**Palindromes On The
12 hour Clock**

** **

**Ken Wais**

**9/3/15**

**What are 12 hour Clock Palindromes?**

** **

**It is common knowledge that the 12 hour
cycle of a clock leads to palindromic configurations on a regular basis.
I am becoming fascinated by how these palindromes occur and would like to
explore the attributes of them. With this in mind let us look at the
structure of palindromes that occur as a 12 hour clock proceeds through its
cycles.**

**I will define two terms to divide the
palindromes on the 12 hour clock set:**

**A full palindrome is one in which the
numbers are all the same, like 1:11, 2:22, 4:44. **

**A half palindrome is one which the numbers
form a palindrome by an alternating sequence of x and y variables, where x and y
are integers from 0 to 9. Examples would be xyx, for instance 1:21, or
3:13, or 4:24. If we don’t allow 0 to be considered an initial place
holder, there are only 2 four digit half palindromes, e.g. 10:01 and 12:21 for
the 12 hour clock.**

**Palindromic Arithmetic**

**There is an ongoing challenge to discover
if any pair of integers when reversed and added will eventually generate a
palindrome. If we take integers larger than 2 digits many quickly become
palindromes, while a few like 196 don’t. In fact 196 is one fundamental
example of a number resisting converging to a palindrome. A term called a
Lychrel number was coined by Wade VanLandingham. VanLandingham is famous
for calculating to 300 million digits, the integer 196 without the occurrence
of palindrome. Visit his site here for a deeper look at this usual integer. http://www.p196.org/**

**a**** Lychrel
number is one that does not converge to a palindrome after extremely long
iterations of reversed addition. The
problem with 196, and other integers that occur like 196, is there is no
deductive methodology to show that a number like 196 will become a
palindrome. That is, there is no constructive proof of the conjecture
that some integers never form palindromes if they are reversed and added
repetitively. A mathematician
would call this a decision procedure.
Happily, in this pursuit, I’m not trying to construct a proof of palindromic
occurrence due to the operation of addition of reverse integers. What I want to investigate is if we can derive
palindromic sets from a 12 hour clock, what happens if we take the output and continue the process? This kind of feedback process occurs in abstract algebra all the time.
**

**Look at the dichotomy of palindromic types
specified above. All the possible
palindromic sequences for the 12 hour clock can be cataloged as an initial
set. Since time-keeping on the 12 hour clock is structured such that we
won’t use zero (0) as a placeholder, many palindromic sequences that could be
used will not, like for instance 03:30, 01:10, 02:20, etc. Still, there
are many possible palindromic combinations. Moreover, since we are really
restricting our domain to a limited set of numbers, due to the nature of addition
for the 12 hour clock, we have a very neat and predictable collection of
palindromic numbers. But, not if we expand the derived sets to include
two forms of addition. From this point
on, I will abandon consideration of palindromic types as specified above, not
relevant at present, but maybe later, since this is a living document.**

** **

**12 hour Clock Additions that Generate
Surjective Sets.**

**Look at any given hour on the 12 hour
clock. For example, 1 o’clock to 2 o’clock. If we tabulate it,
we’ll find that there are a definite set of palindromes in this interval.
If we do this for every hour on the 12 hour clock we have the following
tabulation:**

·
**57
minutes are palindromic for the hours from 1 AM to the following 1 AM **

· **The
above interval is a 12 hour period.
Within this interval the following palindromes occur:**

o
**2
minutes have a palindrome for the hours 10 to 12.**

o
**1
minute has a palindrome for the hour 12 am to 1 am.**

·
**54
minutes are palindromic from 1 am 9 pm, which tabulates to 6 per hour (9
hours).**

**We can see that for any 12 hour period, 57
minutes are palindromic. What is becoming much more interesting now, is
the surjective sets that form, if we take the palindromic output of each hour
starting with 1 am to the succeeding 1 am and under the operation of addition,
combine them with the previous palindromic number. This would be taking
the Fibonacci numbers of the 12 hour clock times. In parallel to this, we
can add the palindromic numbers that occur hour-to-hour. Two different
output sets of palindromes will be derived and may well have some cyclic
properties. These sets are surjective mappings because the original set
is being mapped to elements within it to create a new non-isomorphic set.
It is a mapping ONTO a new set that the operation of addition (+) creates from
the original set. Some algebraic sets are surjective. The natural
and base 10 logarithms are surjective. So is modular arithmetic. It
is important to note that, algebraically this is a surjective map and not an
injective or bijective map. In the latter cases we are relating sets by
different interpretations of set mapping. A bijective function simply
forms an isomorphic map between elements of two sets, like 2↔B.
Where the two elements are in one-to-one correspondence to each other. An
injective map would map UNIQUE elements of one set to another via
isomorphism. Please note a bijective function maps the same elements of a
set to each other more than once without violating the notion of a
function. They just have to be the same elements mapped. 9 Mod 3 =0 is a bijective map that maps 3 to
itself thrice both ways. But a surjective maps does violate the notion of
a function in that it can map the SAME element of one set to differing elements
of the output set. And this is what we should expect on the 12 hour
clock. 3:13 AM is identical 3:13 PM numerically, so its image in a new
set related by addition and would be a mapping to the same elements as on the
original 12 hour clock.**

** **

**I’ve begun an Excel worksheet to do these
operations. The worksheet so far
has derived 14 additive palindromes and 11 Fibonacci palindromes. This is
exciting because there is no way to tell where this operation will go. This
sheet was simple to construct and it was manually tabulated. As mentioned above, we have a small
well-ordered set (not in the ordinal set theory sense of that phrase) so I didn’t
need to write macros to get results. Also, no need to use VB for Excel 2013
yet. If I go through a 2 ^{nd} iteration of the derived
palindromes, there be more? What if I mix the palindromic types by
addition? What if I take the derived palindromes and recombined them with
the originals? The permutations are many
as interested readers might have guessed.
This is an unfinished endeavor, and I plan to go further with the simple
excel sheet in the future. My intuition is iterative operations on these
sets might lead to infinite palindromic reoccurrence.**

**One last comment for now, I’ve purposely
avoided going into an expansive narrative on the history of palindromic
integers in mathematics. That would
distract from what I consider a new approach to palindromic integral
derivations. Those interested in the
wider topic can certainly find enough material about it on the Net. If this short essay encourages that sort of
exploration, then I’m doubly pleased.**

**You can download this file by clicking the
icon below. Update: this file is now a macro-embedded file that has calculated palindromes on the 12 clock to the 3rd iteration, and counting...**

** Excel file for 12 Hour Clock Palindromes Palindromes on the 12 Hour Clock**

** **

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