June 2019 - Solution
There are small solutions, like Johan Van Mengsel's solution: [20, 72, 75, 105, 120 ,162 ,336], [2, 7 ,14 ,40 ,42 ,96 ,126, 150, 400]
Reiner Martin was the first to solve it using the rational approximation of sqrt(2). You can read about it on Thomas Huet's blog https://thomash.fr/2019/06/15/ibm-ponder-this-june-2019.html.
Jingran Lin had a nice solution:
Rewrite the example given on the website:
6 + \sqrt{6} - \sqrt{2} - \sqrt{10} - \sqrt{15} < 1.52 \cdot 10^{-5}
Hence:
(6 + \sqrt{6} - \sqrt{2} - \sqrt{10} - \sqrt{15}) ^4 < 5.31\cdot 10^{-20}
Expand to give:
(*)
Rewrite the following to sum/diff of smooth numbers, if not already:
15281= 15309 - 28
3864= 3888 - 24
2968= 3000 - 32
2328 = 2400 - 72
6092 = 6272 - 180
4740 = 4800 - 60
3920 smooth number
980 smooth number
Expand (*) further:
Take the square of each element in the above to produce two sets of smooth numbers:
{234365481, 1152, 3072,28800000, 236027904, 36000, 28812000}
{784, 30233088, 27000000, 25920, 194400, 230400000, 230496000}
As for the bonus question, 1560674304 = 2^{17} \cdot 3^{5} \cdot 7^{2} is the number of seconds elapsed between the UNIX Epoch time (00:00:00 Thursday, 1 January 1970) and the 108th birthday of IBM (Sunday, June 16, 2019).
We were aiming for a similar smooth number, 612615150 = 2\cdot 3^{6}\cdot 5^{2}\cdot 7^{5}, which corresponds to (Wednesday, May 31, 1989), the 30th anniversary of IBM Academy of Technology.
But we accepted other solutions like the one submitted by Chris Shannon, who found that the smooth number 1210104000, interpreted as a UNIX time stamp, results in May 6, 2008, the date when Fran Allen received the Erna Hamburger Distinguished Lecture Award at EPFL.
Others found that the year 2016 is smooth (2016=2^{6}\cdot 3^{2}\cdot 7) and the last woman to become president of the IBM Academy of Technology was Susan Schreitmueller Smartt in 2016).
and other solutions discovered that 2000 is smooth as well (2000 = 2^{4}\cdot 5^{3}), when Frances E. Allen became a fellow of the Computer History Museum.