In the gameshow Countdown there's the numbers round. In summary, the contestants are presented with 6 numbers, and a generated target number Some of the 6 numbers might be duplicates. The contestants are allowed to use each number just once (or not at all), and add, divide (whole number results only, fractions not allowed), multiply or subtract them with each other to get the target.
My puzzle is a variation of this, where players are given different sets of numbers (not necessarily 6 numbers, could be more or less) to choose from but the same target number, I want a way of predicting which set of numbers is most likely to get to the solution with the fewest steps, if a solution is possible at all. I'm asking for your help to establish a reasonable prediction model.
There are a few things that come to mind, but I'm really not a math whizz which is why I come here:
- Sets with more numbers present more options so are more likely to contain a solution.
- I'm not convinced that sets with less duplicate numbers are more likely. I think squaring or double a number is just as useful as multiplying or adding two separate numbers.
- The set of numbers that has the most common factors with the target will give more opportunity for solutions involving multiplication, so therefore more likely to have a solution at all?
- The set of numbers with the least common factors amongst themselves means less overlap; does this imply a wider ranger of opportunities within the set?
- Does having the highest chance of solution also imply the highest chance of having the shortest solution?
- If multiplying all the numbers together gives a lower number than the target, then there is no hope at all for that set.
- Is proximity relevant? For example if the target is 101 and one of the numbers in a small set is 100, is that set really any more likely to contain the solution than a set with more but lower numbers?
Is there already a field of maths that covers this kind of problem? I'm open to any advice. Thanks.