[ *December 27, 1999* ]

Find more **smallest** Squares and Cubes whose digits occur a same number of times ?

E.g. **10011**^{2} = 100220121 has exactly __three__ zero's, __three__ ones and __three__ two's !

Note that the number of different digits in the decimal expansion

may differ from their required frequency.

The next table already give the solutions for frequencies 1 to 7 for the squares.

Though the numbers are definitely 'base-related' they surely display a notable beauty !

Some questions I would like to ask are for instance :

Are there always solutions possible for any frequency ?

What are the possible shortcuts one can make while programming this puzzle ?

Frequency | Square Root | Square |

1 | 0 | **0** |

2 | 88 | **7744** |

3 | 10.011 | **100220121** |

4 | 31.646.191 | **1001481404808481** |

5 | 16.431.563 3.180.566.996 | **269996262622969** 10116006416044464016 is next one |

6 | 667.567.716 | **445646655445456656** by Jeff Heleen - *January 17, 2000* |

7 | 10.715.008.859 | **114811414848448481881** by Jeff Heleen - *January 20, 2000* |

8 | ? | **?** |

9 | ? | **?** |

10 | ? | **?** |

For the Cubes and higher Powers solutions

please visit my newest webpage solely dedicated to this interesting topic.

The title is Numbers whose digits occur with same frequency.