# CTM1764

The significance of "NUMBERS # CTM 1764" has yet to be determined.

The prefix 'CTM' is unknown and 1764 has been considered to be a year, or number of years.

## Contents

### Kaprekar's Routine and Constant

A possible connection to 1764 is that it is the final step in 'Kaprekar's Routine' to finally form the notable 'Kaprekar's Constant' = 6174. When applied to 4 digit numbers, this routine will always reach its fixed point, 6174, in at most 7 iterations:

1.Take any four-digit number, using at least two different digits. (Leading zeros are allowed.) 2.Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary. 3.Subtract the smaller number from the bigger number. 4.Go back to step 2.

Once 6174 is reached, the process will continue yielding 7641 – 1467 = 6174.

For example, choose 3524: 5432 – 2345 = 30878730 – 0378 = 83528532 – 2358 = 61747641 – 1467 = 6174

The only four-digit numbers for which Kaprekar's routine does not reach 6174 are repdigits such as 1111, which give the result 0 after a single iteration. All other four-digit numbers eventually reach 6174 if leading zeros are used to keep the number of digits at 4.

The sequence of Fixed points of the Kaprekar mapping f(n) = n' - n, where in n' the digits of n are arranged in descending, in n in ascending order can be found here - http://oeis.org/A099009:

0, 495, 6174, 549945, 631764, 63317664, 97508421, 554999445, 864197532, 6333176664, 9753086421, 9975084201, 86431976532, 555499994445, 633331766664, 975330866421, 997530864201, 999750842001, 8643319766532, 63333317666664

It can be noted that all numbers (mod 9) in this sequence reduce to zero in a similar manner to other parts of the 432 Mystery.

The figure linked below (similar to that appearing on the cover of the above issue of The Mathematics Teacher) shows the number of steps required for the Kaprekar routine to reach a fixed point for values of n=0 to 9999, partitioned into rows of length 100 (Deutsch and Goldman 2004). In this plot, numbers having fewer than 4 digits are padded with leading 0s, thus resulting in all values converging to 6174: ### 9

Using this site, plugging CTM brings up a digital root of 9. 1764 also brings up a digital root of nine.

Note that "digital root" is misleading, because most numbers with a large number of factors have a "digital root" of 9. This sort of numerological evidence should not be given much weight in the hypothesis space.

### Incomplete Message Theory

It's probable that the message is incomplete, and can be completed by understanding what "NUMBERS #" means. By grouping the message into two, we have (NUMBERS #)(CTM 1764). The creator gave the second half of the message. In this format, it would suggest that NUMBERS=CTM and #=1764, meaning that CTM could be a numerical value in letter form. It's not yet been determined what numerical base should be used to understand this (aside from using the digital root method above). If understood, the first half of the message could be solved which would reveal the entire message.

An example of how  letters can be represented numerically. In that link, we see that base 34 = CTM in a specific numeral system for the positive integer 14480.