by Don Gugan, Bristol Croquet Club

Background

There has been a lot of discussion about the ease of running croquet hoops, particularly since the 'Omega' adjustable hoops were introduced a few years ago. These have great advantages in accuracy and ease of width adjustment, but were rumoured to be easier to run than ordinary hoops and, although new designs of mallet which also make the game easier were readily accepted, Omega hoops met a negative reception in some quarters. As a result of the rumours, the chairman of the CA Technical Committee, Alan Pidcock, carried out experiments in 2001 to test their 'runability', but found no significant difference from ordinary hoops. Rumours still persist, however, and further work is sometimes suggested to find whether they can be supported. The design of better experiments than those which have already failed to find differences is not trivial, and the time and effort needed to carry them out convincingly is more than is perhaps realised. The purpose of this paper is to discuss the design criteria which would be necessary.

'Runability' and Test Criteria

The success rate in running a single hoop, i.e. 'the runability', R say, is meaningless unless the conditions are defined, but an approximate average value for a typical game can be estimated for A-class players, and these are the players most concerned about the issue, since they not uncommonly complete a game by running and peeling 24 hoops without error. The best players may do this in about half of their serious matches, so that for them one can estimate the runability as (R)²⁴ = ~½, i.e. R = ~0.97 as a very approximate average value. Because of their skill, most of these 24 hoops will have been from easy positions, but a few will be less ideally placed, especially at the start of a break; the success rate for these will be lower than for the rest of the hoops, but their overall average success rate will still be close to 97%. Running hoops is easy for most A-class players, even for tightly set hoops with clearances which may be 1/32" or less, but for meaningful comparisons between hoops which distinguish real differences from random fluctuations or subjective impressions, it is desirable that the test conditions be such that the success rate is only about 50%. This implies that strongly angled hoops must be used, which is in any case necessary since if hoops do differ in runability, it can only be as a result of the details of the collision between the ball and a wire.

The results of trials subject to random variations are described by the binomial distribution of statistical analysis, and for a success rate over N trials of N/2 (i.e. a runability of 50%) then the standard deviation from the mean is given by (sqrt(N))/2. For a mean value of 50% to be reliable to within ten percent requires N = ~100 trials, but even then the random uncertainty is such it is likely that an identical hoop would be run less than 45 or more than 55 times in about one in three similar tests. There is enough evidence already to that the differences between hoops are not large, and it is important to realise that a large number of trials will needed if differences are to be established reliably.

Five outcomes are possible when attempting to run a hoop:

1. a clear run with space for an unimpeded hoop-stroke;
2. hoop run, but with an impeded stroke;
3. ball strikes wire hard, but rebounds with enough spin to roll through;
4. ball between the wires; and
5. complete failure.

Outcomes (i)-(iii) all score a point of course, but as it is probable that only (i) is relevant for the intercomparison of hoops, we then require an unambiguous criterion to define what is an 'unimpeded hoop stroke'.

The Variables Involved

The runability of a hoop depends on many different factors, some intrinsic, i.e. to do with the nature of the hoop itself, and others extrinsic, not specific to the hoop, but as there is no theory which allows a calculation of runability, it is necessary to resort to experiment to find out if differences exist. For valid comparisons of intrinsic runability it is necessary to remove the effects of all extrinsic factors. This reduces the generality of such comparisons, because even if one hoop were shown to be more runable than another under some typical game conditions, the opposite might be true under others. Nevertheless, removal of extrinsic factors is the first essential; however, one should bear in mind that if differences are established under some conditions, a further examination under other conditions might then become necessary.

The variables intrinsic to the hoop include:

• the material it is made from;
• its structure (e.g. cast, machined, or welded);
• its design (e.g. one-piece, or adjustable multi-piece, including in the case of Omegas, the tightness of the thread-packing);
• the smoothness and the nature of the surface coating of the wires;
• the size and weight of the carrots; etc.

Each of these has been suggested as a reason for a difference in runability between hoops, but, while all are plausible, there is no quantitative evidence at present about any of them. It would be interesting to obtain reliable evidence, but this is not the major concern of this paper.

The extrinsic variables include:

• player skill (e.g. aim, especially for angled hoops; control of strength and roll on ball);
• firmness of hoop in ground;
• width of hoop;
• nature of balls used in the tests (material; surface milling; roundness);
• nature of rolling surface (length and wetness of grass; freedom from slopes and smoothness of grass; development of tracks after repeated trials); etc.

These variables are examined in more detail below.

Control of the Variables

Player skill, (i) aim of stroke: A critical distance at a hoop is the clearance, i.e. the amount by which the separation of the wires exceeds the maximum diameter of any ball in use on the court, and this is often set as little as 1/32" (and sometimes even to zero) which implies that experimental tests need to have ball paths which can be repeated at the hoop to considerably better than 1/32" - something like ±1 /100" would be a good target figure. Precision at this level over a ball path of ~10" is equivalent to always roqueting less than 1" off-centre over the full length of a court, and is not realistic for a large succession of single strokes, so some way to improve the accuracy in direction of ordinary strokes must be devised. Fortunately, it is a well-known fact that in a peel stroke the croqueted ball tends to move along the line of the centres of the balls, and although there is 'pull' (i.e. deviation from the line of centres) which depends on the friction between the balls and with the ground, and on the type of stroke used, nevertheless, the aiming error with a straight stop-shot peel is probably less than one tenth of the error in the aim of a single ball stoke. This is the technique which was used by Pidcock in his experiments.

For a successful run, the edge of the croqueted ball must miss the near wire, but at any appreciable angle to the hoop the ball must strike the far wire before it can pass through, as illustrated in figure 1. Pidcock devised a wooden aiming jig which fixed closely round the near wire at ground level, with an edge against which the two balls were placed to define the line of aim; the near wire clearance, and thus the distance between the Iine of aim and the centre of the far wire, were varied by interposing thin spacers between the edge of the jig and the balls, as shown in figure 2, while the angle of aim was set to give a suitable value of the runability, and when this was obtained, the jig was fixed firmly at that angle.

Croquet balls are not perfectly uniform spheres. Typical variations in the diameter of Barlow GT balls (±1/100") could lead to aiming errors of ~±1/20" over a 10" path using the peeling jig: for a large number of trials this would probably average out, but it would be a source of random noise best avoided by choosing only balls very well matched in size, and also using them always in the same orientations. The peeling jig is simple in construction and convenient in use, and its use appears to be fully adequate to define reproducibly the line of the ball towards the hoop, even for players of only average aiming skill.

Player skill, (ii) strength and roll of shot: The peeling jig is limited to rather gentle ball speeds which are not exactly reproducible, and it allows no control of the spin imparted to the ball, both of which factors are controlled by skilful players when running difficult hoops. Suggestions are sometimes made that a ball-striking mechanism could be constructed which would allow control of the direction, strength and spin of a stroke, but this would not be a simple task and would result in a complicated and rather bulky piece of machinery hardly suitable for general use. Alternatively, balls could be rolled down an inclined ramp, as in one method of measuring the speed of a lawn, but problems arise of a consistent exit path from the ramp, and it is not clear that the direction followed by the ball would be as well defined as with the peeling jig, while control of strength and spin would be no better. Despite its limitations, the peeling jig gives a very consistent way of aiming a ball towards a hoop in order to test its runability, and since at present there is no hard evidence that hoops differ significantly in their runability under any conditions, it would be premature to design more elaborate procedures.

Firmness of the hoop in the ground: This is widely thought to be a critical variable, although there appears to be no quantitative evidence about its effects on hoop runability: such evidence would certainly be valuable. Some initial measurements to quantify the firmness of hoops have been made by Alan Pidcock and Tal Golesworthy; they applied forces up to 25N (i.e. ~5 lb. weight) at right angles to the cross bar with a calibrated spring, and measured the deflection (typically ~1/10") with a dial gauge for a succession of increasing and then decreasing forces. The results for hoops held rigidly in a vice were closely as expected for elastic behaviour of steel, but when the hoops were tested in ground typical of croquet courts, the deflections were several times greater (though still reasonably elastic), and varied considerably from position to position. Whether the deflection under this semi-static loading is truly comparable with that under the dynamic impact of a croquet ball (which takes only about a thousandth of a second) is not certain, but as even a moderately slow hoop shot can give an impulsive force of ~200N to a wire, significant movement of the hoop is to be expected, and measurements of this sort need to be made to check the equality of hoop firmness when comparing values of runability. Replacing different hoops in the same carrot holes might perhaps yield the same firmness values, but variation can arise when re-using carrot holes, as discussed later. For tests on the intrinsic behaviour of different hoops it might be desirable to use rigid mounting for them all, though whether this would be regarded as acceptable by players is questionable, and in any case there would be problems with using the peeling jig under these conditions. Tests on an artificial surface with the carrot holes set into concrete but with suitable carrot packings to give controlled amounts of yield might also be practicable. There is evidently a lot of work to be done on the subject, particularly if there is to be any prospect of compensating for different degrees of firmness by altering the width of the hoop so as to obtain a constant value of runability, as is sometimes suggested to be possible.

Width of the hoop: This can be set to ~0.005" with the Omega adjustable hoops, though since the carrots are rotated in the ground, a final firming of the hoop in the ground is necessary. Conventional hoops are not easily adjusted, and in practice firmly set hoops are usually hammered into the ground after some only approximate initial guidance; if adjustment is necessary this involves judicious scraping of the carrot holes, a hit and miss procedure which tends to leave a ragged carrot hole which is likely to have a reduced firmness unless the hoop is hammered further into the ground. One method of hoop comparison is to make runability tests on an ordinary hoop in newly made carrot holes without attempting to set the width of the hoop to a particular value, but to measure the width with vernier callipers during the tests (to ensure that it remains unchanged under the repeated impacts from the ball), and then to replace the hoop by an Omega hoop, either in the same or in new carrot holes, and to repeat the runability tests making sure that the width is adjusted to exactly the same value as before. Using the same carrot holes allows them to remain in good condition, and probably means that the firmness of the hoop in the ground is unaltered, but it has the disadvantage that rolling patterns have been made on the surface due to the repeated passage of the ball, which could affect the runability. If new carrot holes are made the question of firmness arises, and its value would need to be checked. However, it is unsatisfactory to make a series of comparisons between hoops always in the same order since, (i) their sequence should be randomised in order to avoid unsuspected systematic trends, and, (ii) also be 'blind' so that the person making the strokes does not know which hoop is being tested, and cannot have any influence on the results. Broadening the scope of the tests to include a comparison at different values of the width would also be necessary if one were attempting to some generally valid assessment of runability.

Nature of the balls used: All croquet balls are much softer than the material of the wire, and it is known that they suffer easily measurable deformation during the time of impact, though as this only lasts about a thousandth of a second and recovery is very rapid, there is no sign of it to the unaided eye. Most types of modern ball are of a uniform composition, and they are probably similar in their mechanical properties, both elastic and anelastic, nevertheless, since the running of an angled hoop depends on the details of the impact at the far wire and this in turn must depend crucially on the deformation and energy loss of the ball, it is necessary to control the sort of ball which is used. Barlow, Dawson and Sunshiny balls may all give similar results, but one should not take this for granted, and certainly, one would be surprised not to find significant differences if Jaques 'Eclipse' balls were to be used since they have a two piece construction with an outer plastic skin.

The effect of the surface milling of the ball appears to be unknown: it is sometimes thought to be responsible for 'pull' when peeling, but the effect arises because of frictional forces at the surface of the ball, and whether the milling makes a significant difference is not proven. Unmilled balls would avoid the complication and have been manufactured, but they never proved popular and they do not appear to be available for use in tests, even if players would accept their use as relevant. However, most modern balls are extremely durable, so provided that the same balls are used in the comparisons, the milling should not alter, and relative values of runability should be reliable. The materials the balls are made from have mechanical properties which may depend strongly on temperature, so that it is necessary for the weather conditions to be the same during the tests. The shape of the balls has already been mentioned in connection with the peeling jig, but it should also be remembered that good roundness is not the only requirement since for a ball to roll truly it must be symmetrical about its centre of gravity, and this needs to be confirmed e.g. by a floating test.

Nature of the rolling surface: Beside its effect of the firmness of the hoop, the nature of the rolling surface will have an important effect on the path of the ball. It needs to be very level and without local pits or bumps such as often occur on croquet lawns near to hoops, and the texture needs to be very uniform. A grass lawn is not the ideal surface for runability tests for several reasons; even very fine mowing is likely to leave a nap to the grass, surface dampness can be expected to affect the rolling friction of the ball over its path, and repeated strokes with any sort of aiming jig will tend to leave an impression on the grass, all things which may cause systematic changes of runability during a series of measurements. Unless the sequence of the testing is devised so as to eliminate such systematic variations, it would be better to carry out tests with an artificial surface similar to an indoor carpet chosen to have a coefficient of friction (i.e. a 'lawn speed') typical of grass, and, since lawn speeds vary greatly with weather and length of grass, it would also be desirable to be able to control the speed of the artificial surface and to repeat the experiments with values typical of both slow and fast lawns.

Control of the intrinsic variables: While of considerable interest in itself, it is not the concern here, except that for proper use of Omega hoops it is necessary to avoid excess yield due to any slackness of the packing of the screw thread in the carrot. One should usually need a spanner to rotate the carrot of a properly packed thread (this requires three or more layers of plumbers' tape to be wrapped around the thread), and it should be repacked every few months when the carrot becomes easy to rotate, or if the wire becomes loose.

Summary and Conclusion.

The control of all the extrinsic variables which may affect the runability of hoops is a non-trivial problem which must be addressed before one can make any certain general statement comparing the runability of different hoops. From the discussion given above it seems inevitable that a comprehensive test with hoop firmness, rolling surface, and direction of aim all tightly controlled, could only take place with a great deal of labour and under laboratory conditions. This is clearly not justified at present, if ever, but a preliminary program suitable for ordinary croquet courts and ordinary players can be envisaged to see whether more elaborate examination of specific points is necessary.

Provided that careful attention is paid to: (i) reproducible aiming so as to produce a runability of about 50%, and a sufficient number of trials for this to be statistically reliable; (ii) ensuring that hoops are set at exactly the same width, and in ground which has the same firmness; (iii) making sure that the grass at the hoop is short, smooth and free of wear or other defects; and (iv) using well-matched modern balls, then a comparison between hoops could give a preliminary indication of relative runability under the conditions of the test. If differences were to be found, then, depending on the statistical significance of the results, it could be possible to reach a provisional judgement on the issue, or more likely, it would indicate that more measurements under better controlled conditions were necessary order to reach a firm conclusion.

The experiments carried out by Alan Pidcock in 2001 conformed closely to the four conditions above; they led to the conclusion that there are no significant differences between Omega (adjustable) and conventional hoops, and did not suggest that further experiments would be desirable. Those who doubt this conclusion need to carry out further experiments which are at least as well controlled as Pidcock's, and to report them publicly so that others can comment on their results and procedure.

Received 20.vii.04