Steinhaus' Problem

In his popular book, One Hundred Problems in Elementary Mathematics, the famous Polish mathematician Hugo Steinhaus posed the problem (#6 in my Russian translation) of finding 10 numbers in the closed interval [0, 1], such that the first two are in different halves of the interval, the first three are in different thirds, ..., and all of them are in different tenths of the interval. He also asked (problem #7) whether number 10 can be replaced by a larger number.

The applet may help you experiment with the problem. To the left of the axis that represents the segment [0, 1] there is a box from which one can drag the indices and place them on the line. Their positions on the line will be automatically displayed above the line.

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Examples

References

H. Steinhaus, One Hundred Problems in Elementary Mathematics, Dover Publications, 1979

In the solution section, Steinhaus proved a result by Schinzel that for n = 75 no such sequence of numbers exists. Finally, in a footnote he mentioned that M. Warmus has proved that n = 17 is the largest number for which the problem has a solution.

Berlekamp and Graham proved in 1970 a generalization that implies M. Warmus' theorem. R. Guy has included an example in his 1990 Mathematics Magazine article.

The story with a solution for n = 14 has been included into the section devoted to number 17 in J. Roberts' Lure of the Integers.

For n = 10, Steinhaus gives two solutions:

.95, .05, .34, .74, .58, .17, .45, .87, .26, .66
.06, .55, .77, .39, .96, .28, .64, .13, .88, .48

The latter sequence can be augmented by .19, .71, .35, .82 to give a solution for n = 14. A rearranged system also solves the problem for n = 14:

.19, .96, .55, .39, .77, .06, .64, .28, .88, .48, .13, .71, .35, .82.

From [Guy] we get a solution for n = 17:

.71, .09, .42, .85, .27, .54, .925, .17, .62, .355, .78, .03, .48, .97, .22, .66, .32.

References

E. R. Berlekamp, R. L. Graham, Irregularities in the distribution of finite sequences, J. Number Theory, v 2 (1970), pp. 152-161
R. Guy, The second strong law of small numbers, Math Magazine, v 63 (1990), pp. 3-20
J. Roberts, Lure of The Numbers, MAA, 1992, p. 132
H. Steinhaus, One Hundred Problems in Elementary Mathematics, Dover Publications, 1979

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