World!Of
Numbers

WON plate
111 |

[ July 16, 2001 ]
Analogical Reasoning and The Power of Patterns
Klaus Brockhaus (email) and pdg

At my occupation with sequences of the smallest multiplier of n
which produces a palindrome with 'even' digits (e.g. A061797, A061916,
A062293, A061915), I obtained a result for n = 81 (beyond the last entry
shown in the database) by "analogical reasoning" (i.e. guessing) which
may be of interest to you.

without restriction
nmultipliermultiple
27
81
37
12345679
999
999999999
with EVEN digit restriction
nmultipliermultiple
27
81
329144
10973936897122085048
8886888
888888888666888888888

Of course it is only a conjecture, that 888888888666888888888 is
indeed the smallest palindromic multiple of 81 with EVEN digits,
but the above pattern
- trebling the digits when passing from 27 to 81 -
makes it a very plausible one.

Inspired by finding the real smallest palindrome with EVEN digits
divisible by 81 or wanting to confirm the above conjecture
I wrote Klaus the following discovery:

I can now say that your (though nice) conjecture doesn't hold
any longer as I found solutions - that are just a tiny bit smaller.
I derived them from some pattern exploration and was lucky :

81 *  8.504.801.097.146.776.406 = 688.888.888.868.888.888.886
81 * 10.727.023.319.368.998.628 = 868.888.888.868.888.888.868
81 * 10.949.245.541.591.220.848 = 886.888.888.868.888.888.688
81 * 10.971.467.763.813.443.048 = 888.688.888.868.888.886.888
81 * 10.973.689.986.035.665.048 = 888.868.888.868.888.868.888
81 * 10.973.912.208.257.885.048 = 888.886.888.868.888.688.888
81 * 10.973.934.430.480.085.048 = 888.888.688.868.886.888.888
81 * 10.973.936.652.702.085.048 = 888.888.868.868.868.888.888
81 * 10.973.936.874.922.085.048 = 888.888.886.868.688.888.888
81 * 10.973.936.897.122.085.048 = 888.888.888.666.888.888.888

Note how the three 6's come closer together to form the Number of the Beast.
The pattern gets more and more beautiful! Let me replace the digits 8 by
dots and do away with the thousand points to reveal the pattern:

6.........6.........6
.6........6........6.
..6.......6.......6..
...6......6......6...
....6.....6.....6....
.....6....6....6.....
......6...6...6......
.......6..6..6.......
........6.6.6........
.........666.........

"The power of patterns" never to be underestimated !

The new smallest palindrome with EVEN digits divisible by 81 now became
688888888868888888886
Beware however that it is not proven that this is effectively the smallest
possible solution. There is still more research to do!

If you liked the topic of this WONplate then no doubt you'll
be interested also in the next two dealing with the same subject :
WONplate 36
WONplate 96

A000111 Prime Curios! Prime Puzzle
Wikipedia 111 Le nombre 111
```

```