In a paper of more than 40 years ago: "Irregularities in the Distributions of Finite Sequences", E.R. Burlekamp and R.L. Graham , Journal of Number Theory 2,152-161 (1970), the authors define a number s=s(d) in the following way.

"For n>=1, 0<=k< n, define

B$_n,_k$=[$k/n, k/n+1/n$).

Fix an integer d>=0 and suppose ($x_1,x_2,...,x_s+_d$) is a sequence with x$_i$ belonging to [0,1) and with s=s(d) chosen to be maximal such that for each r<=s and each k < r, B$_r,_k$ contains at least one point of the subsequence
($x_1,x_2,...,x_r+_d$),"

s(0) is the answer to a question of Hugo Steinhaus and is known to be 17. The calculation to find s(0) takes only seconds. I believe that it is possible to calculate s(1) in a reasonable amout of time, perhaps weeks. I'd be willing to make my Mathematica program available to anyone who's interested.

What is the value of s(1)?

[Sorry that the plus signs in subscripts don't get rendered at the same height as the subscrip letters. I don't know how to fix that.]