The method is to pencil in a line (bridge), mentally or physically, between a candidate in a cell to the same candidate in another cell, only once for each candidate, and each cell being visited by only two lines (bridges). The candidates that only appear in two cells should be bridged first because such bridges cannot be avoided (they are obligatory).
If there are no more than two obligatory bridges to any cell, then a complete circuit should be diligently sought. If successful, there will be n bridges bridging n candidates in a circuit around the n cells, with only two bridges visiting each cell. If a complete circuit is possible, it reliably identifies the n ordered multiple as undividable. It cannot be divided into sub-multiples.
Fig 1. Undividable 8 ordered multiple.
If there are more than two obligatory bridges to any cell (below), then one of the obligatory bridges can be led to an ‘isolated cell’ that does not form part of the circuit. The remaining candidates should be bridged in the usual way that bridges each candidate only once and each remaining cell is visited by only two bridges to complete a circuit. If successful, there will be n bridges bridging n candidates but the circuit itself will not visit the isolated cell. There will be (n-1) bridges in the actual circuit. If this circuit is possible, it reliably identifies that n ordered multiple as undividable into sub-multiples. If four obligatory bridges visit a cell, then there will be two isolated cells and the circuit will visit n-2 cells, etc.
Fig 2. More than two mandatory bridges to one cell
Fig 3. Isolated cell and a completed circuit. Undividable multiple
If the circuit cannot be completed (Fig 4) then the n ordered multiple is reliably identified as dividable into sub-multiples. The points of failure often indicate the likely submultiples. Circuits should then be completed, mentally or physically, for each of the sub-multiples.
Fig 4. No circuit can be completed. The 8 ordered multiple identified as dividable into sub-multiples.
The only candidate not bridged is 5, but bridging the 5 will identify the {2,5} hidden double. Completing the obligatory bridges first would have identified the {2,5} {2,5} double earlier.
Fig 5. Eight ordered multiple divided into a double and sextuple. m+p=n. 6+2=8.
After the unavoidable bridges have been made, for example, 2 and 5 above), continue with a cell containing the fewest candidates since then the ‘return’ candidates for that cell are known and can be reserved till last. For example, above, start with the 1 in the sixth cell {1,4,8}, then avoid using 4 or 8, reserving them for the return path to the sixth cell. There are submultiples of order 2 and 6.
Fig 6.
The bridges for 7 and 8 (above) are obligatory. Further bridges can be attempted, but at this point it can be shown that the full circuit cannot be completed. Continuing with the 1, 3 or 4 will make it impossible to bridge the 6 candidate (since only one 6 will remain ‘available’), and bridging the 6 candidate now will complete a circuit of only 6, 7 and 8; not the full circuit, identifying 6, 7 and 8 as a hidden triple.
Once the other candidates are removed from the 6, 7, and 8 triple, the candidates 4 and 5 are each present in two cells only, but the same two cells. These two bridges are unavoidable (mandatory) and these cells make a hidden double. The remaining cells make a naked triple (1, 2, and 3. It has always been there).
Fig 7.
Three Helpful Facts. Firstly, a single can be considered a multiple of order 1. Secondly, if an n ordered multiple can be divided, it will be divided into two submultiples of order m and p, where m + p = n. So, the search is aways for two submultiples, and their orders will add up to the original multiple’s order. And finally, if there are sub-multiples then there must always be at least one naked submultiple present. Submultiples cannot all be hidden because hidden multiples do not allow their candidates to exist in other cells, and so cannot ‘hide’ anything. Only a naked multiple allows its candidates to exist in other cells and hide other multiples.
The Theory of the Method will be added below, once this post has been uploaded to the Forum.
