Can the lottery code be cracked?

I just came across this article about a Canadian who broke the Scratch Lottery Code.

The article s about Mohan Srivastava who came across two unscratched Tic-Tac-Toe tickets near his desk. Not a big lottery fan but with nothing else to do, he scratched them. Lost on the first, won $3 on the second. He went to cash in his winning at the nearby gas station, but all the while he started thinking about how these tickets are created. Because the lottery corporation needs to keep careful track of how many winning tickets get printed, the computers can't just generate random numbers onto each card, winning and losing tickets need to be carefully created while giving the illusion of being completely random.

The more he thought about it, the more he became convinced there could be a way to determine whether a ticket was a winning ticket or not without having to scratch it. Here's what a Tic-Tac-Toe ticket looks like:

You have 8 3-by-3 grids with visible numbers. You then scratch the 24 numbers on the left. If any of t3 of the 24 numbers are lined up in any direction in any of the 8 grids, you've won!

Well Mohan bought many tickets, scratched them all, and ultimately found a relatively simple rule to determine ahead of time whether a lottery ticket is a winning ticket or not.

The trick? In your lottery ticket you have 72 numbers visible in the grids (8 * 3 * 3). Because the numbers go only up to 39, so will have to occur multiple times. Mohan kept track for each number how many times it occurred on the ticket:

He was especially interested in singletons that appeared only once. His finding was that if three singletons lined up, then you had a winning ticket!

I am not a 100% about why that is and how Mohan came to this conclusion, but intuitively I think the algorithm works something like this (at a very high level)

• randomly separate numbers from 1 to 39 into two sets, 24 singletons and the 15 duplicates
• fill in the grids with duplicate numbers, and with at most 2 singleton numbers
• list the 24 singleton numbers on the left hand side

This will guarantee that the ticket is a losing one. You can make it a winning one by adding three singletons in a row, column or diagonal in any grid. This will guarantee that the ticket is a winning one with only one gain.

This is overly simplistic but it does allow a simple way to generate the tickets. Creating random grids and then testing whether they are winners or losers before printing them would take to much time, the above process creates them in one shot without testing.

On the example given in the article we can see that the 24 numbers given on left come out a total of 35 times in the grid (average 1.45), whereas the other 15 numbers not listed come out 37 times (average 2.47). This hints that the duplicates are indeed something like background noise, and the 24 numbers selected on the left are actually much less likely to occur in the grid. And this is where we are fooled: If we reveal 24 numbers, we should find a lot more of them in the grid made out of 39 different numbers shouldn't we?

The article also talks about the lottery industry in general and how the mob uses the lottery for money laundering.

And as to why Mohan revealed the trick instead of making a fortune? He simply calculated that if he were to spending all his time identifying winning tickets in each store across town, scratching them, and redeeming them, he would make less than his current full time job!

All in all, a rather interesting read, although I already gave out the end...