W's move up a table and even numbered E-W's move down a table -- same movement every round. The boards move in the opposite direction to the E-W pairs. In the move for the middle round (move to round five with eight tables or to round seven with twelve tables) the boards are exchanged with the table half the number of tables away (for eight tables, 1 and 5 exchange, 2 and 6, ...). This movement is of my own devising and freely available to anyone who wants to use it. Giving just the E-W positions and the board positions, the eight-round "Cardell movement" is:

12345678
ABCDEFGH

21436587
HCBEDGFA

74163852
CHEBGDAF

47618325
FEHGBADC

56781234
BADCFEHG

65872143
GDAFCHEB

38527416
DGFAHCBE

83254761
EFGHABCD

XI. Conclusion

   Dr. Heaton's movements would have appeal if the time savings turned out to be substantial. His movements, combined with his handicapping system, seems to be of interest as a way to involve novice players without using flighted events. His suggestions for marking boards has some appeal but is unlikely to be adopted. His suggestion that more boards be played in partial movements rather than playing fewer than than 22 boards has already been largely adopted.

2. From Olof Hanner, Sweden:

   Heaton's criticism of the usual form of Howell movement is exaggerated. For instance, one might have to walk 17 miles in a 31-round, 16-table movement, but no sensible director would arrange such a contest.
   His alternative would not be an improvement, as far as it is decipherable (he keeps some secrets to himself, such as how to deal with late arrivals and the cures for the points he marks #). However, his guide-less Howell is a true Howell, a round-robin with one pair stationary and the others moving cyclically. His description avoids the use of numbers for pairs or board sets, but to describe the Heaton movement here I follow the usual notation.
   In a (standard) guide-less Howell, for each successive round each East-West moves up one table, except that the East-West pair at the highest table switches to North-South; each North-South moves down one table, except that the North-South at Table 2 moves to East-West at Table 1. For example, for six tables, we could give the starting positions as (where each number triple stands for table number -NS number-EW number): 1-12*-1, 2-2-11, 3-3-10, 4-4-9, 5-5-8, 6-6-7. The asterisk indicates that pair 12 is stationary throughout. Pair 1 follows pair 11; the others follow the next lower-numbered pair.
   Although the player changes for these movements are so simple they can just be announced, for the boards there is needed some sort of guide card at each table, telling where the boards just used will be played next (though not necessarily on the next round). [In an ordinary Howell, the boards are moved to the next lower table. Heaton shows the board movements for various numbers of tables. For example, for six tables the flow of boards can be written as 1 to 4 to 3 to 6 to 2 to 5 to 1. In other words, a board played at Table 1 is next played at Table 4, then at Table 3, etc.]
   Heaton does not explain when boards arriving at a table should be played there. In a Howell, there are boards that are not being played, presumably placed under the active boards. After the round, each North moves the played boards to become the bottom boards at some new table. Hence, Heaton needs a guide card for each table, telling how many board sets to place there at the start and where played boards should be moved. (He gives an actual such card only for Table 1 in a 10-table movement. The card should say, in effect, 4#7, meaning start with four board sets and move the played boards to Table 7.) The appropriate numbers for these cards can be reconstructed from the way the boards should move and the fact that no pair should play a board twice. For instance, in a six-table game, a board played at Table 1 in Round 1 will be played thus (table-round-NS-EW): 1-1-12-1, 4-3-6-11, 3-7-9-5, 6-8-2-3, 2-9-10-8, 5-11-4-7. If we number the board sets 1 through 11, this scheme shows the movement of Board 1. If the boards are to be played in numerical order at each table, Table 4 ought to start with Board 10 (so that Boards 10, 11 and 1 are played there in the first three rounds). Similarly, we can deduce the starting boards for the other tables, yielding this starting position in the usual notation (table-NS-EW-boards): 1-12-1-1, 2-2-11-4, 3-3-10-6, 4-4--9-10, 5-5-8-2, 6-6-7-5. For instance, Table 3 starts with Boards 6, 7, 8 and 9. After the first round is played, Board 6 is sent to Table 6 and Board 10 arrives from Table 4. Thus, the guide card at this table shiuld read 4#6.
   This form of movement is not unknown to moderns; it is sometimes called "Endless Howell" or "Flower Howell." It may have certain advantages; its main flaw is that board sharing is needed for five tables.
   The main fault with Heaton's implementation is that there are few possibilities for the director or the players to check activities or to correct errors. What if the boards are not moved or are sent to the wrong table? Heaton complains of slavery to guides, but guides allow verifying that boards and opponents are correct.
   Another flaw is that boards may pile up. In a four-table contest, with four boards per round, Table 1 will have 12 boards.
   There is no way to adjust for balancing. Compass directons are fixed by the necessities of the movement, and cannot be reorganized to take inter-pair comparisons into account.
   The only positive festure of Heaton's movements is that pair travels will be short. However, that is implied by the use of an Endless Howell. Here, I give variants of Heaton's movement for 4 through 12 tables (except 5). Those for 4, 6, 8 and 10 tables have perfect balance; the others have reasonably good balance. In each case, the notation is table-NS-EW-board set:

4 tables: 1-8-1-1, 2-2-7-6, 3-3-6-4, 4-4-7-5.

6 tables: 1-12-1-1, 2-2-11-4, 3-10-3-6, 4-4-9-10, 5-5-8-2, 6-6-7-5.

7 tables: 1-14-1-1, 2-2-13-4, 3-3-12-6, 4-11-4-10, 5-5-10-2, 6-6-9-7, 7-7-8-3.

8 tables: 1-16-1-1, 2-2-15-7, 3-3-14-2, 4-4-13-8, 5-5-12-14, 6-6-11-12, 7-10-7-3, 8-9-8-6.

9 tables: 1-18-1-1, 2-2-17-11, 3-16-3-6, 4-4-15-2, 5-5-14-5, 6-6-13-12, 7-7-12-9, 8-8-11-16, 9-9-10,5.

10 tables: 1-20-1-1, 2-19-2-11, 3-3-18-13, 4-17-4-5, 5-5-16-18, 6-6-15-4, 7-7-14-19, 8-8-13-12, 9-12-9-15, 10-11-10-7.

11 tables: 1-22-1-1, 2-2-21-11, 3-3-20-6, 4-19-4-10, 5-5-18-7, 6-6-17-4, 7-7-16-21, 8-8-15-9, 9-9-14-13, 10-10-13-5, 11-11-12-8.

12 tables: 1-24-1-1, 2-2-23-20, 3-3-22-13, 4-4-21-3, 5-20-5-9, 6-6-19-17, 7-18-7-10, 8-17-8-2, 9-16-9-18, 10-10-15-11, 11-14-11-4, 12-12-13-19.

   For a 5-table Endless Howell, one must allow either board-sharing or an irregular board circulation. To arrange for board-sharing, one may use (note the duplicated board-set number):

5 tables: 1-10-1-1, 2-2-9-6, 3-3-8-6, 4-4-7-3, 5-5-6-6.

   For the five-table case, Heaton gives an irregular board circulation with board sharing only in one round. However, board sharing can be avoided completeliy with the following irregular movement. (Here, for each of the nine rounds I give 15 numbers: NS at Table 1-EW at Table 1-Board set at Table 1; NS at Table 2-EW at Table 2-Board Set at Table 2; etc. Note that tables are separated by semicolons.)
Round 1: 10-1-1; 2-9-6; 3-8-2; 4-7-5; 5-6-7.
Round 2: 10-2-2; 3-1-7; 4-9-1; 5-8-6; 6-7-8.
Round 3: 10-3-3; 4-2-7; 5-1-4; 6-9-2; 7-8-9.
Round 4: 10-4-4; 5-3-9; 6-2-5; 7-1-6; 8-9-7.
Round 5: 10-5-5; 6-4-9; 7-3-1; 8-2-4; 9-1-8.
Round 6: 10-6-6; 7-5-2; 8-4-3; 9-3-5; 1-2-9.
Round 7: 10-7-7; 8-6-1; 9-5-3; 1-4-2; 2-3-8.
Round 8: 10-8-8; 9-7-4; 1-6-3; 2-5-1; 3-4-6.
Round 9: 10-9-9; 1-8-5; 2-7-3; 3-6-4; 4-5-8.

   In an Endless Howell, with one exception all pairs move. Heaton points out that for each round it is possible to let half the pairs sit. To see this, let us take the 5-table movement and rewrite the pairs (NS at 1, EW at 1, NS at 2, EW at 2, etc.) for each round this way:

Round 1: 1-10; 2-9; 4-7; 6-5; 8-3.
Round 2: 1-3; 2-10; 4-9; 6-7; 8-5.
Round 3: 1-5; 3-10; 4-2; 6-9; 8-7.
Round 4: 1-7; 3-5; 4-10; 6-2; 8-9.
Round 5: 1-9; 3-7; 5-10; 6-4; 8-2.
Round 6: 1-2; 3-9; 5-7; 6-10; 8-4.
Round 7: 1-4; 3-2; 5-9; 7-10; 8-6.
Round 8: 1-6; 3-4; 5-2; 7-9; 8-10.
Round 9: 1-8; 3-6; 5-4; 7-2; 9-10.

   When changing to an even-numbered round, the East-West pairs move up one table, as in a Mitchell. When changing to an odd-numbered round, one East-West pair (number 10) does not move; instead, the North-South pair at that table will be part of the moving to the next higher table.
   I see no advantage to using this sort of transformation, because the board movement would have to be irregular and, as the movement of the pairs would be knitted to their compass directions, there would be no possibility for comparison balancing.

3. From Don Kersey, Kingston, ON:

Part One: The Guideless Howell Movement

   (a) Heaton's basic idea is to simplify the movement of the players, at the cost of making the movement of the boards more complex, with the further idea of making the director rather than the players primarily responsible for the movement of the boards. In the Guideless Howell movement, the pairs move in the same way after each round: Each EW pair moves up one table and stays EW, except that the EW pair at the highest numbered table (the pivot) changes to NS at the same table. Each NS pair moves down one table and stays NS, except that NS at table 1 is stationary and NS at table 2 moves to EW at table 1.
   (This is the translation of Heaton's table card instructions: "EW go to next higher-number table, EW; NS go to next lower-number table, NS" and "At T-1, NS becomes EW; go to highers. Last T; EW pivot to NS; go to lowers.")
   To see how to read the first table (the one with all the "to"s), let's isolate the first three columns (where the number triple X-Y-Z match columns headed T-to-&):

1-4-2
2-3-
3-1-1
4-2-
(why Heaton used * for 1 in this table seems inexplicable, but I assume that is the meaning of his cryptic note "*, instead of Fig. 1, means: one board here.")
   These columns show the board movement for a 4-table Guideless Howell. The column headed "T" refers to table number, the column headed "to" shows the table to which boards must be moved, and the column under "&" shows the number of bye-stands at the table in question. So, e.g., in this movement, table 1 has a double bye-stand, and boards from table 1 move to table 4 (a double bye-stand means that boards are idle for two rounds when they reach this table).

   So here is the full Guideless Howell movement for 4 tables:
NS at table 1 are stationary; the other pairs move in this closed cycle:
1EW -> 2EW -> 3EW -> 4EW -> 4NS -> 3NS -> 2NS (-> 1EW)
Boards move:
Table 1 -> Table 4
Table 2 -> Single bye-stand at Table 3
Table 3 -> Double bye-stand at Table 1
Table 4 -> Table 2

   (b) In essence, this 4-table movement differs only cosmetically from the standard 4-table Howell found in Groner; it is necessary to make only the following modifications to Groner's movement:
Interchange the physical locations of tables 2 and 3.
Switch arrows to interchange NS and EW at these two tables.
   To verify this, first note that the pair movement in Groner's 4-table Howell has NS stationary at table 1, and the other pairs move in this cycle:
1EW -> 3NS -> 2NS -> 4EW -> 4NS -> 2EW -> 3EW (-> 1EW)
In this cycle, if we interchange 2 and 3, and then further interchange 2EW with 2NS and 3EW with 3NS, then we get exactly the same cycle as in (a). Furthermore, the movement of boards in the standard 4-table Howell is:
Table 1 -> Table 4
Table 2 -> Double bye-stand by Table 1
Table 3 -> Single bye-stand by Table 2
Table 4 -> Table 3
and, once again, interchanging tables 2 and 3 produces the board movement described in (a).

   A Guideless Howell movement can always be "Howellized" by rearranging tables (for instance, the 6-table Guideless Howell has this board movement:

Tbl 1 -> bye -> Tbl 4 -> bye -> bye -> bye -> Tbl 3 -> Tbl 6 -> Tbl 2 -> bye -> Tbl 5 (-> Tbl 1)

and a normal Howell movement, in which the boards move down one table each round, can be produced by physically rearranging the tables in this order:

1 - 5 - bye - 2 - 6 - 3 - bye - bye - bye - 4 - bye.

   On the other hand, most standard Howell movements cannot be made guideless. In a normal (full) Howell, there is one table with a stationary NS, one table at which the difference between the EW and NS pair numbers is 1, another at which it is 2, another at which it is 3, etc. The Guideless Howell movement insists that the table with pair difference 1 be the highest numbered table, the table with pair difference 2 be table 2, the table with pair difference 3 be the second highest table, the table with pair difference 4 be table 3, and so on, and we can always physically rearrange the tables in a standard Howell to meet these conditions. However, the Guideless Howell movement also further insists that pair 2 start at table 2, pair 3 at table 3, pair 4 at table 4, and so on (up to pair n, where n is the number of tables; for higher numbered pairs, pair n+i will play at table n+1-i), and most Howell movements will not satisfy this additional condition.
   Heaton was able to devise appropriate board assignments for all number of tables from 4 to 16, except 5; for 5 tables there is no perfect Guideless Howell movement, and he was forced into an ugly ad hoc movement, which he gives as an auxiliary table (the one with "AxA" in its midst) to the first table.
   This means that there are 9 sets of boards, labelled A-I; the table numbers are across the top, the round numbers go down, so, e.g., in round 3 table 4 plays board set D, while in round 5, tables 2 and 3 relay board set A). The movement of pairs is standard Guideless Howell.

   (c) Heaton suggests several reasons to prefer his system to standard Howell movements. Let's take them one at a time:

   1 - Slavery to guides. Certainly the pair movement is much easier in the Guideless Howell.

   2 - Cost of full printed cardboard sets. It is true that one could in theory have a single set of table cards which would cover all numbers of tables from 4 to 16 (except 5); for instance, each table card could have a pocket into which the director would insert the number of the table to the boards should be moved. On the other hand, one probably would still want customized cards for each possible game size, since for the prevention and straightening out of missteps, it is handy to have the round-by-round data on the card (NS pair, EW pair, board numbers).

   3 - Walk-collide beyond next tables. The change might be somewhat more orderly, but still NS and EW pairs would be walking in opposite directions, and N players (or the director) would be darting around the room to move the boards to the appropriate table.

   4 - Extra time loss at changes. Unlikely to be much improvement here, given the complexity of the board movement, but only a trial would tell. However, if, as Heaton hints, the director is responsible for moving all the boards, then the changes might well be slower with the Guideless Howell than with an ordinary Howell.

   5 - Heaton doesn't claim any benefit for the Guideless Howell under this point.

   6 - Time loss in starting. This presumably refers to the necessity to rearrange the game to accommodate late pairs, and would perhaps be a small advantage for the Guideless Howell, since the game can always be started on time and extra pairs added as they arrive during the first round (necessitating re-numbering the pairs, re-arranging the boards, etc, but not disrupting the players unduly).

   7 - Again, Heaton doesn't claim any advantage for the Guideless Howell under this point.

   8 - Frequent handicaps. Heaton's handicap method is simple enough - seed the pairs from weakest to strongest, and, during the game, switch arrows so that each round at each table, NS has the higher seeding number, thereby minimizing the number of times the weaker players compete for matchpoints against the stronger pairs. Of course, exactly the same thing could be done if desired in a standard Howell.

   Some disadvantages of the Guideless Howell, which Heaton doesn't mention, are:

   Howell movements for 7 tables and up can always be chosen to have a single assembly table, which holds all the bye-stands, while the extra conditions for Guideless Howell movements always force multiple bye-stand tables (up to 8 for 16 tables!). A single assembly table is a big convenience.

   One of Heaton's basic premises (that it is better to have the inanimate objects perform the complicated action while the presumably intelligent customers perform the simple action) is surely open to question.

   If the players are responsible for moving the boards, much of the gain is lost, while if the director is responsible for moving all the boards, particularly when there are many tables, it seems likely that the players will frequently be ready to play but have no boards. (For instance, what happens when the director has to make a ruling at the end of a round?)

   The number of boards played in a Howell can be adjusted by adding extra stationary pairs, resulting in a 3/4 Howell (in the limit, make half the pairs stationary, and you have a Mitchell). This adjustment can't be made in a Guideless Howell (at least not in an obvious way) without abrogating Heaton's basic principle of having pairs move only to adjacent tables.

   In summary, with many tables in play, either a Mitchell or a standard Howell with an assembly table seems a better solution; with few tables in play, the Guideless Howell might be nicer, but for very small games, the gain is minimal (or non-existent, e.g., 5 tables). It might be an improvement for 6 tables, but that is the worst size of all for a Guideless Howell, since the number of boards played will be 22 (too few) or 33 (too many) or else the movement must be curtailed (leaving some boards played more often than others, forcing factoring, which nobody likes). Maybe next time I direct a 7-table game, I'll try out a Guideless Howell.

Part Two: The Heaton System

   Heaton's article doesn't give many details of the full Heaton system, clearly because he wanted to exploit the commercial possibilities(!), and it may not be possible to decipher what he had in mind. However, the pair movement at least is easy to figure out. His idea was to designate one particular East-West pair as "Key", and to have a normal Mitchell pair movement (East-West up one table, North-South stay put) for the most part, except that every other round the key pair would stay put, sending the North-South pair they had just played to play East-West at the next higher table, while telling the arriving East-West pair to sit North-South against them. Since any East-West pair could be designated as Key, and the unusual movement could be after either the odd-numbered rounds or the even-numbered rounds, there are several choices, which lead to equivalent systems. The choice that is most convenient, given Heaton's method for numbering the pairs, is to have East-West at Table 1 be Key, and to have the unusual movement after the odd-numbered rounds. Using these choices, here is the pair progression for a 4-table Heaton (in each round, tables are separated by semicolons; NS-EW pairs are listed):

Table number: 1; 2; 3; 4
Round 1: 1-8; 2-7; 3-6; 4-5
Round 2: 5-8; 2-1; 3-7; 4-6
Round 3: 5-6; 2-8; 3-1; 4-7
Round 4: 5-7; 6-8; 3-2; 4-1
Round 5: 5-1; 6-7; 3-8; 4-2
Round 6: 5-2; 6-1; 7-8; 4-3
Round 7: 5-3; 6-2; 7-1; 4-8

   Note the pattern: each pair from 1 to 7 plays the other normal pairs in cyclic order; when a pair is scheduled to play against itself, it plays against pair 8 instead. In this way, "each-meet-each" results. The same pattern works with any number of tables.
   This pair movement places severe constraints on possible board movements, and no completely regular board movement (that is, a movement without relays in which boards from a particular table always move to a particular location, either another table or a bye-stand) is possible. To see this, note that in a regular board movement, each table gets all the boards in the same cyclic order; but in the 4-table Heaton pair movement, the set of boards played at table 4 in round 1 can never be played at table 1 (since pair 5 has already played the boards in round 1, and is stationary at table 1 from round 2 through round 7). The same objection serves to show that a fully regular board movement is impossible in a Heaton game of any size. I believe that Heaton's intention was to gather all the boards together after each round and have the director redistribute them to the appropriate tables for the next round (this is my translation of Heaton's "Boards are stacked and dealt each round to adapt." Only someone like Heaton would talk of "dealing" boards when he meant "distributing" them.)
   To return to the 4-table Heaton movement, I think that here at least we can see what Heaton had in mind, since there are only two possible board ovements that do not involve relays. Lettering the board sets A, B, C, . . . in the order they are played by the Key pair, these movements are (in each round, the board sets at each table are given in the form W-X-Y-Z; W is played at Table 1, X at Table 2, etc.):

Round 1 A-E-B-C
Round 2 B-D-C-F
Round 3 E-C-G-D
Round 4 A-D-F-E
Round 5 F-G-E-B
Round 6 G-C-F-A
Round 7 D-A-B-G

and

Round 1 A-D-G-F
Round 2 B-G-A-E
Round 3 A-C-F-B
Round 4 G-D-B-C
Round 5 D-C-E-A
Round 6 E-B-F-D
Round 7 C-F-E-G

   Aside: It should perhaps be pointed out that there is a way to modify what I believe to have been Heaton's movement to produce a semi-regular progression, in which the movement of the pairs and boards depends only on the parity of the round number. To do this, make the Key pair stationary at table 1, and adjust the rest of the game accordingly. The next table shows the result if this is done to the first 4-table Heaton movement above (in addition, I have switched arrows at table 1 after each round). Here, for each round, the table are separated by semicolons and for each table the NS number, EW number and board set are shown:

Table number: 1; 2; 3; 4
Round 1: 8-1-A; 2-7-E; 3-6-B; 4-5-C
Round 2: 5-8-B; 2-1-D; 3-7-C; 4-6-F
Round 3: 8-2-C; 3-1-G; 4-7-D; 5-6-E
Round 4: 6-8-D; 3-2-F; 4-1-E; 5-7-A
Round 5: 8-3-E; 4-2-B; 5-1-F; 6-7-G
Round 6: 7-8-F; 4-3-A; 5-2-G; 6-1-C
Round 7: 8-4-G; 5-3-D; 6-2-A; 7-1-B

   Here, the instructions for the movement after each odd-numbered round are:

   "East-West pairs move up one table; Table 1 switch directions; boards follow this pattern:
Table 4 -> Table 3
Table 3 -> Table 1
Table 2 -> bye-stand at Table 4
Table 1 -> double bye-stand at Table 4"

while the instructions after each even-numbered round are:

"North-South pairs move down one table; Table 1 switch directions; boards follow this pattern:
Table 2 -> Table 3
Table 3 -> Table 1
Table 4 -> bye-stand at Table 2
Table 1 -> double bye-stand at Table 2"
(note that this board movement is what happens if we interchange Tables 2 and 4 in the odd-numbered board movement.)

   Here are the instructions for a similarly modified 6-table Heaton movement:

After odd-numbered round:
"East-West pairs move up one table; Table 1 switch directions; boards follow this pattern:
Table 6 -> bye-stand at Table 3
Table 5 -> triple bye-stand at Table 2
Table 4 -> Table 2
Table 3 -> Table 1
Table 2 -> Table 4
Table 1 -> bye-stand at Table 5"

After even-numbered round:
"North-South pairs move down one table; Table 1 switch directions; boards follow this pattern:
Table 6 -> Table 4
Table 5 -> Table 1
Table 4 -> Table 6
Table 3 -> triple bye-stand at Table 6
Table 2 -> bye-stand at Table 5
Table 1 -> bye-stand at Table 3"

(Note that the odd-round movements and even-rounded movements are related to each other by interchanging Tables 2 and 6, and Tables 3 and 5.)

   One real problem of these Heaton movements is that they are severely, perhaps maximally, unbalanced. To adjust to a reasonable comparison among pairs, arrow switches would be needed.

ESOTERICA

This section is devoted to weird, wild and wacky material. For bridge friends, lovers of arcana, pursuers of special interests, and anyone intrigued with a particular facet of the game of bridge.