Challenge Level

Here is a little journey taken by one creative explorer:-

They decided on an easy start by taking:

The pattern went as follows:-

2 | 8 | |||||||

2 | 2 | 8 | ||||||

1 | 8 | 2 | 8 | |||||

1 | 4 | 6 | 2 | 8 | ||||

1 | 1 | 7 | 0 | 2 | 8 | |||

9 | 3 | 6 | 2 | 2 | 8 | |||

7 | 4 | 8 | 9 | 8 | 2 | 8 | ||

5 | 9 | 9 | 1 | 8 | 6 | 2 | 8 | |

4 | 7 | 9 | 3 | 4 | 9 | 0 | 2 | 8 |

This was quite nice, after the first number [the chosen one] all the digits in the units column are 8, the tens are 2 and the digital roots go:- 1, 3, 1, 3, 1, 3, 1, 3, 1.

So what happened when the starting number is changed?

4 | |||||||

3 | 6 | ||||||

2 | 9 | 2 | |||||

2 | 3 | 4 | 0 | ||||

1 | 8 | 7 | 2 | 4 | |||

1 | 4 | 9 | 7 | 9 | 6 | ||

1 | 1 | 9 | 8 | 3 | 7 | 2 | |

9 | 5 | 8 | 6 | 9 | 8 | 0 |

And the digital roots now go:- 9, 4, 9, 4, 9, 4, 9, 4

Whoa! That's quite something. The units now go, after the first chosen number, 6204 and repeat.

Well let's summarize what the rest of the exploration gave :-

RULE | UNITS | DIG.ROOTS | OTHER COMMENTS |
---|---|---|---|

3 x 8 + 4 | 8.... | 1 3 1 3 1 3 ... | Tens > 2 Hunds > 2860.... |

4 x 8 + 4 | 6204 ... | 9 4 9 4 9 4 ... | |

5 x 8 + 4 | 4260 ... | 8 5 8 5 8 5 ... | |

6 x 8 + 4 | 2046 ... | 7 6 7 6 7 6 ... | |

7 x 8 + 4 | 0462 ... | 6 7 6 7 6 7 ... | |

8 x 8 + 4 | 8... | 5 8 5 8 5 8 ... | Tens > 6480 |

9 x 8 + 4 | 6204 ... | 4 9 4 9 4 9 ... | |

10 x 8 + 4 | 4620 ... | 3 1 3 1 3 1 ... | So we could predict that the next ones go:- |

11 x 8 + 4 | 2 ... | 2 ... | Dig. Roots should be 2, 11 but 11 is 2 so all are 2! |

12 x 8 + 4 | 0462 ... | 1 3 1 3 1 3 ... | |

13 x 8 + 4 | 8 ... | 9 4 9 4 9 4 ... | Checked, and Tens > 0648.... |

Well reading this has been very exciting and led Bernard to make a further change, using 7 instead of 8.

RULE | UNITS | DIG.ROOTS | OTHER COMMENTS |
---|---|---|---|

2 x 7 + 4 | 8024... | 945378612 .. | Tens > 1310 ... |

3 x 7 + 4 | 5973... | 786129453 .. | Tens > 2750 ... |

4 x 7 + 4 | 2804... | 537861294 .. | Tens > 3200 ... Hunds > All even |

5 x 7 + 4 | 9735... | 378612945 .. | Tens > 3740 ... |

6 x 7 + 4 | 6... | 129453786 .. | Tens > 4280 ... Hunds > 3200... |

7 x 7 + 4 | 3597... | 861294537 .. | Tens > 5720 ... |

8 x 7 + 4 | 0428... | 612945378 .. | Tens > 6270 ... |

9 x 7 + 4 | 7359... | 453786129 .. | Tens > 6710 ... |

10 x 7 + 4 | 4280... | 294537861 .. | Tens > 7251 ... |

11 x 7 + 4 | 1... | 945378612 .. | Tens > 8701 ... Hunds > 5000 Thous > Even |

There is so much here that is fascinating.

I hope that your explorations have gone well!