Their "arbitrary tests" are based on well-known psychological trends. For example if asked to pick a random number from 1 to 20, 17 and 7 are far and away the most popular choices. It is also well-known that people have a tendency to avoid numbers with recognizable visual patterns. Hence the avoidance of 00, 11, 22, 33, 44, 55, 66, 77, 88, and 99 is also grounded on known psychological grounds.

That said, you're right that election fraud is a serious charge. We'd like to have as much evidence as possible before leveling it. Unfortunately Iran chose to keep people from having evidence that would allow a good judgment either way. For example there were no external election monitors. Going into the election the consensus of what available polling there was said that the election was close. Then they then announced a landslide that was sufficiently absurd that their own population has been engaged in widespread protests.

We are then left with a situation with imperfect data. While we'd like to set a high standard for declaring the fairness or unfairness of the election, we simply lack sufficient data to do so. However if we lower the bar to look at what the data suggests, all lines of data that we have point to an unfair election. Those lines include their unwillingness to allow external election monitors in, large discrepancies between (admittedly limited) pre-election polling and results, large popular protests, and the statistical arguments that the Washington Post makes. What is interesting about the Washington Post article is that it provides reasonably strong evidence that the method of cheating was assigning numbers rather than something more subtle such as, for instance, stuffing the ballot boxes.

Let me repeat that and make it clear. Even without the Post's argument, evidence pretty strongly suggested that the election was rigged. Along that line the Post only offers another line of evidence that confirming what we already had reason to believe. But what it offers that no other line of evidence does is evidence about what method was used to rig the election.