induction_chart

m copies of the original n-die set |
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0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 18 | 20 | 24 | 30 | ||

n | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

1 | - | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | |

2 | - | - | 6 | 4 | 6 | 12 | 4 | 12 | 6 | 4 | 6 | 12 | 4 | 12 | 6 | 4 | 6 | 4 | 6 | 4 | 4 | |

3 | - | - | 6 | 8 | 6 | 24 | 6 | 24 | 6 | 8 | 6 | 24 | 6 | 24 | 6 | 8 | 6 | 6 | 6 | 6 | 6 | |

4 | - | - | - | - | 90 | 144 | 20 | 720 ? | 45 | 80 | 36 | 20 | 20 | 20 | ||||||||

5 | - | - | - | - | 90 | 288 | 20 | 1440 | 45 | 160 | 36 | |||||||||||

6 | - | - | - | - | - | - | 840 | 8640 ? | 945 | ? | ? | |||||||||||

7 | - | - | - | - | - | - | 840 | 17280 ? | 945 | ? | ? | |||||||||||

8 | - | - | - | - | - | - | - | - | ? | ? | ? |

Bold numbers highlight that those numbers along the diagonal match the sums of the rows of the triangular array of Cotesian Numbers. Why this is so is not clear.

If you have an *n*-player, permutation-fair set, this chart can help you construct an (*n+1*)-player, permutation-fair set. To do so, you will need to make *m* number of copies of the alphabetic string representing the original set, and in between those copies insert some number of repeated copies of the new letter representing the (*n+1*)^{th} side being added. How many copies of that new letter you need to insert at various points are indicated by the schematics found by clicking on the numbers in the cross-referenced *m*-*n* cell in the chart above (the numbers in the cells represent the lowest total number of sides you will need to insert, and the linked schematics will all add up to that number).

Suppose you start with the 3-player permutation-fair set: `cbabccccbabc`

(d2+d4+d6). Looking along the *n*=3 row of the chart, you see you can make 2 copies of that set and insert 6 copies of the letter `d`

between those copies (this is in column *m*=2 of row *n*=3 where a “6” exists). Clicking on that “6” shows you the schematic of *how* to insert those six copies of `d`

. Specifically: `1+4+1`

.

Meaning, the two copies of the original set go where the plus signs are in the schematic, and the numbers indicate how many sequential `d`

s are in between those copies. In this case the outcome is:

`d cbabccccbabc dddd cbabccccbabc d`

which is now a 4-player permutation-fair set (d4+d6+d8+d12).

Most entries in the chart link to a list of possible schematics (not just a single one like in the example above). Any of the schematics may be used (and not all are palindromic). The listed schematics all total to the number in the chart cell, and this is the lowest number for a given *m*-*n* combinations.

induction_chart.txt · Last modified: 2023/09/18 13:47 by paulmeyer