How Can a Croquet Referee
Detect a Double-hit?
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
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
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
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
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.
Normal single hit
Another double hit
Hit with the edge of the end face of the mallet. This is
a fault if the shot is hampered
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
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.
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
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
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
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.
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.)
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.)
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.
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.
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
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
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.
If the angle between the paths
of the two balls is 45 degrees or more, the stroke is fair.
If the angle between the paths
of the two balls is less than 40 degrees, a double hit is likely, but
this test is not reliable if the mallet was tilted forwards.