---
title: 'Palindrome Ages - Numberphile'
source: 'https://youtube.com/watch?v=fwLsCgibGw4'
video_id: 'fwLsCgibGw4'
date: 2026-06-30
duration_sec: 553
---

# Palindrome Ages - Numberphile

> Source: [Palindrome Ages - Numberphile](https://youtube.com/watch?v=fwLsCgibGw4)

## Summary

In this Numberphile video, a mathematician recounts a puzzle from his mother: their ages (62 and 26) form a reversible pair. He breaks down the math behind palindrome ages, showing that the age gap must be a multiple of 9 and that such reversible ages occur every 11 years. The video offers a clear, accessible explanation of a fun numerical curiosity.

### Key Points

- **The Palindrome Age Puzzle** [00:00] — The speaker's mother texted him 'Happy palindrome year' after noticing their ages (62 and 26) are reversible. This sparked a mathematical investigation.
- **Three Core Questions** [01:06] — The mother asked: Why did this happen? Will it happen again? Does it happen to everyone? The speaker set out to answer these on a train with just a notebook.
- **Fixed Age Gap of 36** [02:15] — The speaker and his mother have a constant age difference of 36 years (the mother's age when she had him). This gap is key to finding reversible pairs.
- **Difference Formula for Two-Digit Numbers** [03:47] — For a two-digit number N = 10A + B and its reverse 10B + A, the difference is 9(A - B). This is always a multiple of 9.
- **Age Gap Must Be a Multiple of 9** [04:44] — For reversible ages to occur, the age gap must be a multiple of 9. The speaker's gap of 36 works (36 = 9 × 4). Not all parent-child pairs share this property.
- **Digit Difference of 4** [05:40] — With a gap of 36, the digit difference A - B must equal 4. For the speaker and mother, A=6, B=2 gives 6-2=4.
- **Reversible Pairs Every 11 Years** [06:48] — Adding 11 to both digits yields the next pair: 73/37, then 84/48, 95/59. So reversible ages occur every 11 years once both ages are two-digit numbers.
- **Past Pairs and Edge Cases** [07:30] — Going backwards, the pair 51/15 also works. Including 40/4 is debatable but fits the digit difference of 4. The speaker allows it.
- **General Conjecture** [08:04] — If two people have an age gap that is a multiple of 9, they will have reversible ages every 11 years. The video teases further exploration in other number bases.

### Conclusion

The video elegantly demonstrates that palindrome ages occur when the age gap is a multiple of 9, and then repeat every 11 years. It turns a personal anecdote into a fun, accessible math lesson that anyone can explore with their own family.

## Transcript

Uh, I want to tell you a puzzle that my
mom showed me and some really nice maths
behind it. If that's all right.
>> Your mom, is she a mathematician?
>> No. No, she's not, but she loves
puzzles. Um, and she really in
particular loves throwing complicated
puzzles at me and seeing how I react to
them. Um, so story starts. I'm on a
train and I get a text from my mother uh
saying, "Happy palendrome year."
Wasn't anything particularly palendroic
about the date itself. I think it was
like 2023 or something. Um the day there
was nothing reversible going on there.
So text my mother, what are you on
about? And after a bit of back and
forth, I realized that both of us had
just had birthdays quite recently. Um
and she was 62 and I had just turned 26.
And this is the bit where I apologize
for telling people my mom's age, but
she'd noticed that our ages made this
reversible pair. She had a few questions
that she wanted me to answer as the the
family mathematician. Um, namely, why
has this happened? Uh, is it going to
happen again? And is there something
special about me and mom or does this
happen to everyone? Pretty normal
questions that you'd have, I guess,
about about this. So, I was stuck on a
train with nothing but a notebook. Um, I
could have Googled it, but that's
boring. So, I decided to take up the
challenge and and see what was going on
here.
Key thing to note about this is we have
two ages and mom is older than me. I
think that's pretty well established. I
can say that we're definitely going to
have one number bigger than the other.
Otherwise, we would both be, you know,
the same age, 11, and that solves my
problem. Because it's our ages, there's
a fixed distance between the two
numbers, right? Realistically, me and
mom, we're very fortunate that outside
of I think one or two weeks, we are
exactly the same age apart throughout
the year. So, this works for most of the
year round, which is nice. that's going
to is that an assumption we're going to
work with that that your kind of your
birthdays are on the same day sort of
thing
>> basically. Yeah. I think we could do it
where you know half the year we're one
age gap apart and half the year we're
the other but this is a nicer story and
makes the maths easier. So assume two
people who have fixed ages and say their
birthday is the same. So for me and mom
uh the difference between our ages is 36
which I I worked out by hand and then
realized that's the age mom was when she
had me and I could have just asked. So
in particular, I'm looking for when do
twodigit numbers, we're going to assume
we throw out the hundreds and anything
bigger than that. When do two-digit
numbers with a difference of 36 have
this reversible um property,
>> this palendrome, this 62 26?
>> We could start doing it by writing out.
So we're looking for digits 62 and 26.
And I could say that six is going to be
a and two is going to be b um and say
that we're looking for numbers a ba. But
as a mathematician, if I write a b next
to each other, I think I'm going to be
multiplying them. So, we need to be a
bit more formal in in how we write out
the numbers. So, 62
is the same as if I was going to expand
this out. 6 * 10
+ 2, right? Pretty confident with my
maths there, Brady.
>> Looks good.
>> Thank you. Um, and then similarly, 26, I
could do the same thing and I would get
2 * 10 + 6. Yes. So doing this
generally, we're looking for a number n
which I can write as uh a * 10 + b. And
then I'm looking for the reverse of that
number,
which is b * 10 + a. So we've swapped
the digits around and then I'm calling
them n and reverse of n. I don't know if
that's a well established mathematical
function, but it's the one I'm going to
use. And then in particular what I'm
looking for is when is the difference
between these two numbers or what values
of a and b work for the difference of
36. So if I do that computation so the
difference between these two numbers if
I subtract them and because I'm saying
this one is a so a is bigger than b I
can do it this way around b * 10 + a. If
I expand this out and solve it we get 10
a minus a. So I get 9 a and then I get b
minus 10 b. So minus 9 b or 9 a minus b
to write it nicely. So what this means
is I haven't even got to the the 36 the
age gap yet. But for a number a
two-digit number in base 10 and it's
reverse the difference between those two
numbers will always be a multiple of
nine which is quite cool I think at the
very least. And and what this tells me
is that any two people if their age gap
is a multiple of nine, they will have
some reversible ages.
>> 36.
>> Yeah. Or it could be um 18 or I'm trying
to remember my nine times tables here.
27, 36, 45. I'm going to stop there
before I get it wrong. But for me in
particular, me and mom, it was 36. I
hope you don't mind. I did check your
age. we do not have this property
because I thought that would have been
really fun but uh unfortunately we don't
have a multiple of nine difference
between us so we probably won't share uh
reversible ages.
>> When you first told me the problem I was
thinking I wonder if this is true for
all mothers and their children at some
point in their life. But but no
>> nope no only if uh your age gap with a
person is a multiple of nine. Uh and in
particular in my case
this is equal to 36. So the difference
between these two numbers is the other
factor of 36 in this case. So for me and
mum, we're looking for when does the
difference in the digits of our age
equal four, right? When this is equal to
36. It's 9 * 4. So we're looking for
values of a and b that have a difference
of four, which we can check here. 6 and
two does in fact have a difference of
four, which is quite nice. Am I allowed
to get a new bit of paper? Yes, you are.
>> So, how often does it happen? Um, or in
particular, when is it going to happen
again? So, we've answered the why and
we've answered the who. But next is the
when. Again, we're looking for numbers n
with digits AB and
reversible numbers n or the reverse of
that number. And we know that 62 and 26
work because those were our ages. We
proof by lived experience, I guess. But
we're looking for particular the
difference between the digits being
fixed, right? It has to be equal to
four. So whatever we do to one digit, we
just do to the other. That's quite nice.
Um so if I plug in, say I add one to
each of these digits and I get 73, that
would make my age 37 just by flipping
them. And we can check that the
difference between these two numbers is
in fact 36. And you can do that maths
yourself. The exercise to the viewer.
Um,
>> you added 11 to both of them.
>> I did.
>> Yeah.
>> So it's plus 11. So I can do it again.
Um, and I'd get 84 and 48 and 95 and 59.
Yeah.
>> So it happens every 11 years
>> once you reach double digits because
when you were born obviously and when
you were
>> when you were 11.
>> Yeah. Hang on. So when does it kick in?
>> So we can go backwards to check. So if
we go from 62 and we take one off each
side. Uh I got to really This was a fun
bit of the conversation with mom was
being like well actually mother we've
already had a palendrome year. You
you've missed one out. Um because when
she was 51 I was 15. And this is the bit
where people might have opinions. I
would say because we're talking about
twodigit numbers.
I would allow 40 and four. But I
appreciate people are going to have
opinions about that. But I think it's
really nice if we include this one
because four is precisely the difference
between the digits that work in this
case of me and mom.
>> Okay.
>> So these are all of our possible
palendrome years that could happen. And
as you say, it's every 11 years. So uh
this was the closest they think I'm a
applied mathematician so I don't do
theorem proof style maths and this was
my closest conjecture of if two people
have an age gap that's a multiple of
nine then they will have reversible ages
every 11 years.
>> Right. Yeah.
>> Yeah. Check out the links below to a
video on number file 2 where cat
continues this conversation and cracks
out the code and looks at other bases.
>> This is where the powers of coding come
in. Um, so that's exactly what I did.
Would you like to see the results?
>> Also down below you can find links to
cat and her work and find out other
stuff she's up to.
>> Find out what the representation of that
number is in our given number base. find
its reverse and then see if the
difference between those two numbers is
36 base 10. Okay.