Until, on a random whim, Gary Briggs came along and ruined it all.

Mr. Briggs, a Rand software engineer, spent his spring rifling through the million digits and discovered that while the numbers inside are indeed quite random, the venerated book is not quite right.

“It’s this seminal 65-year-old piece that we all herald and revere,” says Mr. Briggs, an 11-year veteran of the Santa Monica, Calif., research organization, “so the idea that I’m finding errors that we’ve ignored for 65 years is upsetting.”

Before modern computers, he says, “it was really hard to get high-quality random numbers.” The book changed that for a generation of pollsters, lottery administrators, market analysts and others who needed means of drawing random samples.

An engineer might be charged with inspecting welds on a rusting bridge, in an example Rand cites. Rather than check every weld or just easy-to-reach ones, the engineer would number the welds and use the book’s random-number tables to decide which to inspect.

While the engineer could pull numbered slips of paper from a hat, the book provides huge quantities of digits of guaranteed randomness, eliminating the risk that the engineer might bias the inspection by, for instance, not shaking the hat sufficiently.

Rand legend has it that a submarine commander used the book to set unpredictable courses to dodge enemy ships.

Mr. Briggs, 40, creates Rand computer models for the U.S. Air Force. In his free time, he obsesses with puzzles and projects. He made a chain-mail hoodie to wear to a comic-book convention, taught himself to knit, learned to juggle.

In May, he attended an online presentation by Rand’s archivist, who said work on the million digits had stretched for years before publication in 1955. Mathematician Bernice Brown spent the late 1940s conducting mathematical tests to ensure the numbers contained no predictable patterns.

In her 1948 paper, “Some Tests of the Randomness of a Million Digits,” Mrs. Brown announced that “none of the tests contradicts the assumption of randomness.”

he died at 99 in 2003. Her analysis held until Mr. Briggs fixated on replicating her work, leading him down a three-month rabbit hole from which he hasn’t fully emerged.

“We were in quarantine, so that didn’t help,” says Mr. Briggs’s wife, romance and science-fiction writer Elizabeth Briggs. “He had to do something.”

He started by looking at how Rand collected a million digits. Douglas Aircraft Co., instrumental in Rand’s creation, provided a simple machine that registered random fluctuations in voltage and converted them into strings of ones and zeros. “I cannot overstate how much I want one of these things sitting on my desk,” says Mr. Briggs, whose next project is to build one.

A circuit board converted sets of ones and zeros into digits zero to nine, which a third machine translated into holes punched into as many as 20,000 computer cards.

Technicians fed the cards into an IBM data-processing machine, which generated a million-digit number filling 400 pages of tables.

At one point, the Rand team noticed the Douglas machine was producing a suspicious imbalance between even and odd numbers. They unplugged it, let it cool and turned it on again; the bias disappeared, according to Mr. Briggs.

Rand called the first edition “the largest table of random digits ever published.” The book included instructions to compensate for a user’s tendency to pick numbers by opening to the middle and pointing to the page’s center, which renders chosen digits not-very-random.

Mr. Briggs obtained the original numbers and wrote programs interpreting them the way the original IBM did. He soon found his results didn’t match what appeared in the book.

The first sign of something amiss came when he examined the eighth bloc of 50,000 digits. In the book, that bloc contained 5,003 zeros and 5,163 twos. When he ran the numbers, he got 5,004 zeros and 5,162 twos.

It was a minuscule deviation that, Mr. Briggs thought, could be the result of cosmic rays messing with the computer’s memory. Such an error would mean nothing for users: No bridge would fall, no submarine sink.

He conducted a follow-up test, counting how many two-digit sequences appeared. In a bloc of 50,000 random digits, he would have expected to see about 500 sequences of 0-1, about 500 of 0-2, and so on—and that his numbers would match the book’s.

Instead, he came up with extra 4-4 and 9-1 sequences where the book showed 9-4 and 4-1.

He spent weeks trying to figure out why.

In the 1940s, researchers wrote code during work hours, then gave punch cards to technicians operating the IBM overnight.

Mr. Briggs hypothesized a technician dropped cards and put them back in the wrong order. He envisioned running computer simulations to re-create the error by moving a card or two out of place.

But he didn’t know what the original cards looked like.

In 1949, the Rand newsletter cheerfully announced the Numerical Analysis Department, in a spring-cleaning frenzy, had sold 8,435 pounds of used IBM cards for scrap, bringing in $60. Rand’s archivist suspects the random-digit cards were in the recycling.

That left Mr. Briggs in the dark about whether the cards each had 50 digits or 72 or 80. The answer was critical to understanding whether shuffling them could have caused the error.

So he wrote a program that assumed a variety of formats.

His findings: If each card had 72 digits, there were eight different ways one card could be out of place and produce the exact error he found. If each had 80 digits, any of three cards could have been moved to eight different spots and caused the same mistake.

Elated but cautious, Mr. Briggs examined sequences of repeated numbers as a final test.

In a group of 50,000 random digits, mathematicians would expect 4,050 sequences of two identical digits in a row—77, for instance. They would predict 405 spots with three identical digits in a row, such as 555. There would be about 40 cases of four identical digits in a row. And four or five places with five identical digits together.

His results were “soul crushing,” Mr. Briggs says. The book contains 48 runs of four digits instead of 40, an astoundingly wide divergence in statistical terms that eluded any explanation he could conjure.

It’s not that the digits in the book aren’t random, he says. They just don’t seem to be exactly the right digits in exactly the right order, given the impulses the Douglas machine generated.

The mystery deepened when he noticed the book’s 1955 edition described its randomness as “obviously satisfactory.” That phrase had disappeared by the 2001 edition, leading him to wonder if some editor noticed the error and tempered the claims.

A Rand spokesman says the missing line is the result of a random typo and will be restored to the online edition. “It may be time for a new edition of the book,” he said. “The digits are fine—Gary’s sleuthing attests to their continued randomness—but now we have new fodder for the foreword.”

That said, “A Million Random Digits” became less relevant as powerful computers generated instant randomness. Mr. Briggs acknowledged as much this summer when he told stunned Rand staff about the errors in their beloved publication.

“Nobody,” he told them, “has actually opened the book in many years.”

Write to Michael M. Phillips at michael.phillips@wsj.com