Counting the Palindromes 

All the digits are palindromes (1,2,3,...,9).

There are also 9 palindromes with two digits(11,22,33, ...,99).

You can find to every two-digit number one, and only one number with three digits and with four digits.
For example: For the number 34 there are 343 and 3443.
You can conclude that there are 90 palindromes withthree and also 90 palindromes with four digits.

You can find to every three-digit number one, and only one number with five digits and with six digits.
For example: To the number 562 there are 56265 and 562265.
You can conclude that there are 900 palindromes withfive and 900 palindromes with six digits.

You have 9+9+90+90+900+900 = 1998 palindromes up to one million. That's 0,1998 %. About every 500th number is a palindrome. 

Position of the Palindromes top
But they are not spread over all numbers regularly. This shows the following picture, which includes the first 1000 numbers.

And so it goes on.
Part of the 1000x1000-diagram:

Multiples of 9  top

Strange Equations top

Products with the Digit 1 top

I suppose that all products with the digit 1 are palindromes, if one.factor has at the most 9 digits. 
All palindromes have the shape 123.....321.

Squares among the Palindromes top

121=11² 484=22² 676=26² 10201=101² 12321=111²

14641=121²  40804=202²  44944=212²  69696=264²  94249=307²

698896=836²  1002001=1001²  1234321=1111²  4008004=2002²  5221225=2285²

6948496=2636²  123454321=11111² . . ..

Cube Numbers among the Palindromes top
343=7³                1331=11³       1030301=101³           1367631=111³ 

Prime Numbers among the Palindromes top
All palindromic primes with 3 digits:

101 131 151 181 191

313 353 373 383 .

727 757 787 797 .

919 929 . . .

There are no primes with 4 digits. They all have the factor 11. (Example:4554=4004+550=4x1001+550=4x91x11+11x50=11x(4x91+50)
There are 93 primes with 5 digits.
There are no primes with 6 digits. They all have the factor 11.
There are 668 primes with 7 digits.

Products of Successive Numbers  top

16x17 =  272 77x78 = 6006 538x539 = 289982 1621x1622 = 2629262 2457x2458 = 6039306

77x78x79 = 474474 . . . .

Products  top

12x42 = 24x21 12x63 = 36x21 12x84 = 48x21 13x62 = 26x31

13x93 = 39x31 14x82 = 28x41 23x64 = 46x32 23x96 = 69x32

24x63 = 36x42 24x84 = 48x42 26x93 = 39x62

34x86 = 68x43 36x84 = 48x63 46x96 = 69x64

Pairs of Squares top

12² = 144 and 21² = 441  13² = 169 and 31² = 961 . .

102²=10404 and 201²=40401 103²=10609 and 301²=90601 112²=12544 and 211²=44521 113²=12769 and 311²=96721

1012²=1024144 and 2101²=4414201 1112²=1236544 and 2111²=4456321 1212²=1468944 and 2121²=4498641 2012²=4048144 and 2102²=4418404

Palindromic Dates top
The only palindromic year was 1991 in the last century. 
The only palindromic year is 2002 in this century. If you type in 2002 to a calculator and turn over, 2002 will stay ;-). 

196-Problem   top
Pick a number. Add the number, which you must read from the right to the left (mirror number), to the original number. Maybe the sum is a palindrome. If the sum isn't a palindrome, add the mirror number of the sum to the sum. Maybe you have a palindrome now, otherwise repeat the process. Nearly all numbers have a palindrome in the end. 
Example: 49       49+94=143       143+341= 484 !
There are some numbers, which have no palindromes. The lowest one is 196. But the proof is still missing. 

References   top
(1) Walter Lietzmann, Sonderlinge im Reich der Zahlen, Bonn, 1947
(2) Walter Sperling, Auf du und du mit Zahlen, Rüschlikon-Zürich, 1955
(3) Erwein Flachsel, Hundertfünfzig Mathe-Rätsel, Stuttgart 1982, Seite 138 f. 
(4) Martin Gardner, Mathematischer Zirkus, Berlin 1988, Seite 259 ff.

Palindromes on the Internet     top

English

Chip Burkitt
Reversible Factors and Multiples

Eric W. Weisstein (MathWorld)
Palindromic Number

Jason Doucette
196 Palindrome Quest

John Walker
Three Years Of Computing (Final Report On The Palindrome Quest, May 25th, 1990) 

MathPages
On General Palindromic Numbers

Patrick De Geest
Palindromes

Wikipedia
Palindromic number,  Palindromic prime,  Emirp, Lychrel number, Palindrome

Hans-Jürgen Caspar
Palindromzahlen

Karl Hovekamp
Palindromzahlen in adischen Zahlensystemen

Silvan Maaß
Palindrom aller Art

Wikipedia
Zahlenpalindrom, Primzahlpalindrom, Mirpzahl, Lychrel-Zahl, Palindrom