The “196 algorithm” is the process of starting with a positive integer and iteratively:
• reversing its digits,
• adding the reversed number to the previous number, and
• checking for a palindrome (the digits in the sum read the same left-to-right as right-to-left).
Most numbers lead to palindromes fairly quickly, like 235 (235 + 532 = 767), but some take a while (89 takes 24 steps). The process is named after 196 because that is the smallest positive integer not known to lead to a palindrome. The current record is 375 million digits! By that standard, my achievement of calculating 1000 iterations is positively paltry. But, here ‘tis nonetheless, in all of its 411-digit glory:
35346644392413689785837714402912114362859098083414
08344020861450405992918328457190349563871687958004
63971545914548326676428378028814710683108505496412
73388365259932008237493462055424091251579012001668
76923521977766210101074152201325440264395822899140
06246477437313605494900387117318730883824675524836
40965506947400858697069355944181744933808299515044
25811945379423290791058264410120303427728587887404
29334664452
(Scroll to the end of this post to see what this number is in words.)

One characteristic of this process is that it’s dependent on the base in which the number is expressed--using another base will give different results. In binary (base 2, digits 0 and 1), it turns out that twenty-two (10110) is the smallest number non known to lead to a palindrome.

What about negative numbers? By using a negative base, all integers can be represented without using a negative sign. For example, in base -10 (negadecimal), the places are not ones, tens, hundreds, thousands, etc., but ones, negative tens, hundreds, negative thousands, etc. So the number 29 in negadecimal is equal to -11 in decimal (2 × -10 + 9 × 1 = -20 + 9 = -11). Using this as starting point:
• 29 negadecimal = -11 decimal
• Reversing, 92 negadecimal = -88 decimal (9 × -10 + 2 × 1)
• Adding gives 1901 negadecimal or -99 decimal
• Reverse (1091 negadecimal, -1089 decimal)
• Add (-1188 decimal, 2992 negadecimal, palindrome!)
In base -10, 29 leads to a palindrome in two iterations.

And why stop there? How about complex numbers? Using certain complex bases allows one to express any complex number with integer real and imaginary parts (Gaussian integer) as a single unsigned string of digits. For example, base -1+i, (where i2 = -1), works with the digits 0 and 1, like binary. The decimal string “6” can also be expressed in binary as “110.” Interpreting this in base -1+i gives the complex number -1+i. Reversing the string to "011" gives 0+i. Adding yields -1+0i, or "11101." Continuing this process doesn’t seem to lead to a palindrome. After 100 iterations, the binary string is:
11101001011101001111001010011011010111011000100011
10010101101100010110001010100111011110110110001011
11001101000001101001111110110101001101011101011111
010010000
which converts to about -3.2119823593426 × 1022 + 1.10520083032793 × 1023 i.

More fun can be had by using others bases or by subtracting instead of adding. I'll talk more about that another time.

Oh, and here’s the (American) English version of that 411-digit number from above, courtesy of the “Name of a Number” site: