The statistician who read scratch tickets without scratching them: Mohan Srivastava and the singleton flaw

A $3 win and a second look

Mohan Srivastava estimates gold. He is a geological statistician — the co-author of a standard textbook on applied geostatistics — whose job is to take a handful of drill-core samples and say how much metal a whole deposit holds. In 2003, waiting for files to download, he scratched a tic-tac-toe lottery ticket someone had given him as a joke and won $3. On the walk to collect it he started thinking about how the ticket had been made.

A scratch ticket cannot be a true lottery draw. Its prizes are printed before anyone buys it, and the printer has to deliver a fixed prize structure — so many $3 winners per pack, so many $100 winners per million — so a computer decides, ticket by ticket, which ones win. Srivastava’s question was the one a geostatistician asks about any sampled process: if the outcome is designed, does the design leave a trace on the surface?

On the tic-tac-toe ticket it did. The game printed eight small tic-tac-toe boards on the face of each ticket, every one of their 72 spaces showing a number from 1 to 39. The player scratched a latex panel to reveal their own set of numbers and won if those numbers made a line on any board. The visible numbers were supposed to be decoration — random filler the hidden numbers happened to match. They were not.

The singleton rule

Srivastava counted how often each visible number appeared across the 72 spaces. Most repeated — a number might show up three or four times on the same ticket. A few appeared exactly once. He called those singletons, and when he scratched tickets to compare, the singletons were almost always among the hidden numbers under the latex. The repeated numbers were the filler; the ones that stood alone were the ones the generator had placed deliberately.

• Count every visible number. Across the eight boards, mark each number that appears only once on the ticket.
• Look for three singletons in a line. A row, column or diagonal of singletons on any single board is a winning line that has not yet been revealed.
• Buy that ticket. In his tests, tickets picked this way won about 90% of the time. Nothing had to be scratched to know.

The method had limits he was candid about. It worked only on “extended play” tickets whose face shows information before scratching. It did not say how much a winner paid, and most winners paid small prizes. And it needed the ticket in hand: a buyer has to see the face before paying, which is possible at a counter and impossible from a vending machine.

The numbers themselves couldn’t have been more meaningless. But whether or not they were repeated told me nearly everything I needed to know.Mohan Srivastava, quoted in Wired, “Cracking the Scratch Lottery Code”, 2011

He priced it, and it was not worth it

What makes the story unusual is what he did with the trick, and the reason is arithmetic. Reading a ticket took him about 45 seconds. Multiply the win rate by the prize sizes and the number of tickets a person could plausibly inspect at store counters in a working day and the method was worth, by his estimate, about $600 a day. That is a good day for most people and a poor one for a consulting geostatistician, and he said so plainly: he had done the math and beating the game was not worth his time.

So he called the Ontario Lottery and Gaming Corporation. After some telephone tag with officials who did not believe a scratch ticket could be read, he did the thing that ends such conversations: he bought 20 unscratched tickets, sorted them into predicted winners and losers using only the numbers on their faces, and mailed the stack to the lottery’s security office with his predictions. The game was pulled from Ontario’s retailers within days. The lottery described the problem as a design flaw in that one game.

Figures as reported in Wired (Jonah Lehrer, 2011) and the NPR interview with Srivastava (February 2011).

He did not stop at one game. A variation of the singleton count at least doubled his chance of picking winners on several similar tickets, including games from other lotteries whose tickets were mailed to him to test. The printers knew the risk by name: an executive of one of the largest ticket manufacturers told Wired that “breakability” — whether a ticket can be read before it is scratched — is among the main things they worry about.

Why a fixed prize count leaks

The deeper lesson is about where randomness lives. A ticket that was genuinely random on its face would show no useful singletons. Draw 72 numbers independently from 1 to 39 and repeats are unavoidable: each number has about a 15.8% chance of appearing exactly once, so a random ticket would carry roughly 11 singletons scattered without structure, and whether any of them matched the hidden numbers would be a coin toss the player could not improve. The tic-tac-toe ticket was not that. Its visible numbers had been written by the same program that placed the prize, and a program that must put a winning line on this ticket and not on the next one has to coordinate the face and the latex. Coordination is information.

This is the same shape of flaw as Phil Ivey’s edge-sorted cards or the superuser account that could see hole cards: the game advertised one source of uncertainty and actually ran on another, and the gap could be read by anyone who looked. For a printed lottery the gap is structural. The operator wants two things at once — a guaranteed number of winners per million tickets, and tickets that look random — and the first requirement defeats the second unless the face is generated independently of the prize. Modern tickets are designed exactly that way; the 2003 game simply was not.

A scratch card with nothing on its face

Betkyo’s omikuji is a scratch card too — nine cells, a fortune glyph in each, three of a kind pays — so it is fair to ask whether it has a face to read. It does not, for a reason that is the opposite of the lottery’s: no prize count is fixed in advance. Each card is an independent draw. The tier is decided by one roll against fixed weights at the moment the round is created, from the committed server seed, the player’s client seed and the nonce, and only then are the glyphs painted into the cells. There is no pack, no quota of winners per thousand cards and no generator trying to make this card and the next one add up to anything.

Two things follow. First, the previous card tells you nothing about the next: a thousand losing cards in a row do not make 大吉 “due”, because nothing is being counted down. Second, the card can be audited the other way round. Srivastava had to buy tickets and scratch them to prove the pattern; on a provably fair card the player can recompute the whole card from the revealed seed after the round and confirm the tier roll was the one the weights say it should be. What the verifier checks, step by step is the same for every original.

It is worth being honest about what the design does not change. Our card has its own near-miss structure — 46% of losing cards show two 大吉, by the pigeonhole of nine cells and seven glyphs — and a 3.45% house edge that no amount of reading improves. Srivastava’s trick made a bad ticket readable; a committed seed makes a fair card unreadable. Neither makes the card pay more than its weights.