The Billboard

So perhaps it's no great surprise that Google utilizes unusually mathematically oriented recruitment techniques. In fact, these practices have received widespread coverage in the last few weeks and months following the erection of a mathematical billboard in Silicon Valley on southbound Highway 101 near Ralston, California in July 2004. The billboard poses the question of finding the first 10-digit prime number occurring in consecutive digits (i.e., the decimal expansion) of the mathematical constant known as e, which is a transcendental number whose first few digits are 2.7182818284.... The unusal nature of the billboard prompted coverage through numerous mainstream media outlets such as National Public Radio, The Boston Globe, and the Oakland Tribune, as well as (of course) on the internet.




Ed mentioned the puzzle on his MathPuzzle website on July 13, 2004. Minutes after its posting, Wolfram Research CEO and A New Kind of Science author Stephen Wolfram sent the solution as a single line of Mathematica code:

Select[FromDigits/@Partition[First[RealDigits[E,10,1000]],10,1],PrimeQ,1]

{7427466391}

The Billboard, Level 2
Upon determining this value and typing the corresponding URL http://7427466391.com into a web browser, a potential Google employee (or curious MathWorld news story reader) is taken to a web page congratulating him (or her) and providing instructions for the second level of the puzzle, which involves finding the next term of the following sequence:

f(1)= 7182818284
f(2)= 8182845904
f(3)= 8747135266
f(4)= 7427466391
f(5)= __________


If the first two terms look familiar, that's because they are 10-digit portions of the decimal expansion of e given above. In fact, a little extra analysis shows that they are precisely those 10-digit portions summing to 49. After having determined that, it is easy to find the next number, again using Mathematica:

FromDigits/@Select[Partition[First[RealDigits[E,10,1000]],10,1],Total[#]==49&,5]

{7182818284,8182845904,8747135266,7427466391,5966290435}
This is sequence A095926 in Neil Sloane's On-Line Encyclopedia of Integer Sequences.

While we could likely continue this discussion at least a little further, we prefer at this juncture to leave additional levels of the billboard puzzle to the enterprising reader.

Son of Billboard: The Google Labs Aptitude Test
On September 30, Google concocted an even more challenging recruitment device: the Google Labs Aptitude Test. Hardcopies of this test were also distributed to University of Illinois students in the October 12 edition of The Daily Illini. While some of the questions on the test relate more to computer knowledge and general creativity, many of them are highly mathematical. And for the these problems, Mathematica clearly shows its extremely high mathematical aptitude by solving them easily, especially when guided by a little research on MathWorld and other online resources such as the On-Line Encyclopedia of Integer Sequences.