# 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

The applet may help you experiment with the problem. To the left of the axis that represents the segment

What if applet does not run? |

### References

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

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Copyright © 1996-2018 Alexander BogomolnyIn the solution section, Steinhaus proved a result by Schinzel that for

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

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

From [Guy] we get a solution for

- .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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Copyright © 1996-2018 Alexander Bogomolny68278817