This paper reports some experiments that were performed to test how the law on double taps can be applied by a referee.
 The work was done by Max Hooper and Peter Tavender, mainly at Canberra Croquet Club, but also at Sydney Croquet Club.

 What do the laws say?:

 The laws on double taps have changed over the years, and referees have to adjust.
 Originally, the law was probably aimed at the player who took two swipes at the ball. Thus in 1870 the law said that it was a foul "to strike a ball twice in the same stroke". This sort of play is now hardly ever seen, and in time the law was taken to refer to several contacts in the one swing.
 An important change concerns the word "audible". For many years, a double hit was a fault only if it was audible. In 1961, the wording was changed to "audible or distinct", perhaps because if the ball is moving slowly the second hit may be easier to see than to hear.
 In 2000, these words were deleted. This was because of increasing knowledge of the relevant physics. If two similar sounds occur within 30 or 40 milliseconds of each other, we hear only the first sound. (This effect is known as the Haas effect, and is well known to sound engineers.) This creates a problem for the referee. If he can't hear two sounds, how can he be sure a double hit occurred? That is what these experiments are about.
 Not all double hits are faults. The exceptions are listed in Law 28(d). The commonest one is when a roquet is made.
 On the whole, the law against double hits has been interpreted more strictly as time went by, but not in croquet strokes. In 2001, the O.R.L.C. commentary suggested that referees should ignore double taps in croquet strokes. In 2008, the law changed so that a double hit is not a fault in a croquet stroke unless the mallet is seen to hit the ball more than once. For these reasons, we did not study two-ball strokes.
 There is no such forgiveness with single ball strokes. This is because most double hits in single ball strokes occur when the previous stroke has been messed up and the striker is trying to play his way out of trouble
 Some referees do not take these events seriously. If the striker's ball hits another ball, there is usually a roquet, and this usually excuses a double hit. But this is not so if the target ball is dead, because a dead ball cannot be roqueted. Players do not often aim at a dead ball, but there are some cases where such a stroke is of real concern in association croquet:
 1. In a cannon stroke when the third ball is dead.
 2. After a failed Irish peel where the croqueted ball has stuck in the hoop. If the striker tries to send both balls through the hoop in a swinging stroke, he is likely to commit a fault.
 3. If a break cannot be continued, the striker may wish to send the striker's ball and a dead ball to a distant boundary in a scatter shot. It is hard to do that legally.
Even if a roquet has occurred, there are some rare cases where a double hit is a fault. This can happen if the striker's ball makes a roquet, hits another object and then contacts the mallet again.

 The tests we made:

 Four different types of stroke were investigated:
 1. Where the striker's ball hits a dead ball full on,
 2. Where the striker's ball hits a dead ball at an angle, with the intention of sending both balls in a different direction.
 3. Where the striker's ball tries to run an angled hoop from a short distance, and
 4. Where a jump stroke or hammer stroke is played.
 In all cases, a piece of carbon paper and a piece of white paper were fastened to the end face of the mallet. A full description of the procedure is given by Ian Plummer at http://www.oxfordcroquet.com / under the heading Simple Impact Measurements. The equipment is cheap and easy to get, and anyone can confirm the results. Some examples of the marks made on carbon paper are shown on the diagrams below
 In most cases, the strokes were also recorded on a video camera. This recorded the sounds made as well as the movement of the balls.
 We relied on the carbon paper test to tell whether a double tap had occurred in each stroke. We then compared what a referee could have seen and heard with the result of the carbon paper test.

  The pictures show the result of a single legal hit and several cases of double hits. Note that the second hit always leaves a smaller mark than the first one (because some kinetic energy is lost at each hit). So the mark of the second hit can always be seen, even if it overlaps the first hit.
  An umpire could hardly ask the striker to fasten these materials on to his mallet before a questionable stroke (although it would probably make no difference to his stroke), but we can do experiments where we compare the marks made by the carbon paper with what we see and hear.
 Other scientific tests are possible. Useful information can be gained from a video camera if it is played back one frame at a time. Unfortunately, standard cameras show only 25 frames a second, and this is not fast enough to show all faults, although it shows many of them. High speed cameras show more, but they are very expensive.
 Another series of tests was made by Stan Hall with a storage oscilloscope - also an expensive item - with results that are published on at http://www.oxfordcroquet.com/tech/hall/index.asp           

 How reliable are carbon paper tests?  

  Since we relied on these tests to decide if a double hit had really occurred, it is fair to ask if we were justified in doing so. This raises several questions:
 1. Is a separate mark made on the paper each time the ball contacts the mallet and paper?
 2. Does the ball ever hit a part of the mallet that is not covered by carbon paper?
 3. Can the mark made by a ball be distinguished from other marks on the paper?
 The answer to the first question is yes. If you hold the ball in your hand and gently tap it on the mallet over a site covered by the carbon paper, you get a distinct mark of a ball. It is smaller than the mark made by a ball in a normal stroke, but it is unmistakable.
 The answer to the second question is less satisfactory. We seldom managed to cover all the beveled surfaces next to the end face. A number of second hits were seen on the lower edge of the end face, and it is possible that some were missed. If any second hits were on the side of the mallet head, they would not have been seen at all.
 Various smudges occurred when the carbon paper was handled, but they were not hard to distinguish from ball marks. A little practice was needed to identify some ball marks at the edge of end faces.
 Some other writers have published carbon paper tests that show streaks where the ball has rolled or slid along the surface of the mallet. Nearly all such images occurred in roll strokes. We did not see that pattern, but we did not test any roll strokes.
 In summary, if two ball marks were visible on a paper, we have no doubt that a double hit occurred. If only one mark was seen, we could not exclude a second hit on a bevel or side of the mallet head. However if we saw a single impact when the same stroke was played several times we concluded that no fault had occurred. To put it another way, it is possible that we missed some double hits, but we believe we did not make any mistake when we claimed that a fault had occurred.  

 When the striker's ball hits a nearby dead ball full on.     

   This stroke is more common in golf croquet, but the real reason for dealing with this case first is that it is the easiest one to analyse.
 Some people have claimed that the referee can tell whether a stroke is legal or a fault by noting the distance between the striker's ball and target ball. We found this is not so. The distance between the two balls is one factor, but much depends on the type of stroke.
 Using a swinging stroke, a double hit could be produced even if the balls were 20 cm apart. When they were 25 cm apart, a double hit was avoided only because the mallet head was lifted, so that it passed above the striker's ball.
 A quite different result occurred with other types of stroke. If the mallet was grounded as for some types of stop shots, we found a legal stroke with a gap of 5 cm, but a double tap at 3 cm. Perhaps a more skillful striker could play this stroke fairly at shorter distances.
 After a failed Irish peel, a jump shot appears to be the only legal way to continue the break. These strokes looked legal to the unaided eye, because the mallet routinely hit the ground, but in many cases the carbon paper tests showed double taps. They are discussed later.
        

 What do we expect in theory with this type of stroke?

  In a fair stroke, when the striker's ball hits the dead ball, most of the momentum of the striker's ball is passed to the other ball. The striker's ball has so little momentum left that it moves slowly or even stops dead. If the mallet follows through and hits it a second time, the mallet delivers more momentum to the ball. So we expected to distinguish fair strokes from faults by the distance travelled by the striker's ball after impact. This proved a reliable test so long as no top spin was involved.
 If the mallet handle is tilted forward, the striker's ball acquires top spin. Because of this, it will roll forward more after it hits the dead ball. In addition, the striker's ball may jump, in which case it will deliver less of its momentum to the dead ball. In the extreme case, it will jump over the dead ball without touching it, and fail to deliver any momentum to it.
 Even in a swinging stroke, the striker's ball will gather top spin if there is a big enough gap between the balls, but this is not significant with just a 20 cm gap.     

  Place the red ball 25 cm (10 inches) from the black ball, and use the black ball as the striker's ball. Play a swinging stroke so that the black ball hits the red full on. In most cases, the black ball will deliver nearly all its momentum to the red ball. So after hitting the red, the black ball will travel forward slowly and for a short distance. The red ball will travel forward at something like 80% of the original speed of the black ball.
 Now repeat this test with the two balls 3 cm apart. Both balls will move forward at a similar speed, although the target ball will usually go faster and farther.
 Why the difference? In the second case the mallet hit the striker's ball twice. This has been confirmed with carbon paper tests and slow motion photography. The sequence is as follows: the mallet hits the black ball; the black ball hits the red; the black ball stops dead (or travels very slowly while the red speeds away; the mallet follows through and hits the black ball a second time.

 This picture shows four stages of one stroke.
 The mallet hits the red ball, the red hits the black and comes to a stop, the mallet follows through and hits the red a second time, causing it move on further.

What should the referee look for with this type of shot?

    It depends on the way the stroke is played. We found the following rules reliable with a swinging stroke or a stop shot action:
 1. If the striker's ball travels only a small distance after impact, the stroke is fair.
 If the striker's ball travelled a long way after impact, the stroke was a fault, provided the mallet handle was not tilted forward.
 This test does not work if the mallet was tilted forwards. We found cases where the striker's ball travelled further because of top spin, and the carbon paper showed only one hit.
 2. Sometimes the mallet was grounded before it travelled far enough to hit the striker's ball a second time. If this happened with a stop shot action, the stroke was fair. If it was a jump shot, the carbon paper test showed a fair stroke if the tilt was small enough, say less than about 40°. (This is discussed later.)         

 What if the balls start off very close together?   

  We tried an experiment with the two balls placed 1 mm apart. A firm, swinging stroke was played. As expected, the carbon paper showed no sign of a double tap.
 This may need some explaining. The mallet and ball stay in contact for just under a thousandth of a second. (See articles at the Oxford web site, for example by Hall.) The ball leaves the mallet with a speed of several meters per second at least. A little calculation shows that the two balls come into contact before the striker's ball separates from the mallet. So no double hit occurs.
 But has there been a breach of Law 28(a)(7)? Yes. A fault has been committed because the striker maintained contact between the mallet and the striker's ball after the striker's ball hit another ball. (This is not what we used to say, but it follows a recent ruling of the International Laws Committee.)

 When the striker's ball hits the target ball at an angle:

  The result depends on the angle, but what we learned about full-on hits is still useful, at least if the impact is not too far from full on. The adjacent pictures show the type of result that occurs with a fair stroke and a fault if the aim is at half ball. In a fair stroke, the striker's ball goes off well to the side away from the target ball, but it does not go forwards much. If there is a double hit, the striker's ball still goes to one side, but it travels much further forwards.
 This will not surprise anyone who is used to the relevant physics. The component of momentum of the striker's ball in the forward direction is the same, no matter what the angle of impact was; it is only the component in the direction at right angles that is different.
 However the referee who watches the stroke may have trouble watching the different directions. In practice, it is much easier to look at the angle between the path of the striker's ball after impact and the path of the target ball. If it is greater than about 40 degrees, the stroke is fair. If it is less, the stroke is a fault unless there is reason to expect a lot of top spin. Close to 40 degrees was a borderline area where we sometimes found two hits on the carbon paper and sometimes only one.
 In these angled shots, the distance between the balls could be important, as the diagrams show. After the balls came in contact, the striker's ball moved to one side, so that it might miss the mallet on the follow through unless the balls started off close together.

      We see four stages of a stroke where the red hits the black ball at an angle. After the impact, the red moves to one side, but in this case the mallet still hits the red a second time as it follows through.
  If the balls had started off a bit further apart, or if the mallet had moved slower, the red would have had time to get out of the way of the mallet.

 Does the sound help?  

 Not a bit. It could be quite misleading.
 This answer will surprise a lot of readers, and goes against what a lot of teachers say. I can only say that sounds were recorded with a video camera. They were played back later and checked against the results of carbon paper tests. We found that the sounds varied according to the distance between the two balls, but did not tell us if the stroke was fair. Of course if the target ball was hit full on from 3 cm, we would hear the typical sound of a hit from that distance and could predict that it must be a fault, but the sound gave no extra information.
 This became obvious in angled strokes. Some players did not always hit at the angle they intended. The sounds we heard were consistent with the distance between the balls, but they did not predict the carbon paper findings, whereas the angle between the paths of the balls did.
 If you doubt this, we can only urge you to try the test for yourselves.
 We considered whether the age of the referee made a difference, because older referees lose the ability to hear high pitches. The sounds were played back to younger referees who were no better at picking which sounds corresponded to faults.

Attempting an angled hoop from a short distance:

 We were familiar with the work that John Hanscomb did in 1998. In fact, one of us helped with it. You will find a description at http://croquet-nsw.org/animation/. After getting to this page, click on "The Double Tap".
 John built a device that allowed a mallet to be swung in exactly the same direction as often as desired. It could be set up so that the mallet followed through freely, or else the mallet could be restrained so that it was arrested just after the impact.
 If the aim was just right, the ball would pass quickly through the hoop. Even if the mallet was allowed to follow through, it never caught up with the ball.
 If the aim was shifted a little toward the far upright, the result was very different. If the mallet was restrained from following through, the ball would either stick in the hoop or stop just in front of it. If the mallet was allowed to follow through, it would hit the ball a second time and often send it through the hoop. This was easily detected with a video camera when played back one frame at a time. None of us could detect the fault with our unaided eyes or ears.
 John's work was done without using carbon paper. We repeated it with a few strokes with human players - we did not have a machine - but using both carbon paper and a video camera. We confirmed his results in the sense that when a double hit was seen on the video camera it was confirmed with carbon paper, and vice versa. In none of the strokes could two hits be seen with unaided eyes. One of our helpers judged the strokes by the sounds and was wrong more often than right. The sound as recorded by the video camera might contain two noises in a fair stroke and one in a fault.
 A special type of sound ought to be mentioned, although it did not occur in this series of  tests. If a mallet strikes a ball that is in contact with a hoop, the sound is more discordant than if a ball alone strikes a hoop. In theory, one would expect this sound to occur only if the ball was in contact with the hoop and mallet at the same time, in other words if a breach of Law 28(a)(9) occurred. If this proves to be true, a fault can be declared whenever this sound is heard.
 We should also mention a different type of double tap. We have described what happens if the aim is deviated a bit towards the far upright. We did not study the case where the aim is deviated towards the near upright. In this case, the ball may hit the near upright. If it does and the mallet follows through, a double hit may occur. This sort of double hit is easier to see, so long as the stroke is not played too hard.

What should the referee do about angled hoops from a short distance?

   The main thing is to realise your limitations. If you think you can see a crush or a double hit in a 40 millisecond period, you are being unfair to the adversary who relies on you to do justice.
 Most strikers do their best to avoid following through. Some arrest the mallet with an action like a stop-shot. Others use an action like a jump shot. Such players should cause the referee no trouble. This doesn't mean they never fault, but that any fault is easy to see. Many strikers play this sort of stroke gently, and this allows a referee to pick some faults but probably not all.
 The problem lies with the striker who plays firmly and who also follows through. In general, the harder he hits, the harder it is for the referee.
 If the ball is seen to pause in the jaws of the hoop while the mallet follows through, this is a good test of a fault, but it is hard to be sure of unless the stroke is played gently.
 John Riches has reported another test. You note whether the ball hits the upright almost full on or decidedly to one side. This test needs more study, and it may be the topic of a separate report later. Even if it proves successful, there will always be uncertain or borderline cases.
 Other so-called tests have been published, but they have no basis in fact.
 So what do you do if you are still in doubt?
   1. If a referee is in doubt, he is usually expected to go by the balance of probability, as in Regulation 7. (There are still people who talk of giving the striker the benefit of any doubt, but there is nothing in the laws or regulations to support this, and it is unfair to the adversary.)
   2. A video camera could be used like the third umpire in cricket. This is known to work with strokes played by John Hanscomb's machine, but it has not been tested with really hard hitting. As far as we know, it has not been used in a real game, but our experience gives some ideas on the practicalities.
 A camera is not intrusive or hard to use. However unless the operator is well practiced, he will sometimes miss the shot because of some technical error. So a human referee to watch the shot is still needed.
 The older types of camera - analogue cameras - require extra equipment to play back the stroke, and this can be cumbersome and time consuming. Some digital cameras do not allow play back in full sunlight.
 There is also the risk of theft. A good camera is expensive.
 I would not advise using a camera unless one of the competitors had a reputation for playing dubious strokes, and then only if a tested digital camera was available.   

 There is also an ethical problem. As we said, most serious players play these strokes with as little follow through as possible, to make the referee's task easier. A few players hit these strokes hard and with a full follow through, and may get away with a fault because the referee can't see clearly. Should something be done about this?    

Unusual ways of playing an awkward hoop:

 Some players try an unusual action in the hope of avoiding a fault. Some actions are variants of jump shots and face the same problems. Another action is shown in the diagram.

 We have tested this action with carbon paper and found it fair. (However some players have trouble with aiming. If the aim is poor, the ball may bounce off the hoop back on to the mallet. This causes no problem for the referee, but it is not what the player wants.) 

Jump and hammer strokes

 These strokes are played with the mallet handle tilted forward. The striker deliberately hits downwards as well as forwards. Jump shots are usually played with less tilt, up to 40°, while hammer shots are played with more tilt. Videos taken by Bob Kroeger at 1000 frames per second show a sharp difference between these two types of stroke. (A copy of Bob's video can be obtained from him at bobkroeger@aol.com for US$15 plus postage. It comes in NTSC format which has to be converted to PAL format for Australian TVs, and only some VCRs can do this. In the process, some detail is lost.)
 In hammer shots, Bob's tape shows the ball being compressed between the mallet and the ground. Our carbon paper tests showed double hits consistently. In jump shots, Bob's tape shows the ball staying next to the mallet for a much shorter time, because it had room to move forwards. Carbon paper tests showed that some but not all jump shots were fair.
 Further study showed that a jump shot was likely to cause a double tap if the angle was much over 35°. It was more likely to be fair if the ball actually jumped, because the mallet would pass under the ball during follow through. The mallet was much more likely to hit the ball a second time if the ball failed to jump. This test could easily be applied by the referee.
 The distance between the two balls was significant. With a gap of 5 or 10 mm, the striker's ball could not jump because the target ball got in the way, and all such cases ended in a double tap. If there is a fair way of scoring the hoop from such a position, we did not find it. We found that a fair stroke was more likely to result from a 50 mm gap than 30 mm, but this seemed to depend on whether the jump was successful.     

 Rescue shots after a failed Irish peel

 If an Irish peel goes wrong, the croqueted ball may stick in the hoop while the striker's ball comes to rest short of the croqueted ball. The striker still has a continuation stroke, but the turn will end unless he sends his ball through the hoop. If he can send the other ball through as well, then so much the better.
 But the second ball is now dead. So if he plays a swinging stroke, he has the same problem as we discussed earlier. If the striker's ball travels far enough to go through the hoop, this is very strong evidence of a double hit. If he avoids a double hit by playing with a stop shot action, the stroke may be fair and the target ball may run the hoop, but the striker's ball will travel a much shorter distance and fail to score the hoop, and the turn will end. Some players and some referees do not realise this, and this has caused trouble even at MacRobertson Shield level.
 The striker's only hope is to play a jump stroke, when he meets the problems described in the last section.