Walk into almost any gas station convenience store in the United States or Canada and you’ll likely find something to satisfy your particular vice. They have candy for the late-night snacker, beer if you need a drink, cigarettes if that’s your crutch, and scratch-off lottery tickets if you want to try your luck. For as little as $1, you can buy yourself a small piece of paper, parts of which are covered in a silvery latex. Scratch off that latex area and, if you’re lucky, you can win a few bucks.
And if you’re really good at math, maybe you don’t need much luck. Just ask Canadian statistician Mohan Srivastava — and the Canadian lottery board which made the game seen below.
The game above was issued by the Ontario Lottery and Gaming Commission in 2003, and the rules are pretty simple. On the left is a box which says “Your Numbers.” You scratch the latex covering off that area, revealing 24 different numbers. If those numbers appear on the tic-tac-toe boards on the left, great! You’re on your way to winning some extra cash. They just need to line up properly — you need three in a row, column, or diagonally, just like in tic-tac-toe. If that happens, you win the prize indicated. It’s pretty straightforward.
In June of 2003, according to Wired, Srivastava was cleaning his desk when he came across a couple of these scratch-offs — most likely a gag gift from a friend. (Statisticians typically don’t buy lottery tickets; the math doesn’t add up.) One of the tickets was a winner and for Srivastava, it became a point of professional curiosity. He inspected the winner a bit more and saw a pattern — a pattern which suggested that the game was beatable.
Take another look at the ticket above, and pay particular attention to the bottom-center square on the 4th game as well as the center-left square on the 7th and 8th game. You’ll note that the number 15 appears in all three locations. Srivastava surmised that, unless the Gaming Commission wanted a lot of winning grids on one ticket, it didn’t make a lot of sense for a repeated number to be one of “your” numbers. By extension, Srivastava postulated, numbers which only appeared once among the eight grids — “singletons,” as he termed them — were more likely to be winners. If a card had three singletons in a row, column, or diagonal, Srivastava concluded, it had a very good chance of winning. (The card above was one such example: the middle row of grid 6 is 24-12-29, each of which is a singleton.)
Srivastava grabbed $60 and ran a test. According to the Toronto Star, he “chose 20 tickets currently on sale in Ontario, predicting six would be winners” because they had this triplet of singletons. And while not all six were winners, four of them were. Doing a bit of math, Srivastava estimated that there was only about a 2% chance of this happening. He was rather sure he found a way to beat the odds.
But untold riches weren’t on the horizon for Srivastava. “Once I worked out how much money I could make if this was my full-time job, I got a lot less excited,” Srivastava told Wired. “I’d have to travel from store to store and spend 45 seconds cracking each card. I estimated that I could expect to make about $600 a day. That’s not bad. But to be honest, I make more as a consultant, and I find consulting to be a lot more interesting than scratch lottery tickets.” (Kids: become math consultants. It’s a good job.)
Srivastava did the next best thing: he told the Gaming Commission about the flaw. At first, they didn’t believe him, but as he told NPR, “I decided to send them 20 tickets that I had separated into winners and losers. I said go ahead and scratch them off. And they did and they realized that they really did need to talk to me.” The Commission fixed the game, ruining the trick for everyone else in the process.
From the Archives: Alive and Scratching: When it came to this guy’s scratch-off ticket experience, the re-enactment was better than the original.