Here is a little journey taken by one creative explorer:-
They decided on an easy start by taking:
3 x 8 + 4
The pattern went as follows:-
28
228
1828
14628
117028
936228
7489828
59918628
479349028
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:-
28 >1 228 >3 1828 >1 14628 >3 117028 >1 936228 >3 7489828 >1 59918628 >3 479349028 >1
So what happened when the starting number is changed?
4 x 8 + 4
4
36
292
2340
18724
149796
1198372
9586980
And the digital roots now go:-
4 36 >9 292 >4 2340 >9 18724 >4 149796 >9 1198372 >4 9586980 >9
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!
Bernard.