I just sort of winged it on scale. 30 pegs for each power of 10 seemed like a reasonable number. But I decided to go back and investigate, and looked at all sizes between 30 and 50. (50 horizontal lego units = 400mm.) Strangely, there does seem to be a scale which is a much better than others within this range! I assumed that each number was rounded to the nearest brick (although there are easy ways to get half-bricks). The metric I used was root mean square error for numbers 1-10, 12, 14, 15, 16, 18, 20, and 21. (i.e., all numbers <= 10, and all non-prime numbers up to 21. I picked 21 as the cutoff because even at scale 50 the difference between 22 and 21 is just one brick.)

A scale of 40 lego units for each power of 10 produces RMS of 0.015, while almost all other scales within this range are above 0.06. 46 has error 0.053, and 50 has error 0.027. So 40 is by far the best choice.

This scale looks like:

Number | Position |
---|---|

1 | 0 |

2 | 12 |

3 | 19 |

4 | 24 |

5 | 28 |

6 | 31 |

7 | 34 |

8 | 36 |

9 | 38 |

10 | 40 |

11 | 42 |

12 | 43 |

13 | 45 |

14 | 46 |

15 | 47 |

16 | 48 |

17 | 49 |

18 | 50 |

19 | 51 |

20 | 52 |

21 | 53 |

22 | 54 |

23 | 54 |

So 23 is the first ambiguous number. 24 and 25 occupy separate positions, but all numbers past 26 share a position.

If we label every ambiguous point with its largest number, then of the products 2x2 through 9x9, we get 6 wrong: 4x7 = 29, 5x7 = 36, 6x7 = 43, 7x7=51, 7x8 = 57, 7x9 = 64. Of these we can fix 4x7, 6x7, 7x7, and 7x8 by relabeling (though 7x7 comes at the cost of getting 5x10 wrong.) Changing 5x7 and 7x9 won't work because that point is also the result of 6x6 and 8x8.

7 seems to be a problematic number because log(7)*40 = 33.8, so we have to round by 0.2, while most other approximations are smaller than this, and most are rounded down instead of up.

TODO: figure out a scale size at which all single-digit multiplications work.

TODO: the book describes an alternate system of slide rules based on triangular numbers. Does this convert into Lego better or worse?

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