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Ball In A Box Productions
Ball In A Box Productions
Covering Designs (Abbreviated Wheels)
From a complete range of numbers, a certain quantity of numbers are randomly chosen.
Take the case of there being 6 numbers randomly selected from a much larger range.
These 6 randomly chosen numbers are referred to as being picked, chosen or drawn numbers. Let them be denoted as K.
In trying to guess which of the 6 numbers will be randomly chosen, drawn or picked, we are permitted to select a range of numbers greater than those randomly chosen and less than the complete range of possible numbers.
Let our set of guess numbers be denoted as V. Another term for them is a 'pool' of numbers.
Using this pool of guess numbers, we create a list of all possible combinations, taking a certain amount of numbers at a time. The amount of numbers for each line in our list usually equals the amount of randomly chosen numbers K.
Each line can also be referred to as a ticket or block. Let the number of lines in our combination list be denoted as B.
Eg: from a guess pool of 14 numbers, write down every combination of 6 numbers. The list should be 3003 lines long. B=3003.
With V numbers in our pool of guesses, we hope K of them will match the randomly drawn numbers.
From the example V=14 and K=6, if all 6 random numbers are in our pool of 14 numbers, somewhere in the list of 3003 combinations there will be a match. This is because our list is composed of every 6 number combination from our V pool of 14.
But what if a list of 3003 lines is too many?
The combination list of B blocks can be shortened if we are not concerned with exactly guessing the K random numbers drawn. However, we want a guarantee that if all K randomly drawn numbers are within out V pool of guesses, there must be a minimum match amount somewhere in our combination list.
Let the minimum amount of numbers to be found in any block that match the K randomly drawn, which are also listed in our V pool of numbers, be denoted as T.
So now we can say, no matter which K numbers are drawn, if they are all in our V pool of guesses, there will be one line in our B blocks which matches T of the K numbers.
Eg: 6 randomly drawn numbers are within our pool of 14 guesses. In at least one line of our abbreviated list of combinations, there will 4 of the randomly drawn numbers.
This is written as C(v, k, t)=b where b is the number of blocks in the combination list and is a variable length for different values of v, k and t.
Worded, it is expressed as a guarantee of 4 if 6 of 14.
The list can be further shortened, hopefully, by requiring fewer drawn numbers to match the guess list.
Let M be the minimum amount of numbers from the drawn numbers K which match any of our V list of guess numbers.
We still want to guarantee that at least one block or line in our combination list has a certain amount of numbers shared by the drawn numbers K and the guess list V.
In this case, no matter which K numbers are drawn, if M amount of them are within our V guess list, there will be at least one line in our B blocks of combinations which contains T of the matched M numbers.
This is written as C(v,k,t,m)=b.
Eg: C(14, 6, 4, 5)=30
From a pool of 14 numbers, if 5 of 6 randomly chosen numbers match, 4 of the 5 matching numbers will be found at least once in a 30 line list.
"Guarantee 4 if 5 of 6 in 14."
Draco Merest Nov 2010