The Serpentine
This exercise works on: suppleness, straightness, contact,balance, and the half halt.
 
The serpentine traditionally consists of two or three half circles connected, however the size, the number and the shape of the loops  can change for added difficultness, emphasis on certain weaknesses in the horse and rider and increased agility in the horse.  The loops must stay round, and even through each.  This is what works the horse evenly on both sides, allows the horse to learn to shift and balance, and not worry about falling on his nose.

    The two loop serpentine if performed in a 20x40 meter arena consists of two 20 meter circles which join at X.  If started at A the first loop begins, just like any normal 20 meter circle, the horse following the track of the circle this works on the lateral suppleness and straightness.  At X the rider straightens the horse for one stride, applies the half halt to rebalance the horse, change the bend (and changes the posting diagonal) and continue on the second loop.  This one stride (sometimes two while teaching it to the horse) teaches the half halt, as even if the horse does not know the half halt the natural instinct to rebalance and shift the weight will teach the horse to associate that with the half halt.  The one straight stride also works on the longitudinal suppleness by asking the horse to shift the weight, and change the bend.

    If the Serpentine is started at A it will end at C.

On the left the correct two loop serpentine in a 20x40 meter ring.  The right shows it done incorrectly.
 
    The three loop serpentine has the same basics as the two loop serpentine.  If done in properly the loops are the same size.
The traditional three loop serpentine called for in dressage tests etc.
There are many variations on the three loop serpentine for schooling the horse.  One variation is to  not go wall to wall in the loops but from the quarter line to quarter line.  This makes it so the loops can be circles in a shorter ring asking for greater precision from the rider and horse as they do not have the wall of the arena for a guide.

    For more suppling the serpentine can be changed so that more loops are added, this requiring smaller loops for it to work asks the horse to bend more becoming and showing more lateral suppleness.  The added changes of direction asks for longitudinal suppleness of the horse as the horse is required to shift and shorten and change the bend more often.  The variation shown below gives many loops, and allows the loops to be carried on the curve for longer then the traditional serpentine.  This variation is NOT correct for riding tests, dressage, it is purely a schooling exercise.  For even added difficulty the loops can be carried further, making the angle more severe between loops allowing for even more loops to be added into the ring.

The serpentine changed to work on more suppleness in schooling the horse.  Added loops take it from the traditional form.
 
     No matter which way the rider changes the serpentine, it must be done deliberately with the shape of the loops in the riders mind.  The rider must make sure to bend the horse accordingly on the loops (but with out over bending), must straighten the horse properly on the straight away applying the half halt and changing the bend.  The serpentine if done properly will put the rider and horse onto the outside rein, giving an even contact, but this will only occur if the rider stays with the exercise.