[ *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 |

n | multiplier | multiple |

**27** 81 | 37 12345679 | **999** 999999999 |

with EVEN digit restriction |

n | multiplier | multiple |

**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