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In a paper of more than 40 years ago: "Irregularities in the Distributions of Finite Sequences", E.R. Berlekamp 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 $n \ge1,0 \le k\lt n$ , define

$B_{n,k}=[k/n, k/n+1/n)$.

Fix an integer $d \ge 0$ 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 $r \le s$ and each $k \lt r$, $B_{r,k}$ 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 amount 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.]

In his book of problems from elementary mathematics Hugo Steinhaus asked the following:
does there exist for every positive integer N a sequence of real numbers $x_1,x_2,...,x_N$ such that for every n in {1,...,N} and every k in {1,...,n} we have

$(k-1)/n\le x_i<k/n$ for some i in {1,...,n}. If not, what is the greatest possible N?

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

"For $n \ge1,0 \le k\lt n$ , define

$B_{n,k}=[k/n, k/n+1/n)$.

Fix an integer $d \ge 0$ 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 $r \le s$ and each $k \lt r$, $B_{r,k}$ contains at least one point of the subsequence
$(x_1,x_2,...,x_{r+d})$,"

$s(0)$ is the answer to the 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 amount of time, perhaps weeks. I'd be willing to make my Mathematica program available to anyone who's interested.

What is the value of $s(1)$?

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 $n \ge1,0 \le k\lt n$ , define

$B_{n,k}=[k/n, k/n+1/n)$.

Fix an integer $d \ge 0$ 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 $r \le s$ and each $k \lt r$, $B_{r,k}$ 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 amount 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.]

In a paper of more than 40 years ago: "Irregularities in the Distributions of Finite Sequences", E.R. Berlekamp 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 $n \ge1,0 \le k\lt n$ , define

$B_{n,k}=[k/n, k/n+1/n)$.

Fix an integer $d \ge 0$ 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 $r \le s$ and each $k \lt r$, $B_{r,k}$ 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 amount 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.]

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.]

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 $n \ge1,0 \le k\lt n$ , define

$B_{n,k}=[k/n, k/n+1/n)$.

Fix an integer $d \ge 0$ 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 $r \le s$ and each $k \lt r$, $B_{r,k}$ 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 amount 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.]