As Ye Have Done It Unto One of the Least (Gazette Singles Book 2)

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In order to give an element of randomness to who plays whom, the players will seat themselves anywhere they like in the two rows and, when they are all seated, then I will place chess boards between neighbouring pairs of players. Of course I want to ensure that when I do this there are no players left isolated on their own as they are not permitted to move once seated. So given two rows of n players 2n in total to play n games at the same time, how many ways are there for me to place the n chess boards between neighbouring pairs of players so the everyone is playing a game?

Here are the 3 ways to orient 6 players sitting on opposite sides of 3 chessboards: It is clearly always possible to place the boards no matter how long the rows are can put all the boards between a player in the top row and the corresponding player in the bottom row. But how many was are there altogether if there for 4 games 8 players? When all the points are so paired, it is called a Perfect Matching. The answer is again the Fibonacci numbers.

Can you explain why? In mathematics, this is a problem of finding all paths on part of a triangular network the nodes have two or three edges meeting at them. On an earlier page we met Fibonacci's rabbits. But Fibonacci neglected to say how those rabbits found their food and recently I found out!

I had planted out two long rows of lettuces but found fairly soon that overnight a rabbit had discovered my new juicy salad plants and had eaten its way through some of them, leaving a tell-tale trail of paw-prints! After replacing the eaten ones it returned and the same thing happened. Watching the trails over several days, I noticed my nocturnal visitor seemed to follow the same pattern in its nightly rampages through my lettuce patch: So, if you are one of Fibonacci's filching and ravenous rabbits, paws for thought and let us "lettuce" see if you can find the answer to this question: List your paths by the number of lettuces that were eaten on that night.

For instance, here are the three paths through my lettuce patch where the rabbit has eaten 3 of my lettuces: How many paths are there if it eats 4 lettuce? How many for 5? In mathematics, this is the problem of finding a certain kind of path in a lattice a rectangular collection of lines around squares and hence the pun on lettuce. Directed lines from 0,0 to n,0 Reuben C. Drake, Fibonacci Quarterly vol 8. Fibonacci Quarterly , vol 8. What about the number of sequences of n coin tosses that end with three Heads together? Does this have any relationship to the Fibonacci numbers? You are putting stamps on a parcel to make a value of 10 pence but all you have are some 1p and 2p stamps.

The stamps are placed in a single row at the top of the parcel. What if we are interested in collections of coins rather than sequences? Mathematicians call a collection that sums to n a partition of n. They have many applications in mathematics. Can you find a simple link between answers to the Change puzzle and your answers to the Stepping Stones puzzle?

In mathematics, this is a problem of finding the number of combinations of numbers with the same sum. In this puzzle, it is combinations that we are counting, but we only allow the use of 1s and 2s and no higher numbers in the sums, so they are restricted combinations. We could just phone everyone ourselves, so 14 people to share the news with would take 14 separate calls. Suppose each call takes just 1 minute, then we will be on the phone at least 14 minutes if everyone answers their phone immediately. Can we do better than this? We could use the speakers on the phone - the "hands free" facility which puts the sound out on a speaker rather than through the handset so that others in the room can hear the call too.

For the sake of a puzzle, let's suppose that 2 people hear each call. That would halve the number of calls I need to make. My 14 calls now reduces to 7. Can we do better still? Well, we could ask each person who receives a call to not only put the call through the loudspeakers but also to do some phoning too.

So if two people hear the message, they could each phone two others and pass it on in the same way and so on. Here's what it looks like if I have 14 people to phone in this system as the calls "cascade". In the first minute, my first call is heard by A and B.

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Yes - if all the people got together in one room, it would only take one minute! So let's assume that I cannot get everyone together and I have to use the phone. Now here is your puzzle. The phones in my company are rather old and do not have an external speaker and no "conference call" facility - only one person can hear each call. So I decide that I will phone only two people using two separate calls. I shall give them the news and then ask that they do the same and phone just two more people only. What is the shortest time that the news can pass to 14 people? Draw the cascade tree of telephone calls, or the telephone tree for this problem.

It begins like this: How does the tree continue? What is the maximum number of people in the office that could hear the news within N minutes using this method? Why is the answer related to the Fibonacci numbers? This is a great book! Suppose we restrict how we may move permute each object to either fix it, leaving it in the same position or flip it with a neighbour - two items next to each other swop places they cannot now be moved again.

However, not all permutations are made of just these two kinds of transformation. Here are some examples of both kinds permutations on 4 objects: ABCD Nothing has moved - all 4 items are fix ed! In the fourth A has moved more than 1 place and in the last two C has moved 2 places. How many fix-or-flip permutations are there for 4 objects?

So a natural question is what happens if we have double glazing which has two sheets of glass separated by an air gap, that is, 4 reflecting surfaces? Hang on a minute Let's look at that first! For three surfaces for example two sheets of glass resting on each other what happens depends on whether we are looking through both sheets of glass the rays of light come in on one side of the window but exit from the other or whether we are looking at our own reflection from the sheets the rays of light enter and leave from the same side of the window. We can ignore the reflection off the top surface - the light bounces off and we get one reflection.

The other cases are the interesting ones - where all the reflections are internal reflections. It is usually far better to use the continuation shot to get a perfect rush on the other ball near your hoop. When you take the bisque you then start with a simple roquet.

This will give you a much easier approach shot to the hoop. How do you decide which shot to go for? You do it by referring to your picture of yourself and deciding which shot stands the best chance of succeeding. Are you good at positioning? Can you rush accurately? Can you do accurate short croquet shots?

Which are you best at? Decide and make your shots accordingly if you are bad at all of them then it's back to the practice ground! Finally, you may have done too well and got within an inch or two of the hoop. The same points apply as above, with the added complication that you now have a hampered shot. You must ask a referee to watch this; see the chapter on how to deal with hampered shots. The position of this ball is determined by the croquet stroke which put your ball in front of the hoop.

Getting in front of the hoop is obviously the most important objective. This does not mean that the position of the other ball is unimportant. It is, because there will be two things that you will want to do when making this roquet. The roquet should be an easy one. You will want to put the croqueted ball a convenient distance past the go through a hoop and slightly to one side. What is a convenient distance?

It depends very much on how hard you normally go through a hoop. Most high bisquers tend to over estimate their hoop control and do not send the croqueted ball far enough. I suggest that you try for about six feet past and see how that works, modifying the distance with experience. Slightly to one side? About 2 feet is a sensible distance. You will want to rush to a good position. This position should make creation of a new pioneer and the approach to the pivot easy. In general terms this means that you want a rush towards the peg, where the pivot ball should be.

The following table shows the position that you should try to attain for each hoop. Position of the croqueted ball just prior to the running of each hoop assumes starting position of Figure 2. Shows the side of the hoop that you should send the croqueted ball. This is as seen when looking at the playing side of the hoop. Shows which will be the pioneer hoop when you have run this one. Thus, when you have run hoop 1, the pioneer will be hoop 3.

Notes Assumes that six feet is your distance - if not adjust accordingly. Assumes that the pivot ball is in the middle, if not adjust accordingly. If you don't have a pivot at all, then you haven't got a four-ball! I mentioned earlier that the rush should allow you to create the pioneer easily. What do I mean by an easy shot? If you watch top players you will rarely find them making complicated rolls and splits. The only times that they will do so is to get out of trouble or when doing complex breaks involving peels.

By far the easiest croquet stroke to play is the straight drive. This is where your ball and the croqueted ball both go in the same direction, and you take your shot from a normal stance with a normal swing. In this type of shot the croqueted ball goes about 4 times the distance of your ball. So you rush to a position which is on a line from your pioneer hoop to the pivot ball and a little short. See the diagram below for an illustration. The balls do not have to be dead in line. In fact it is a slight nuisance if they are!

As long as they are close to a line, the shot will be a simple one. As an example, Figure 2. There is not a lot to say here. Remember not to rush it too far from the middle. If it is off centre, place your ball so that you rush back towards the middle of the lawn again. A lot of people worry about take-offs. There is no need to. There are several ways to line up a take-off, this is mine.

Look down at the two balls in contact your ball on the left. Then imagine that you have a pencil and draw two circles round the balls. Where the two circles meet and touch, you can see two arrowheads. The one going away from you points to where your ball will go in a take-off shot.

Remember that you must move or shake the croqueted ball. So make that arrowhead point to where you wish your ball to go. Then strike just to the right of this direction. If your ball is on the right hand side, you must of course hit slightly to the left. Remember that you are virtually only hitting one ball, so don't over hit it. With care you can make these shots very accurate and will be able to good position for the last shot in this sequence. There will be only two reasons why you will want to stop. Either you don't want to go further or you cannot because you have reached the peg.

You will want to stop before the peg when you do not want to take the risk of your ball being pegged out. How far you go will depend on the situation. Here is a rough guide against differing opponents. This assumes that your other ball the backward ball has more than 6 hoops to go. ALSO you do not have enough bisques left to be reasonably certain of getting the other ball round.

If you do have enough bisques left - go to peg always. With less than 6 hoops to go with the backward ball, decide according to the merits of the situation. All of the above seems to leave a risk of being pegged out. Yes it does, but it is a calculated risk and you have to take it sometimes. If you creep up hoop by hoop to the peg, you will give your opponent too many chances of a hit in. In any case as you will see in a later chapter, being pegged out is not the end of the world. OK, so you have decided where you want to stop before you have got there!

There are hundreds of varieties of finish. I am only going to talk about one because more would confuse, and this one is simple and effective. The leave is this, illustrated in Figure 2. Put one of your opponent's balls at the first hoop that you want with your backward ball. Put the other either at the next hoop, or if that is too close to another ball, the next hoop but one. Put your own balls as far away as possible from the opponent's. Leave a rush for the ball which is not at your hoop but make sure that you do not leave a double.

If you cannot have a rush without leaving a double, forget the rush. For picking up the break afterwards assuming your opponent misses! There are so many things that could go wrong that a list would be enormous. I have simplified them into two categories. Here things have gone so badly wrong that to recover, even using bisques, will not be worth it. It is always worth recognising fatal errors for what they are, because you may lose the innings but save the game. This is where you miss a 10 ft roquet and go 10 ft past, take a bisque and miss again, etc.

I have seen 4 bisques used in this way, and still no roquet made. If you miss after the first bisque, give up unless you are very, very close. The four-ball has crumbled to nothing and you only have a few bisques left. Use one bisque for a tidy finish and play the game tactic appropriate to only a few bisques Section b later. You have bounced off twice from the same hoop, or you have failed twice to approach it properly. Give up and close your turn fully.

Recovery from a mistake will involve the taking of a bisque or more. Remember that a bisque is an extra turn. This means that you can roquet all of the balls again. This is a very important point. Suppose that you have just failed to get in front of your hoop with a continuation shot left, as described earlier. If you have a perfect break, then all that you need to do is use a bisque to get good position next time. Suppose though that your pivot ball has gone 10 yards away from the middle.

Why not use the continuation shot to get near to the pivot ball. Then take a bisque and rush it to the middle and take, off to the ball by your hoop. That way you will have restored the break and still have made the hoop. This all sounds too perfect - and it is! The problem is of course that you are making five strokes before running the hoop instead of three. So weigh up the situation. What are the benefits? How difficult is what you are proposing?

What will happen if it goes wrong? Weigh up, make a decision and go for it. Do not start thinking about what you might have done; concentrate on executing your chosen option correctly. As I have said, the combinations of positions are endless. Here however are a few useful do's and don'ts.

I am still assuming that you have many bisques. If you win the toss, you should go in second. The reason for this is that you intend to get going as soon as possible. By going second you get, on the fourth turn, the first opportunity when there are four-balls on the lawn. Also it prevents the very good player from having a hit in chance on fourth turn and going round.

There are many start variations and all have their pros and cons. Some guidance is needed, so here are a few ideas. Do not treat them as anything more than a first introduction on how to start. The descriptions first ball, second ball, etc. I shall begin with the situation where you have lost the toss and your opponent has put you in first. First and third balls are yours - send the first ball near to the peg. Normally the opponent will then go for option i, ii, or iii below see next paragraph if this does not happen. Whatever they do, put your third ball in 4th corner, or if occupied, the 2nd.

Unless remarkably brave or foolhardy your opponent will not shoot at your ball in the middle. If you can make hoops without difficulty, do so. Otherwise put your opponent's ball in front of hoop 2 and lay up near 2nd or 4th corner. Remember that you may start from the A- or the B-baulk, whichever is most advantageous to you. Fourth ball is your opponents - It is virtually certain that your opponent will shoot. The shot will be at a ball which if missed, leaves the fourth ball in or near a corner. Whatever the situation, you will eventually get to play turn number 5. Your objective should be to get to peg and only have used half of your bisques, or less.

This is aggressive croquet - the best kind - and there are some risks. The advantages both tactical and psychological are enormous though, if you get it right. There are many combinations possible at this stage. As an example see Figure 2. Assume that your opponent put one ball into Corner 2 and shot and missed your ball in the middle with the other.

The ball then landed in the B-baulk near Corner 3. So you have your balls in the middle and in Corner 4, while your opponent is in Corners 2 and 3. This may seem an impossible situation but it is not! Shoot at Corner 3 not the ball with your ball that is in Corner 4. Take a bisque and rush the ball at Corner 3 a few yards towards hoop 6. You should give yourself a clear view of hoop 1.

Croquet the ball to hoop landing with your ball near the pivot at the middle. Roquet the pivot ball and take-off to the ball in Corner 2. Roquet that and send it to hoop 2 with a normal drive shot.

Take position near the ball at hoop 2 with the continuation stroke. Take a second bisque and simply take-off via the pivot ball to your ball at hoop 1. You have a four-ball break in two bisques and no difficult shots except the first, which requires some care. When should you hold back and play a waiting game? If you are really tense, perhaps because you have just lost a previous game due to bad play on your part, then wait.

If conditions bad weather or lawn make playing a lottery, then use bisques sparingly. Otherwise go for it! A short tice will be laid.

Put your second ball into 4th or 2nd corner whichever seems safest 4th is normally best. You may be able to do this AND have a shot at the tice. If you can, and hit, put both balls into the middle, but don't leave a double. Fourth ball is yours - the objective should again be to get to peg using no more than half of your bisques. There is NO situation possible where you should not be able to approach hoop 1 after three bisques. Here is the most difficult. A ball in each of corners 2 and 4, and one half-way down the east or west boundaries.

As an example, consider Figures 2. Your opponent went first and laid a longish tice. You put your ball to 4 and your opponent shot at and missed the tice landing in Corner 2. Shoot at the ball in Corner 2 from B-baulk. If you hit, carry on; otherwise take a bisque and make the roquet. Send that ball to the middle, putting your own ball near the original tice. Send it just in front of hoop 1 with a straight drive. Shoot at your ball in Corner 4. If you miss, take a bisque and roquet your ball in Corner 4.

With a strong half roll, send the forward ball to hoop 2 and your own ball to the middle. Roquet the ball in the middle, take-off to hoop 1 and away you go! If you should happen to hit the ball in Corner 4, straight drive the forward ball to hoop 2. Then trickle to the middle and take the second bisque. This will put your ball near the middle. Because you have taken a bisque, you can roquet the middle ball again.

Do so and take-off to your ball in Corner 4. Roquet it and carry on as before. If something goes wrong - see " Oh Dear! If you do it right, and your opponent does not hit in, the second ball round is a virtual carbon copy of the first. The exception is the peg out - see the Chapter " The End Game ".

Peeling is too difficult at your level to be worth it. The ball is of more use to you still on the lawn. If your opponent does hit in on third and fifth or fourth and sixth turns and finishes, then your opponent deserves the victory. If you have 10 or more bisques then you will be playing someone who is good by your standards. It is probable, although not certain, that once your bisques have gone, you will lose unless you play very carefully indeed. Of course nothing is certain in croquet! Aim to get both balls to peg and still have bisques left.

Of course, once you can do that, it means that you are better than your handicap. You will consequently lose some of your bisques, life is so unfair! However, that is the object of the exercise. You want to become good enough to play people on level terms and still beat them. If you have a reasonable handful of bisques 5 or more take one ball round with them. You may get a good break and still have some left. It will however pay you to wait a while for a suitable opportunity to get going with the four-ball break.

You cannot afford to use three or four bisques to get going as in a. When you only have one or two bisques to start with, or remaining, use them at the appropriate time s. By this I mean be ever watchful for the situation where your turn plus one bisque will give you a perfect break.

Your opponent has a four-ball but blobs hoop 4 when using your blue leaving you an easy roquet. Using blue make the roquet. Send that ball as a pioneer to 2-back, landing near the ball at 5. Roquet that ball, and put it to the middle. Attempt to get a rush to 1-back using the ball already in the middle your opponent's old pivot ball. If you get it right, well done. If you do not, roll up to 1-back, take a bisque and you have a four-ball. Do not hang on to bisques too long just as a reserve. Bisques used in this way are rarely spent well. However the effect of a couple of bisques still standing is sometimes devastating to an otherwise aggressive player.

They will in any case force most players to be more careful. So if you think that they are of more value standing than spent, leave them as pressure on your opponent. Not too long though or you may find yourself beaten with the bisques still there going to bed with bisques. How embarrassing that will be! I have said little about defensive bisques. These are used sparingly and only to get out of really serious trouble. You might wish to prevent a perfect four-ball break for your opponent. You will want to save a certain loss of the game. Do use a defensive bisque if appropriate. I do not subscribe to the theory that bisques are only used for attack.

If you had some and have used them, then you must now play very carefully. I do not mean play "Aunt Emma". If you had no bisques to start with, then you are playing someone who has the same handicap as yourself. Unless you or they are wrongly handicapped your skill levels are similar so you will have to concentrate just that bit harder. Do not fall into the trap of relying on your opponent to make the mistakes. Remember that you will be making them as well so advantages are cancelled out. Essentially you should behave as in i. Remember the pictures of yourself and your opponent, and build them up as the game progresses.

Base your play on these pictures because they represent the true situation. Play to your real ability, not what you would be! By this I mean don't try for complicated break building which is beyond your capability. Just because you can do beautiful four-ball breaks with bisques does not mean that you can do them without. If you can - terrific, you're on your way down!

To be realistic, at your level, the four-ball break without bisques is more often obtained by accident than design. This does not mean that you should not seize the chance when it occurs, but you must be pragmatic. In this style of game you will normally be looking for breaks of one to three hoops with a safe layup at the end. Here are a few practical tips. What if a shot involves a long split roll to maintain a break and you cannot do long split rolls? Do a take-off instead and accept the possible end of the break.

If you do this, think about how you are going to finish and still retain the advantage. Be aware that your tactics when you have bisques are very different from those when you do not.

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I stress this point strongly. It is only the high bisquer, whose knowledge of tactics is by virtue of being a high bisquer imperfect who has this problem. I have heard many beginners say "It's like playing different games" and in a sense that is true. Only in a sense though, because it IS the same game. Unless you get ready to change your play according to the situation, you can get into trouble from what seemed an easy situation.

I have won handicap games where my opponent has got to peg and penult or similar while I have not yet started. One reason why I was able to do so was because my opponents had had it easy with bisques. They had done well, but forgot that when the bisques were gone, carelessness gets punished severely by A-class players. The maximum number of bisques that you can give is 5, so your opponent will only be in category b ii. Your tactical play will therefore be essentially the same as for c. Remember to try not to give your opponent the opportunity to pick up an easy break. The most obvious pitfall to avoid is joining up near your opponent's hoop.

At this stage, nothing else. Remember this is just the start of tactics and that t you at the start of the handicap range. I do not believe in teaching too much at the beginning, because to do so will only confuse. So, concentrate on getting these first tactics correct. In a very short time the handicappers attention will be drawn to you - usually by your beaten opponents!

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You will then be ready for the next chapter. If you feel unsure about any of the above points, you should study or practice as appropriate before proceeding further. By now you should be aware of the importance of attitude and approach to the game. You should have a good idea of your own capabilities and weaknesses. You will have played quite a number of games. Probably they were mostly in your own club. If you haven't already done so, you should now be considering some competitive croquet at other clubs.

Apart from being good fun and good company, it will enable you to play different people. This will widen your experience of playing styles. Most clubs take part in local league play of some sort. There is a wide range of national events held throughout the season under the auspices of the Croquet Association CA. You should consider joining the CA now. You will be supporting your sport. Also you will glean a wealth of information from the magazine and through CA sponsored events such as coaching weekends.

Contact your Secretary for details. In this handicap range you will be able to pick up a four-ball break with one or two bisques. You will also now be starting to make breaks of several hoops without bisques. You will have a basic knowledge of tactics. Because of the confidence that you have gained by using your bisques properly, your shooting and hoop running will have improved considerably.

They almost certainly have but your expectations have gone up as well. Good, because this means that you are ever striving to become a better player. You will still make errors of judgement, some of which will be due to an over estimation of your ability. Many though will he due to a simple lack of thought and planning. In fact planning is the theme of this chapter.

You are for 1-back with a rather untidy four-ball which has the pioneer in good position but the pivot well off centre. You mess up the approach to the hoop. However the break is untidy. This means that your positioning of the ball which goes beyond 1-back is important. This is because it is the ball that you intend to rush to a good recovery position having run the hoop.

You hadn't thought of that! Now the pioneer to 3-back is difficult. You just manage it but leave yourself a long way from the poorly positioned pivot. To roquet it you go firmly - and miss. So there you are, suddenly from a commanding position to a disaster. You had a continuation shot left before you took the bisque.

That shot could easily have been to near the pivot ball with a rush to the middle. A simple take-off shot from there to the ball at 1-back and the four-ball is back again. I have capitalised "every" because that is exactly what I mean. If you need further convincing of this, go and watch top players in a four-ball break. These players can do such a break with their eyes shut. They can easily recover from poor placing.

That is not what happens. They play every stroke with care, every shot lined up and every position considered. Because the discipline of a tight and careful break is part of what makes a good player good. A sloppy break player is a player who will make mistakes and mistakes cost games. You will often hear a sigh of exasperation from an A-class player when a rush goes 2 feet to the side of a hoop and not in front. You think "I would have been happy with that". This is partly why you are not yet an A-class player! You might think at this stage "What has this got to do with my picture of myself?

Well, now that you know your ability, you must plan to it. It is no use knowing your skills if you do not use them properly. You have now added planning to your own picture. Watch for these things and use the information for your own plan. You have bisques, but tactically it is better to make one or two hoops on a three-ball before picking up the odd ball. Let us first of all consider the mechanics of a 3-ball break.

In operation correctly it is simpler than a four-ball. You have a pioneer and a ball at your next hoop, but no pivot. When you have run a hoop you make the next roquet. The subsequent croquet stroke has not only got to create a new pioneer, it has got to get your ball to the ball at the next hoop. The trick is to make that croquet stroke as simple as possible.

To achieve THAT needs accurate placing of the ball past the hoop that you are about to run plus accurate hoop run. When a three-ball break is in progress, you will have the following. A ball at your next hoop and a ball beyond the hoop that you are running. Your ball will be in front of its hoop. You should have a fairly good idea of running hoops but will probably still have some problems with control. Do not worry too much about that at this stage. What you should know, is how far you go through a hoop when you run it with your normal strength.

I mentioned this aspect in Chapter 2 and gave a nominal six feet.

Ones and Twos

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