For example, it is obvious that if you play all possible combinations of numbers you will win all possible prizes 100% guaranteed. There is no question about it! However this would be expensive as there are millions of combinations. 6/49 lottery has close to 14 millions of combinations.

Lottery wheel helps you to bring down the number of combinations to play, and still guarantee the winning. A wheel is made having three possible goals in mind:

1. Your goal is to match fewer amount of winning numbers, for example 3. You can make a wheel that will guarantee you matching 3 numbers at least at one ticket you have played.
2. Your goal is to win the jackpot, but instead playing all, you play a subset of numbers. For example, if you play 20 numbers, you will win the jackpot 100% guaranteed only if officially drawn numbers are among the numbers you have played.
3. Finally, you can combine two goals by playing subset of numbers, and aiming for lower prize. For example, you can play 35 numbers and aim matching 5 numbers. If officially drawn numbers fall in group of your numbers you will have at least one matching ticket with 5 numbers.

What is full, and what is abbreviated wheel?
The full wheel always gives the jackpot guarantee.  The guarantee for an abbreviated wheel is selected by the player.

What is filtered wheel?
Filters reduce number of combinations (games) in full or abbreviated wheels. For example, a filter can be set to remove combinations with all odd numbers, to include your favorite number in every combination, etc...

100% guarantee? What's the catch?
Your selected prize goal winning is guaranteed 100% only if the officially drawn numbers all appear in the group of numbers you wheeled. If you used a filter, the guarantee is valid if the winning combination meets the requirement set by the filter.

Is there any mathematical background to lottery wheels?
Yes, there is. Lottery wheeling is based on Covering theory, and Covering Design theorem:
Covering Design C(v,k,t,m,l,=b) is a pair (V,B), where V is a set of v elements (called points) and B is a collection of b k-subsets of V (called blocks), such that every m-subset of V intersects at least l members of B in at least t points. It is required that v >= k >= t and m >= t. The case m > k is also a valid case. B can be a multiset.
When applied to lottery:
v   total numbers in the design (amount of numbers you play)
k   amount of numbers drawn
t    min match t   (t of m) 
m  min match of m (t of m) 
l    lambda (min l times t of m)   (it is always 1and usually omitted)
b   number of blocks   (number of games)
For example,  C(44,6,3,5,154) describes wheel with 44 numbers designed for "pick 6" lottery that guarantees 3 matches if 5 officially drawn balls are within the group of 44 numbers you have played. Finally, the wheel size is 154 games.