Suppose there is a lottery such that 6 random balls are chosen from a set of 50. The balls are numbered 1 thru 50. The lottery officials determine that lottery ticket sales are sluggish so they want to make it easier to win so they decide to allow "off by 1" for each of the 6 numbers drawn. For example, suppose with the original lottery the winning numbers drawn were (sorted) 5, 11, 12, 18, 37, and 44). Of course the only way to win would be to match those 6 numbers (in this example case) exactly. However, with the off by 1 variation, each number chosen by the lottery contestant can be off by as much as 1. So for example, to match the digit 5, the player can either have a 4, 5, or a 6 and that would be considered a match. Let's also consider that the player's ticket has randomly chosen numbers as well.

So the question is, how much easier is it to win with the "off by 1" variation compared to the original version?

Things to be careful of are things like matching the 11 and 12 as in the example. If the player chooses say 11, it cannot match both the 11 and the 12 actual numbers drawn. However to match both the 11 and 12, there are several ways the player can do that... (10,11), (10,12), (10,13), (11,12), (11,13), (12,13).

We will say that the player cannot choose the same number more than once per play so they cannot choose the number 11 twice for example in the same 6 number game. As with the original version, all numbers chosen much be unique.

$Update$: For clarity, I should mention that the chosen numbers must "map" to the drawn numbers with both in sorted order. For example, if the chosen numbers are 5, 10, 15, 16, 20, and 44 and the drawn numbers are 5, 10, 16, 17, 20, and 45, then the pair of 16s will not "map" to each other. The numbers get mapped in sorted (ascending) order. This example is a winning combination (all 6 numbers "match" using off by 1 rule).

In my simulation program which picks 6 random unique numbers, sorts them, then checks how many possible ticket combinations can match those 6 numbers, I have buckets to record how many ways there are and I tally them up. It appears 486, 648, and 729 ways are common and the lowest I've seen so far is 72 and the highest is 729. This may be useful information for those analyzing this problem. These results were acquired from only 1000 simulated drawings. I can let the simulation program run overnight and get a larger sample then I can post the results.

I can now easily and quickly simulate millions of randomly generated tickets and check how many possible winning tickets there are to cover those numbers. The long term average I appear to be getting is 503. I wish someone else would write a simulation program too to help verify my results. Also, mathematically, there are a lot of "buckets" indicating there are many different cases that contribute different probability "boosts" vs. the original game (where all chosen numbers must match exactly). So, this may be a pure simulation type problem as the "on paper" complexity appears to be too high. Amazing how only little variation makes the problem go from "cake" "on paper" to unwieldy due to complexity.

So to clarify my findings... whereas the original lottery had only 1 winning ticket combination (all 6 numbers must match exactly), the off by 1 simulation is showing me 503 winning tickets (on average) for any given randomly drawn 6 balls (out of the 50 possible).

I think why this would be so hard to solve on paper is because of the many different scenarios such as a ticket with 1 "neighbor" (such as 5, 10, 15, 20, 21, 25), 2 neighbors, 3 neighbors... Also 1 and 50 are special cases cuz they can only go in one direction. It would be interesting in itself to find out how many different cases there are which contribute different boost to the final answer. I suspect there are maybe 100 or so classes/categories (buckets) that all the tickets fall into. I could try running maybe 1 billion decisions overnight and count them up. With 1 million decisions I am already seeing over 110 buckets. Half of those appear instantly on my screen and the other half take a while to pop up.

I am currently running 1 billion simulated decisions and so far have 137 buckets so this problem would not be easy to solve on paper.