Monday, May 13, 2013

On palindromic numbers, part two

In my last post on palindromic numbers, I looked at the frequencies of valid & invalid palindromic numbers and found that almost all invalid palindromes are odd. To explore why this might be, I'll now look at the differences between even and odd palindromes. (By this I mean even and odd numbers, $$n$$, that have palindromic representations in a base less than $$n - 1$$.)

Of evens and odds

Of the 4997 valid even palindromes through 10,000, over 62% have a length of 3 digits and just under 20% have a length of 2 digits. Palindromes of lengths ranging from 4 to 9 comprise the nearly 18% remaining. This is somewhat similar to the odds: just over 70% have length 3, a little less than 14% have length 2, and the bit more than 16% remaining have lengths ranging from 4 to 14. So the odds are spread out a bit more as far as lengths go, but they and the evens are fairly similar in that regard.

What about the frequency of bases? Do the evens have more or less of certain bases or ranges of bases? Well, as far as number of bases, the evens have a slightly higher (a little over 5.5%) diversity of (valid, lowest) bases over the first 10,000 counting numbers. I don't think that's enough to be significant, though (I'd love to be proven wrong in the comments!), so let's look at the frequencies of individual bases:

Now here's where it gets fascinating. For the lower bases, the evens have much more odd bases and the odds have much more even bases. (Why do evens have no base-2 palindromes? A binary palindrome would have to end in a '1' — allowing leading zeroes would cause all sorts of problems — and binary numbers that end in '1' are odd.) In general, the larger the base, the lower the frequency, but around base-20, both the evens and odds see an upswing in frequencies of all bases that gradually tapers down to about base-50. It's here that we see a recurrence of the original pattern, albeit at a much smaller scale. (There is also a small uptick at base-28 for both evens and odds, and the high/low pattern makes a brief appearance in the early thirties.)

Can you take me higher?

Obviously, this particular patterning is the result of the limited number set we're working with. Does it continue for larger number groups?

It seems so. (With one order-of-magnitude increase, at least.)

More pointedly, the early alternating pattern continues, as does the overall swelling later and the return to the alternating pattern. But— there are some hiccups in the alternating pattern now, the swelling occurs later (and is now responsible for the highest frequencies), and the later alternation is less pronounced.

I won't check higher numbers because it takes quite some time to compute these numbers (at least with the quick script I wrote for the task, and I don't wish to spend the time to see if I can make a faster program). However, if the trends I see continue, it looks like the alternating pattern becomes most significant as the upper number limit approaches infinity. (I base this on the alternations continuing mostly unhindered and the swelling moving into higher ranges. As we approach an infinite upper bound, it would follow that the swelling would approach infinity as well.) So why are even numbers likely to have their lowest-base palindrome be in an odd base and vice-versa for the odds?

I don't have an answer to that question. I'd be interested if any readers did. In the meantime, I'll look at something I noticed about one of my first lists: the invalid palindromes. For a refresher, let's look at those, this time through 10,000:

• 1
• 2
• 3
• 4
• 6
• 11
• 19
• 47
• 53
• 79
• 103
• 137
• 139
• 149
• 163
• 167
• 179
• 223
• 263
• 269
• 283
• 293
• 311
• 317
• 347
• 359
• 367
• 389
• 439
• 491
• 563
• 569
• 593
• 607
• 659
• 739
• 827
• 853
• 877
• 977
• 983
• 997
• 1019
• 1049
• 1061
• 1187
• 1213
• 1237
• 1367
• 1433
• 1439
• 1447
• 1459
• 1511
• 1553
• 1579
• 1669
• 1709
• 1753
• 1759
• 1907
• 1949
• 1993
• 1997
• 2011
• 2063
• 2087
• 2099
• 2111
• 2137
• 2179
• 2207
• 2287
• 2309
• 2339
• 2417
• 2459
• 2503
• 2657
• 2677
• 2683
• 2693
• 2713
• 2749
• 2897
• 2963
• 3023
• 3089
• 3119
• 3229
• 3253
• 3259
• 3323
• 3371
• 3407
• 3449
• 3547
• 3559
• 3583
• 3623
• 3643
• 3833
• 3847
• 4007
• 4073
• 4091
• 4099
• 4139
• 4157
• 4211
• 4283
• 4337
• 4339
• 4349
• 4391
• 4463
• 4523
• 4549
• 4643
• 4679
• 4729
• 4787
• 4871
• 4909
• 4919
• 4933
• 5011
• 5021
• 5039
• 5059
• 5099
• 5179
• 5231
• 5297
• 5303
• 5309
• 5351
• 5387
• 5417
• 5431
• 5471
• 5503
• 5527
• 5653
• 5693
• 5711
• 5791
• 5827
• 5839
• 5939
• 6047
• 6067
• 6079
• 6089
• 6131
• 6199
• 6229
• 6247
• 6269
• 6277
• 6311
• 6343
• 6359
• 6389
• 6551
• 6599
• 6653
• 6793
• 6871
• 6947
• 6983
• 6991
• 7019
• 7079
• 7159
• 7213
• 7247
• 7283
• 7433
• 7487
• 7691
• 7817
• 7877
• 7949
• 7963
• 8017
• 8069
• 8089
• 8123
• 8147
• 8221
• 8243
• 8287
• 8291
• 8293
• 8423
• 8539
• 8573
• 8669
• 8699
• 8783
• 8863
• 8941
• 9043
• 9059
• 9067
• 9173
• 9209
• 9227
• 9277
• 9337
• 9341
• 9377
• 9419
• 9421
• 9533
• 9587
• 9643
• 9689
• 9739
• 9781
• 9887

I've highlighted the non-prime numbers. There are only three: 1, 4 and 6. Aha! So, not only are all invalid palindromes above 6 odd, they are all prime as well! (Assuming this continues beyond 10,000. I have no proof it does, but so far it works.)

Unfortunately, I have no idea where to go with this information. I can't see any patterns in the primes that do not have valid palindromes vs. those that do. Moreover, I have no clue of the mechanics behind a number having a palindrome in a particular base over another. We have some clues, but no full set of numbers that do or do not have valid palindromes. Hmm….

Sunday, February 24, 2013

The past two years, I've been keeping track of the books I've read. The following is the list for 2012, in the order they were finished. (I suppose it's more of a books I finished in 2012, disregarding when I started them.) This doesn't include magazines or comic books or fictional graphic novels, but it does include novels & other fiction, nonfiction books and nonfiction graphic novels. Basically, I should make a separate list for the more comic-booky books, I guess. But I haven't done that, and I'm not sure I want to make that big list, so take what you can get, I guess.

Making Money by Terry Pratchett
I love Terry Pratchett. What else can I say? This was another good book from the Discworld series.
Absurdistan by Gary Shteyngart
This is a silly (in a good way) and then surprisingly meaningful and emotional (right at the end) book that we picked up from the giveaway bin at the library.
The Book of Three by Lloyd Alexander
I'm pretty sure I hadn't read any of this series until this year. Possibly The High King because I have a copy from when I was young, but I can't remember reading it then. I sought to amend that this year. Anyway, it's a good start to the series, but only the third best out of five.
The Last Testament: A Memoir by God by David Javerbaum
This is straight-up funny and irreverent. It pokes at many religions, especially the Judeo-Christian ones. Don't think Christianity is singled out — in fact, there's quite a bit of love and/or admiration for it.
The Black Cauldron by Lloyd Alexander
The best book of the series. Hands down.
Nurk by Ursula Vernon
I really enjoyed this fun little adventure of a shy, hermit-like shrew opening up a bit and exploring life. You may notice quite a bit of children's/young adult books here. Partly that's where I hang out in the library most of the time (I have three children, none of whom can yet be labeled teenagers); and partly I just really enjoy a good book written for the younger set. This was one of 'em.
The Castle of Llyr by Lloyd Alexander
Perhaps you'll notice it seems I took a break between the books of this series. I didn't. I just often read two books at once (usually one at home and one at work).
Taran Wanderer by Lloyd Alexander
This and the previous book are the weakest of the series, but they are still very good books. If you're strapped for time, though, you can get away with skipping these two, though. They're more side-quests than main story tales. This one's basically (the main character of the series) Taran's coming-of-age book. It's where he goes from being a pig boy to a man.
George Washington, Spymaster: How the Americans Outspied the British and Won the Revolutionary War by Thomas B. Allen
This was a really good, really interesting book. I had no idea how much espionage factored into the Revolutionary War. Highly recommended, especially for older elementary and middle school kids, as that's the intended audience. There's a big list of recommended reading (that I really should check out sometime) with adult books in the same vein.
The High King by Lloyd Alexander
Second-best book of the series. He closes it out well, with the expected vanquishing of evil. I still think The Black Cauldron is the best book, even though this is the one that won the Newbery. This is more Superman, while The Black Cauldron is more Batman, if that makes sense to you.
What Would Buffy Do?: The Vampire Slayer as Spiritual Guide by Jana Riess
Yeah, so my wife and I are big Buffy fans. This is a more popular (as opposed to scholarly) look at the spirituality in the show. I enjoyed it even though (or perhaps because?) it's a bit of a light read. Which may sound strange given the topic, but I did say it was more of a popular look.
Captain Freedom: A Superhero's Quest for Truth, Justice, and the Celebrity He So Richly Deserves by G. Xaviwe Robillard
This is a funny book that I'd recommend to anyone who likes superheroes. The world of the book is well thought out and feels quite realistic as how our world might be if some people did have super powers. (Hint: follow the money!)
And here is a not superhero novel by someone I normally think of as a comic book writer. This actually seems to be one of a series of books about the main character, and I've come in towards the end of his story — meaning he's decided to settle down after going from being a bodyguard to being an international fugitive. I think his graphic novels Whiteout and Whiteout: Melt (which I think I read this year also) were better, but I enjoyed this, and I'm glad I found it and picked it up.
Feynman by Jim Ottaviani
This is, I think, the only graphic novel on this list. I've included it here because it's a biography, and arbitrary blah blah blah. It just seems different, okay? Anyway, this was a great read, very well done. I learned a lot about Feynman, and it made me want to read more about him, his work, and his life. If you don't know who he is, shame on you: He was a great physicist who worked on the Manhattan Project and figured out what happened in the Challenger disaster, among other things. He had a great way of relating to non-scientists, and he saw things in different ways than others: his diagrams are one of the greatest breakthroughs in helping people understand and work with quantum physics. Seriously, a great read.
I used to love Stephen King, partly because of the length of his works. I'm a sucker for an epic, and King's full of them. I still like him, and some of his books are among my favorites, but I've been able to see the flaws in his writing moreso lately, and maybe he's lost a bit in his later years. But I think he's still got it, and I like when he goes for the short form. It doesn't give him as much of a chance to go overboard and weird for weird's sake. This novella about baseball is one of the good ones. King just tells the story and is done.
The Hunger Games by Suzanne Collins
Catching Fire by Suzanne Collins
Mockingjay by Suzanne Collins
This is where I read nothing but the Hunger Games trilogy. And man, it was good. This quickly became one of my favorite book series. Most everything feels like it would actually happen that way; people would actually act and react that way. Very powerful, very emotional, very good. I can't recommend them enough.
Around the World in Eighty Days by Jules Verne
I have a bunch of old classics like this, and I haven't read many of them. (Okay, not so many anymore, but not until the last few years, at least.) This was fun with a clever twist at the end.
The Traitor's Gate by Avi
This is another one we picked up somewhere for cheap because, hey, it's Avi. It's a fun tale of mystery & intrigue set in mid-nineteenth century London. Highly recommended, especially for the younger set who like complex stories & characters and endings sans pretty little bows.
The Willoughbys by Lois Lowry
The Willoughbys is my favorite kind of book. It's very quirky with Monty Python-esque humor, and it's a very fun book. Another book aimed at teens & preteens but very enjoyable for anyone.
The Silent Boy by Lois Lowry
Without realizing it, I read two Lois Lowry books at the same time, this being the second. It was very easy to overlook this fact because the books are extremely different in every way. This is a rather painfully tragic story about a girl and her mentally handicapped neighbor. Well worth the read.
Nation by Terry Pratchett
Here we are again with a Terry Pratchett book, one I've been meaning to read for a while. I am so very glad I did. This was an excellent excellent book with a great meaning for me and an honest ending. (As opposed to the fairy-tale ending that many others would give, even though it wouldn't make a whole lot of sense.) I always enjoy Terry Pratchett's novels, but this is in the more select groups of those I absolutely love.
The Monsters of Templeton by Lauren Groff
This is the second book I read last year that includes the pregnancy of a student by her professor (the first was Absurdistan). I just thought that was interesting. We picked this book up (again, somewhere for cheap) based mostly on the beautiful cover art. It's a story of a young woman finding herself. In the process, she finds out quite a bit about her family history (hint: it's much more twisted — in more than one way — than the neat & somewhat wholesome story she thought). All-in-all, it was an enjoyable read.
Trackers by Patrick Carman
I'm sorry to say I was really disappointed in this book. I'd been meaning to read it for a while after glancing at it in the library. The first page or so gives so much promise — a teenage boy is being questioned about some mission of sorts that went awry. It's implied his friends didn't make it out — alive? healthy? In any case, things did not go well. It's all told in this transcript style, even directing you to a website with some video & other evidence in the case. Which was interesting, though somewhat gimmicky. (For me, anyway, but I could see myself at a younger age or the intended audience today finding it exciting.) The problem, however, is that this book is all exposition with little real payoff. It's the first act, which would be fine if this were a serial medium like comics. But the first book of a series should stand up on its own, and this does not. I got to end wondering where the rest of the pages went. I'm not even sure if I'll read the next book for closure or completeness purposes because of the bad taste this left in my mouth. Also, the very incorrect science in the book (mostly with how computers work) made me cringe throughout, especially considering the degrees to which the author goes to emphasize realism. This is probably the only book I read last year that I'd pass on given another opportunity to read. Seriously disappointing.
The Pearl by John Steinbeck
Where do you go from there? The classics. This had been on my mind (I forget why) and I hadn't read it since high school, so here we are again! What do I say? It's a classic; it ends tragically — what's not to like?
Televised Morality: The Case of Buffy the Vampire Slayer by Gregory Stevenson
Also, from there you go to comfort foods. This is a scholarly look (much more so than What Would Buffy Do?) at the morals of Buffy the Vampire Slayer. I didn't agree with all the conclusions, but it was an interesting read.
The Friendship Doll by Kirby Larson
And now, the last book I finished in 2012. This is another one I'd seen at the library for a while and wanted to read because of the lovely cover. (Yes, covers matter quite a bit. And yes, I like good art & good design.) I convinced my daughter to read it and she nearly begged me to read it, too, afterward so we could talk about it. It's a really sweet story about a Japanese friendship doll and the lives she touches over many years. The book is written in four parts, each about a different girl in a different place and time in America, and each part could stand as a wonderful (and sometimes painful) tale on their own. The doll ties them together, and overall this a sweet book which I greatly enjoyed.

Okay, finally done! That took much longer than I thought. (I started writing this post just after the new year.) Maybe I should write the blurbs as I go this year?

On palindromic numbers

Much of this is taken from my comment to a post titled Can every number be written as a palindrome in some base? on MathFour.com. The original post asks whether every number has a palindromic representation in some base.

What's a palindrome?

First, some definitions, for those who don't read the original post. A palindrome is read the same backwards and forwards. E.g.: racecar or A man, a plan, a canal: Panama. Note that only the letters are considered, not spaces or punctuation. A palindromic number has the same properties: 121 and 98,766,789 are both palindromes.

We commonly use base-10 numbers, but computers use base-2 (binary) and we encounter base-16 (hexadecimal) in computing, often to represent colors. The number of a base determines how many symbols are possible for each digit. (0-9 for base-10, 0 & 1 for base-2, 0-9 plus A-F for base-16, etc.) More to the point, it determines the value of each number-place. The first place is how many singles, the next is how many of the base value, and subsequent places are how many of increasing powers of the base value. In our everyday decimal, or base-10, numbers, this would be ones, tens (the base), hundreds ($$10^2$$), thousands ($$10^3$$), etc.

The original post asked whether all numbers could be written as a palindrome in some base, whether it be base-10 (121 for example) or another base. As it figures, every number can be written as a palindrome in the base one less than the number ($$n - 1$$) such as 3 being 11 in base-2. This is due to the definition of bases, where the first '1' is the base value (here, one less than the number) and the second '1' is the value one. In other words, $$(n - 1) + 1 = n$$. [Also note that in bases > $$n$$ the number will be a single digit. This, too, is a palindrome, albeit a lame one.]

A limited limit

So, what's a useful limiting factor? The original post looks at those numbers which have a palindrome with a base ≤ 10; i.e., one that would be written using only the digits 0–9. For the sake of completeness, let's look at those numbers that are invalid based on that definition (the original post included too many):

I find this to be too arbitrary. Yes, it's easier to look & think about for us humans who're used to base-10 numbers, but where's the fun there? (Also note that none of those invalid palindromes use digits > 9, highlighting the arbitrariness of the imposed limit.)

I tend more towards general solutions to problems, so I think a more interesting question is: What is the lowest base in which $$n$$ is a palindrome? (Obviously, we're omitting base-1 numbers as they are all palindromes.)

Also, since all numbers are palindromes in bases greater than $$n - 1$$ (excluding base-$$n$$), why not limit "valid" palindromes to those less than $$n - 1$$? So the question becomes: Does $$n$$ have a palindrome in a base less than base-$$(n - 1)$$? Is there a pattern to these numbers? Do they occur more, less or equally frequent as we look at larger sets of integers?

Python to the rescue!

Of course, computers make these things so much easier to look at. I made a quick Python script to examine palindromic numbers. The benefits here are I don't have to worry about inventing a character for each digit. I can just leave the decimal value of that digit as-is and place the digits in a list. So, here are some results for integers 1–100:

The full list (1–100)
Number Base Palindrome
1 2 [1]
2 3 [2]
3 2 [1, 1]
4 3 [1, 1]
5 2 [1, 0, 1]
6 5 [1, 1]
7 2 [1, 1, 1]
8 3 [2, 2]
9 2 [1, 0, 0, 1]
10 3 [1, 0, 1]
11 10 [1, 1]
12 5 [2, 2]
13 3 [1, 1, 1]
14 6 [2, 2]
15 2 [1, 1, 1, 1]
16 3 [1, 2, 1]
17 2 [1, 0, 0, 0, 1]
18 5 [3, 3]
19 18 [1, 1]
20 3 [2, 0, 2]
21 2 [1, 0, 1, 0, 1]
22 10 [2, 2]
23 3 [2, 1, 2]
24 5 [4, 4]
25 4 [1, 2, 1]
26 3 [2, 2, 2]
27 2 [1, 1, 0, 1, 1]
28 3 [1, 0, 0, 1]
29 4 [1, 3, 1]
30 9 [3, 3]
31 2 [1, 1, 1, 1, 1]
32 7 [4, 4]
33 2 [1, 0, 0, 0, 0, 1]
34 4 [2, 0, 2]
35 6 [5, 5]
36 5 [1, 2, 1]
37 6 [1, 0, 1]
38 4 [2, 1, 2]
39 12 [3, 3]
40 3 [1, 1, 1, 1]
41 5 [1, 3, 1]
42 4 [2, 2, 2]
43 6 [1, 1, 1]
44 10 [4, 4]
45 2 [1, 0, 1, 1, 0, 1]
46 4 [2, 3, 2]
47 46 [1, 1]
48 7 [6, 6]
49 6 [1, 2, 1]
50 7 [1, 0, 1]
51 2 [1, 1, 0, 0, 1, 1]
52 3 [1, 2, 2, 1]
53 52 [1, 1]
54 8 [6, 6]
55 4 [3, 1, 3]
56 3 [2, 0, 0, 2]
57 5 [2, 1, 2]
58 28 [2, 2]
59 4 [3, 2, 3]
60 9 [6, 6]
61 6 [1, 4, 1]
62 5 [2, 2, 2]
63 2 [1, 1, 1, 1, 1, 1]
64 7 [1, 2, 1]
65 2 [1, 0, 0, 0, 0, 0, 1]
66 10 [6, 6]
67 5 [2, 3, 2]
68 3 [2, 1, 1, 2]
69 22 [3, 3]
70 9 [7, 7]
71 7 [1, 3, 1]
72 5 [2, 4, 2]
73 2 [1, 0, 0, 1, 0, 0, 1]
74 6 [2, 0, 2]
75 14 [5, 5]
76 18 [4, 4]
77 10 [7, 7]
78 5 [3, 0, 3]
79 78 [1, 1]
80 3 [2, 2, 2, 2]
81 8 [1, 2, 1]
82 3 [1, 0, 0, 0, 1]
83 5 [3, 1, 3]
84 11 [7, 7]
85 2 [1, 0, 1, 0, 1, 0, 1]
86 6 [2, 2, 2]
87 28 [3, 3]
88 5 [3, 2, 3]
89 8 [1, 3, 1]
90 14 [6, 6]
91 3 [1, 0, 1, 0, 1]
92 6 [2, 3, 2]
93 2 [1, 0, 1, 1, 1, 0, 1]
94 46 [2, 2]
95 18 [5, 5]
96 11 [8, 8]
97 8 [1, 4, 1]
98 5 [3, 4, 3]
99 2 [1, 1, 0, 0, 0, 1, 1]
100 3 [1, 0, 2, 0, 1]

Now, "invalid" palindromes (note how few there are):

Base ≥ $$(n - 1)$$ (1–100)
Number Base Palindrome
1 2 [1]
2 3 [2]
3 2 [1, 1]
4 3 [1, 1]
6 5 [1, 1]
11 10 [1, 1]
19 18 [1, 1]
47 46 [1, 1]
53 52 [1, 1]
79 78 [1, 1]

Only 10% of numbers through 100 have only the default palindrome. Interestingly, none of these numbers use a digit greater than 8 in their palindromic representation. In fact … the first number to use 10 (or greater) in their lowest-base palindromic representation is 120 (base 11, [10, 10]). (108 is the first to use 9.)

What's the frequency?

How about the frequencies of these invalid palindromes? Here's a look at the percentages for each group of 100 numbers through 10,000:

Frequencies of invalid palindromes
Range %
1–100
10
101–200
7
201–300
5
301–400
6
401–500
2
501–600
3
601–700
2
701–800
1
801–900
3
901–1000
3
1001–1100
3
1101–1200
1
1201–1300
2
1301–1400
1
1401–1500
4
1501–1600
3
1601–1700
1
1701–1800
3
1801–1900
0
1901–2000
4
2001–2100
4
2101–2200
3
2201–2300
2
2301–2400
2
2401–2500
2
2501–2600
1
2601–2700
4
2701–2800
2
2801–2900
1
2901–3000
1
3001–3100
2
3101–3200
1
3201–3300
3
3301–3400
2
3401–3500
2
3501–3600
3
3601–3700
2
3701–3800
0
3801–3900
2
3901–4000
0
4001–4100
4
4101–4200
2
4201–4300
2
4301–4400
4
4401–4500
1
4501–4600
2
4601–4700
2
4701–4800
2
4801–4900
1
4901–5000
3
5001–5100
5
5101–5200
1
5201–5300
2
5301–5400
4
5401–5500
3
5501–5600
2
5601–5700
2
5701–5800
2
5801–5900
2
5901–6000
1
6001–6100
4
6101–6200
2
6201–6300
4
6301–6400
4
6401–6500
0
6501–6600
2
6601–6700
1
6701–6800
1
6801–6900
1
6901–7000
3
7001–7100
2
7101–7200
1
7201–7300
3
7301–7400
0
7401–7500
2
7501–7600
0
7601–7700
1
7701–7800
0
7801–7900
2
7901–8000
2
8001–8100
3
8101–8200
2
8201–8300
5
8301–8400
0
8401–8500
1
8501–8600
2
8601–8700
2
8701–8800
1
8801–8900
1
8901–9000
1
9001–9100
3
9101–9200
1
9201–9300
3
9301–9400
3
9401–9500
2
9501–9600
2
9601–9700
2
9701–9800
2
9801–9900
1
9901–10000
0

There are very few numbers without a valid palindromic representation. In fact, through 10,000 there are only 2.22 per hundred. While there are only 9 instances of no valid palindromes in the arbitrary 100-number blocks above, there are in fact 22 different stretches of triple-digit runs without an invalid palindrome, the longest of these being 203! This includes the 112-number run that finishes with 10,000 and possibly goes longer. Excluding this final, partial run, the mean length of a run is a little over 44 ⅓.

Runs of valid palindromes
Invalid number Valid numbers preceding
1
0
2
0
3
0
4
0
6
1
11
4
19
7
47
27
53
5
79
25
103
23
137
33
139
1
149
9
163
13
167
3
179
11
223
43
263
39
269
5
283
13
293
9
311
17
317
5
347
29
359
11
367
7
389
21
439
49
491
51
563
71
569
5
593
23
607
13
659
51
739
79
827
87
853
25
877
23
977
99
983
5
997
13
1019
21
1049
29
1061
11
1187
125
1213
25
1237
23
1367
129
1433
65
1439
5
1447
7
1459
11
1511
51
1553
41
1579
25
1669
89
1709
39
1753
43
1759
5
1907
147
1949
41
1993
43
1997
3
2011
13
2063
51
2087
23
2099
11
2111
11
2137
25
2179
41
2207
27
2287
79
2309
21
2339
29
2417
77
2459
41
2503
43
2657
153
2677
19
2683
5
2693
9
2713
19
2749
35
2897
147
2963
65
3023
59
3089
65
3119
29
3229
109
3253
23
3259
5
3323
63
3371
47
3407
35
3449
41
3547
97
3559
11
3583
23
3623
39
3643
19
3833
189
3847
13
4007
159
4073
65
4091
17
4099
7
4139
39
4157
17
4211
53
4283
71
4337
53
4339
1
4349
9
4391
41
4463
71
4523
59
4549
25
4643
93
4679
35
4729
49
4787
57
4871
83
4909
37
4919
9
4933
13
5011
77
5021
9
5039
17
5059
19
5099
39
5179
79
5231
51
5297
65
5303
5
5309
5
5351
41
5387
35
5417
29
5431
13
5471
39
5503
31
5527
23
5653
125
5693
39
5711
17
5791
79
5827
35
5839
11
5939
99
6047
107
6067
19
6079
11
6089
9
6131
41
6199
67
6229
29
6247
17
6269
21
6277
7
6311
33
6343
31
6359
15
6389
29
6551
161
6599
47
6653
53
6793
139
6871
77
6947
75
6983
35
6991
7
7019
27
7079
59
7159
79
7213
53
7247
33
7283
35
7433
149
7487
53
7691
203
7817
125
7877
59
7949
71
7963
13
8017
53
8069
51
8089
19
8123
33
8147
23
8221
73
8243
21
8287
43
8291
3
8293
1
8423
129
8539
115
8573
33
8669
95
8699
29
8783
83
8863
79
8941
77
9043
101
9059
15
9067
7
9173
105
9209
35
9227
17
9277
49
9337
59
9341
3
9377
35
9419
41
9421
1
9533
111
9587
53
9643
55
9689
45
9739
49
9781
41
9887
105

Looking at these two frequency graphs, it seems invalid palindromes, while a little more frequent in lower integers, occur otherwise without a discernible pattern.

Where do we go from here?

While we're having fun, let's look at some specific groups of numbers. Of the 222 invalid palindromes from 1–10,000, a staggering 219 are odd, with only 3 even. These are 2, 4 & 6 — the first three even numbers. Now we're getting somewhere — almost all invalid palindromes are odd! So why do almost all evens (assuming there continue to be none beyond 5,000,276 — the highest I've figured out so far through a brute force program running while I type this post — which is not certain) have valid palindromes? Maybe we'll get some insight if we examine the even palindromes. Join me in the next post for just that.