Always impressive. I learnt three in one hand by learning the six ball fountain, which is juggling three in each hand at the same time. Strangely, for a long time I found that I could run six balls for longer than I could run three in one hand on its own.
How to hold three in one hand
There are two common ways of holding three in one hand, the triangle arrangement & the line arrangement. I personally use the triangle arrangement, but as always, try both & see which feels more comfortable to you.
As the name suggests arrange the balls in a triangle shape in your hand. Hold one ball between your little finger & the butt of your thumb, hold the next ball between your thumb & index finger & hold the third between your middle & ring fingers & the other two balls. From this position you throw the balls in reverse order.
This position has all the balls in a single line & is a bit tricky if you are using large balls. Hold one ball between your little finger & the butt of your thumb, the next ball between your ring finger & your thumb & the hold third ball against the second with your middle & index fingers. Throw the balls from the top of the stack (your finger tips) to the bottom (by your wrist).
There are four basic three in one hand patterns, shower, cascade, reverse cascade & columns. Another common pattern is called 'all over the place' but that doesn't really need any explanation. Each one is the same in that the rhythm doesn't change, the balls are thrown & caught in order. All throws are made to the same height, about a foot above your head. The only thing that changes is where the balls are thrown & caught.
This is the most common pattern & it is exactly the same as the two handed Shower except that you only use one hand. The balls follow each other round in a circle. The pattern can be juggled in both directions, either throwing on the inside & catching on the outside (rolling out) or throwing on the outside & catching on the inside (rolling in).
An easy way to learn it is to start juggling a two handed shower & gradually increase the height of the pattern & bring your hands closer together, when your hands are side by side transfer the pattern entirely into one hand. From this you will be able to get the 'feel' of the rhythm. Aim for a really narrow pattern, the distance between where you make a catch & where you make a throw should only be a couple of inches. Making consistent throws which all peak at the same height is also very important to maintain the rhythm. As you make each throw flick your fingers to spin the ball, this will make it curve inwards if you are juggling a rolling out pattern or outwards if you are juggling rolling in.
Develop a nice smooth circling action with your arm. If you catch & jerk your arm to make each throw, your thows will not be accurate & your arm will get tired very quickly. Your forearm should make smooth & even circles (well elipses to be precise).
Cascade & Reverse Cascade
Again the Cascade & Reverse Cascade are exactly the same as their two handed counterparts. For the one handed versions the hand weaves in a figure of eight on its side. Imagine three points on the figure of eight one on each side & one in the centre where the lines cross. For the cascade all throws are made from the centre point & the balls are caught on the two outer points alternately. The reverse pattern is obviously the opposite, so all catches are made at the centre point & the throws are made from the outer points alternately.
Again rhythm is all important so make all throws to the same height. Flick your hand with each throw to spin the ball so that it curves in the air to where you want the ball to go & keep your arm movements as smooth as possible.
Probably the trickiest of the lot. For this version imagine three columns & you throw each ball up in each one in turn, work down the line then flick back to the start like an old typewriter. This variation is quite hard on the arm because it has to make quick & sharp direction changes as it moves down the line of balls rather than the smooth circles of the other variations. Keep the columns as close together as possible to minimise the amount of movement your arm has to make. Yet again keep the throws to even heights & keep a nice even rhythm.
If all that is not difficult enough try juggling three in one hand with Claw or Penguin catches &/or Behind the back or Under the leg throws.
Be sure to learn three in either hand, obviously it is excellent practise for six balls if you ever take up the challenge & is very useful for five ball tricks.