Elite chess may carry an air of respectability and intellectual rigour, but that hasn't stopped players and fans from levelling accusations of cheating and unfair play.
That's how University of Toronto statistician Jeffrey Rosenthal came to be tapped in 2024 to analyze seemingly unlikely winning streaks racked up by five-time U.S. chess champion and grandmaster Hikaru Nakamura, who goes by the player name Hikaru.
The games in question took place on the online platform Chess.com, which hosts some 10 million matches every day.
"When Chess.com asked me to look into it, I was happy to," says Rosenthal, a professor of probability and statistical computing in the department of statistical sciences in the Faculty of Arts & Science. "It's the sort of opportunity I like because it involves some genuine statistical probability issues. And it was something that was of genuine importance - not just to a small number of statistical scientists, but to large numbers of people around the world."
Former world chess champion Vladimir Kramnik had raised suspicions about Hikaru's streaks, pointing to a run of 46 matches in which Hikaru won 45 and drew one. The suspicion stemmed from the belief that online chess carries a higher potential for cheating - for example, by players using chess-playing software - compared to in-person games.
The controversy even made international headlines a few years back, including in the New York Times .
Rosenthal analyzed data provided by Chess.com and, in an August 2024 report featured on the site , concluded that Nakamura's streaks were well within statistical expectations - meaning it was unlikely he had cheated.
Kramnik responded with a video criticizing the findings. Rosenthal then addressed Kramnik's concerns in another report in September 2024 and, in April 2025, published a paper in the Harvard Data Science Review .
In particular, Rosenthal identified two key reasons why Nakamura's winning streaks on Chess.com didn't indicate foul play.
The first reason was that Nakamura's opponents were significantly less skilled. For example, during a particular 116-game streak, Nakamura's player rating averaged 3,017 - a very high score - while his opponents averaged just 1,526.
In other words, because Nakamura was far more skilled than his opponents, long winning streaks were statistically more likely than if he'd faced strong players.
Second, Rosenthal demonstrated that the sheer volume of games Nakamura played increased the likelihood of streaks.
As he explains in his April 2025 paper, if you flip a coin 12 times, the odds of getting 12 heads in a row are extremely low. But if you flip that same coin 10,000 times, the chance of hitting a streak of 12 heads becomes much higher.
Rosenthal demonstrated this by using a Monte Carlo simulation - a computer program that, in this case, flipped a virtual coin 10,000 times. Run 1,000 times for statistical rigour, the simulation showed a nearly 70 per cent chance of producing a 12-head streak, making it far from improbable.
This helped answer one of Kramnik's questions: Why does Nakamura have 21 long winning streaks, while Magnus Carlsen, the Norwegian grandmaster and multiple world champion, had only one during the same period?
Rosenthal's explanation: Nakamura played 57,421 games compared to Carlsen's 5,104 and Nakamura's opponents were rated much lower than Carlsen's.
"Just because something is striking and dramatic doesn't necessarily mean that it has statistical significance," says Rosenthal.
"In order to truly understand what's going on, you have to stop and think. You have to do the work to figure out the probabilities of something like that happening. It's how we understand what is and isn't true."
It wasn't the first time Rosenthal had been asked to look into high-profile cheating claims.
In 2006, CBC journalists asked for his help in examining why Ontario retailers were winning a seemingly disproportionate number of lottery prizes. Rosenthal's analysis confirmed retailers were indeed winning far more than statistical odds would suggest . The revelations ultimately led to criminal charges, jail terms, the dismissal of the Ontario Lottery and Gaming Corporation's chief executive and the introduction of safeguards still in place today.
