From Newsgroup: rec.puzzles


Following David E, I will start with a preamble.

My father had this thing about finding something
"interesting" about certain (often random!) numbers
and one of his peculiarities was "celebrating" when the
odometer of his car hit a number that struck his facncy
as something "special", be it 10001, 11111, 12345,
12321, 96069, 101112, .... etc. If possible, he
would pull over and we would have an impromtu picnic
of sorts.

Some years back, driving my aging car I kept a close eye
on the odometer as I intended to drive the car past
150,000 miles. As the oddmeter clicked past 144,000, it hit me
all of a sudden. The next interesting number, for me,
would be 144,441.

Yes, it is a palindrome, but more interestingly,

144 = 12 * 12 and
441 = 21 * 21

and 12 and 21 are palindromes of each other. This got
me wondering about a more general problem/puzzle.

Are there other number pairs (p, q), with p not equal
to q, such that p and q are palindromes and so are
p*p and q*q?

How many such pairs are there?

And, there is one extension that I will not mention now,
but it related to the problem David posted.

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From Newsgroup: rec.puzzles


Charlie Roberts <croberts@gmail.com> posted:

Following David E, I will start with a preamble.

My father had this thing about finding something
"interesting" about certain (often random!) numbers
and one of his peculiarities was "celebrating" when the
odometer of his car hit a number that struck his facncy
as something "special", be it 10001, 11111, 12345,
12321, 96069, 101112, .... etc. If possible, he
would pull over and we would have an impromtu picnic
of sorts.

Some years back, driving my aging car I kept a close eye
on the odometer as I intended to drive the car past
150,000 miles. As the oddmeter clicked past 144,000, it hit me
all of a sudden. The next interesting number, for me,
would be 144,441.

Yes, it is a palindrome, but more interestingly,

144 = 12 * 12 and
441 = 21 * 21

and 12 and 21 are palindromes of each other. This got
me wondering about a more general problem/puzzle.

Are there other number pairs (p, q), with p not equal
to q, such that p and q are palindromes and so are
p*p and q*q?

How many such pairs are there?

And, there is one extension that I will not mention now,
but it related to the problem David posted.

See https://www.researchgate.net/publication/355779341_Palindromic-Type_Squared_Expressions_with_Palindromic_and_Non-Palindromic_Sums_-I
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From Newsgroup: rec.puzzles

On Thu, 18 Dec 2025 17:26:45 GMT, Ilan Mayer
<user4643@newsgrouper.org.invalid> wrote:

See https://www.researchgate.net/publication/355779341_Palindromic-Type_Squared_Expressions_with_Palindromic_and_Non-Palindromic_Sums_-I

That settles it! Certainly answered the question "How many of these
pairs there are?"

To be honest, I posed this puzzle in alt.math.recreational
about a decade ago. Did get one interesting answer.

For two digit numbers, it is not hard to show that the only ones
that will work are the pairs (12, 21) and (13, 31). I guess the
larger number has to be less than the sqaure root of 1000 is
the key.

While I was on the hunt for three digit pairs, one of my friends
found that (102, 201) and (103, 301) also work. This was pure
hand calculator work. But, once he told me about it, and after
a bit of thought, it became clear to me that "zero stuffing" can be
carried out indefinitely on the base pairs (12, 21) and (13, 31)
to get an infinite number of pairs. Thus,
(10000003, 30000001) is such a pair. This is presented in
Examples 1 and 5 on pages 3 and 4 of the paper.

The next question is whether there are other numbers that
do *not* contain zeros that exhibit the same property. An
answer was suggested in a.m.r. But, I will have to look at
quoted paper to see what it says about such pairs. I guess
there actually are two cases:

1. Numbers with zeros that are not "zero stuffed" as above
2. No zeros at all in the numbers

The paper contains many examples of Case 1 pairs. I would
*guess* there are an infinite number of those as the the
patterns seem to follow a modified version of the simple
"zero stuffing".

Case 2 numbers are shown in Sec 2.3 on page 6. My
friend and I did not ever stumble on (113, 311).

But, the question of whether is there are an infinite number
of such pairs is reamins. Need to give it some thought. The
answer may lie in Ref. 6.

Thanks, Ilan for this unique find. May I ask whatever got you
interested in such matters?





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From Newsgroup: rec.puzzles


Charlie Roberts <croberts@gmail.com> posted:

On Thu, 18 Dec 2025 17:26:45 GMT, Ilan Mayer <user4643@newsgrouper.org.invalid> wrote:

See https://www.researchgate.net/publication/355779341_Palindromic-Type_Squared_Expressions_with_Palindromic_and_Non-Palindromic_Sums_-I

That settles it! Certainly answered the question "How many of these
pairs there are?"

To be honest, I posed this puzzle in alt.math.recreational
about a decade ago. Did get one interesting answer.

For two digit numbers, it is not hard to show that the only ones
that will work are the pairs (12, 21) and (13, 31). I guess the
larger number has to be less than the sqaure root of 1000 is
the key.

While I was on the hunt for three digit pairs, one of my friends
found that (102, 201) and (103, 301) also work. This was pure
hand calculator work. But, once he told me about it, and after
a bit of thought, it became clear to me that "zero stuffing" can be
carried out indefinitely on the base pairs (12, 21) and (13, 31)
to get an infinite number of pairs. Thus,
(10000003, 30000001) is such a pair. This is presented in
Examples 1 and 5 on pages 3 and 4 of the paper.

The next question is whether there are other numbers that
do *not* contain zeros that exhibit the same property. An
answer was suggested in a.m.r. But, I will have to look at
quoted paper to see what it says about such pairs. I guess
there actually are two cases:

1. Numbers with zeros that are not "zero stuffed" as above
2. No zeros at all in the numbers

The paper contains many examples of Case 1 pairs. I would
*guess* there are an infinite number of those as the the
patterns seem to follow a modified version of the simple
"zero stuffing".

Case 2 numbers are shown in Sec 2.3 on page 6. My
friend and I did not ever stumble on (113, 311).

But, the question of whether is there are an infinite number
of such pairs is reamins. Need to give it some thought. The
answer may lie in Ref. 6.

Thanks, Ilan for this unique find. May I ask whatever got you
interested in such matters?

I have always liked mathematics, especially recreational mathematics.
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From Newsgroup: rec.puzzles

On Thu, 18 Dec 2025 11:48:48 -0500, Charlie Roberts
<croberts@gmail.com> wrote:

And, there is one extension that I will not mention now,
but it related to the problem David posted.

Ilan's reference is a treasure trove of information on this
puzzle/problem, but I want to complete the story with
something from about 15 years ago.

I think I mentioned that I had posted this problem on
Usenet back then and the most interesting reply I got
was this:

"There are other patterns which look like they are going to continue
but for example, 12, 121, 1211, 12111, 121111, 1211111 all work but
12111111
doesn't. I think that means although there is no upper limit to posner
numbers the highest one with no zeroes must be 2111111."

As for "posner" -- that was my moniker at that time.

It did not occur to me, at that time, to ask the poster for an
explanation for the statement. Whether it was due to
"mathematical" reasoning or exhaustive search, I do not
know.

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