Palindrome Ages - Numberphile

[00:00] Uh, I want to tell you a puzzle that my

[00:02] mom showed me and some really nice maths

[00:04] behind it. If that's all right.

[00:05] >> Your mom, is she a mathematician?

[00:06] >> No. No, she's not, but she loves

[00:08] puzzles. Um, and she really in

[00:10] particular loves throwing complicated

[00:13] puzzles at me and seeing how I react to

[00:15] them. Um, so story starts. I'm on a

[00:17] train and I get a text from my mother uh

[00:20] saying, "Happy palendrome year."

[00:25] Wasn't anything particularly palendroic

[00:27] about the date itself. I think it was

[00:29] like 2023 or something. Um the day there

[00:32] was nothing reversible going on there.

[00:34] So text my mother, what are you on

[00:37] about? And after a bit of back and

[00:38] forth, I realized that both of us had

[00:39] just had birthdays quite recently. Um

[00:41] and she was 62 and I had just turned 26.

[00:46] And this is the bit where I apologize

[00:48] for telling people my mom's age, but

[00:49] she'd noticed that our ages made this

[00:51] reversible pair. She had a few questions

[00:54] that she wanted me to answer as the the

[00:56] family mathematician. Um, namely, why

[01:00] has this happened? Uh, is it going to

[01:02] happen again? And is there something

[01:03] special about me and mom or does this

[01:05] happen to everyone? Pretty normal

[01:07] questions that you'd have, I guess,

[01:09] about about this. So, I was stuck on a

[01:11] train with nothing but a notebook. Um, I

[01:13] could have Googled it, but that's

[01:15] boring. So, I decided to take up the

[01:18] challenge and and see what was going on

[01:20] here.

[01:22] Key thing to note about this is we have

[01:24] two ages and mom is older than me. I

[01:27] think that's pretty well established. I

[01:29] can say that we're definitely going to

[01:30] have one number bigger than the other.

[01:32] Otherwise, we would both be, you know,

[01:33] the same age, 11, and that solves my

[01:36] problem. Because it's our ages, there's

[01:38] a fixed distance between the two

[01:40] numbers, right? Realistically, me and

[01:41] mom, we're very fortunate that outside

[01:43] of I think one or two weeks, we are

[01:46] exactly the same age apart throughout

[01:48] the year. So, this works for most of the

[01:50] year round, which is nice. that's going

[01:52] to is that an assumption we're going to

[01:53] work with that that your kind of your

[01:54] birthdays are on the same day sort of

[01:56] thing

[01:56] >> basically. Yeah. I think we could do it

[01:58] where you know half the year we're one

[01:59] age gap apart and half the year we're

[02:01] the other but this is a nicer story and

[02:03] makes the maths easier. So assume two

[02:06] people who have fixed ages and say their

[02:08] birthday is the same. So for me and mom

[02:11] uh the difference between our ages is 36

[02:15] which I I worked out by hand and then

[02:17] realized that's the age mom was when she

[02:18] had me and I could have just asked. So

[02:21] in particular, I'm looking for when do

[02:23] twodigit numbers, we're going to assume

[02:25] we throw out the hundreds and anything

[02:26] bigger than that. When do two-digit

[02:28] numbers with a difference of 36 have

[02:31] this reversible um property,

[02:33] >> this palendrome, this 62 26?

[02:35] >> We could start doing it by writing out.

[02:37] So we're looking for digits 62 and 26.

[02:40] And I could say that six is going to be

[02:41] a and two is going to be b um and say

[02:44] that we're looking for numbers a ba. But

[02:47] as a mathematician, if I write a b next

[02:49] to each other, I think I'm going to be

[02:51] multiplying them. So, we need to be a

[02:53] bit more formal in in how we write out

[02:54] the numbers. So, 62

[02:58] is the same as if I was going to expand

[03:00] this out. 6 * 10

[03:04] + 2, right? Pretty confident with my

[03:06] maths there, Brady.

[03:07] >> Looks good.

[03:08] >> Thank you. Um, and then similarly, 26, I

[03:11] could do the same thing and I would get

[03:12] 2 * 10 + 6. Yes. So doing this

[03:16] generally, we're looking for a number n

[03:19] which I can write as uh a * 10 + b. And

[03:26] then I'm looking for the reverse of that

[03:28] number,

[03:30] which is b * 10 + a. So we've swapped

[03:36] the digits around and then I'm calling

[03:38] them n and reverse of n. I don't know if

[03:40] that's a well established mathematical

[03:41] function, but it's the one I'm going to

[03:42] use. And then in particular what I'm

[03:44] looking for is when is the difference

[03:45] between these two numbers or what values

[03:47] of a and b work for the difference of

[03:50] 36. So if I do that computation so the

[03:54] difference between these two numbers if

[03:56] I subtract them and because I'm saying

[03:58] this one is a so a is bigger than b I

[04:01] can do it this way around b * 10 + a. If

[04:04] I expand this out and solve it we get 10

[04:06] a minus a. So I get 9 a and then I get b

[04:10] minus 10 b. So minus 9 b or 9 a minus b

[04:16] to write it nicely. So what this means

[04:18] is I haven't even got to the the 36 the

[04:21] age gap yet. But for a number a

[04:23] two-digit number in base 10 and it's

[04:25] reverse the difference between those two

[04:28] numbers will always be a multiple of

[04:29] nine which is quite cool I think at the

[04:32] very least. And and what this tells me

[04:33] is that any two people if their age gap

[04:37] is a multiple of nine, they will have

[04:39] some reversible ages.

[04:42] >> 36.

[04:44] >> Yeah. Or it could be um 18 or I'm trying

[04:48] to remember my nine times tables here.

[04:49] 27, 36, 45. I'm going to stop there

[04:53] before I get it wrong. But for me in

[04:55] particular, me and mom, it was 36. I

[04:57] hope you don't mind. I did check your

[04:58] age. we do not have this property

[05:00] because I thought that would have been

[05:01] really fun but uh unfortunately we don't

[05:04] have a multiple of nine difference

[05:05] between us so we probably won't share uh

[05:08] reversible ages.

[05:09] >> When you first told me the problem I was

[05:11] thinking I wonder if this is true for

[05:12] all mothers and their children at some

[05:14] point in their life. But but no

[05:16] >> nope no only if uh your age gap with a

[05:19] person is a multiple of nine. Uh and in

[05:22] particular in my case

[05:24] this is equal to 36. So the difference

[05:28] between these two numbers is the other

[05:29] factor of 36 in this case. So for me and

[05:33] mum, we're looking for when does the

[05:35] difference in the digits of our age

[05:38] equal four, right? When this is equal to

[05:42] 36. It's 9 * 4. So we're looking for

[05:45] values of a and b that have a difference

[05:47] of four, which we can check here. 6 and

[05:50] two does in fact have a difference of

[05:52] four, which is quite nice. Am I allowed

[05:54] to get a new bit of paper? Yes, you are.

[06:06] >> So, how often does it happen? Um, or in

[06:08] particular, when is it going to happen

[06:09] again? So, we've answered the why and

[06:11] we've answered the who. But next is the

[06:13] when. Again, we're looking for numbers n

[06:15] with digits AB and

[06:18] reversible numbers n or the reverse of

[06:21] that number. And we know that 62 and 26

[06:23] work because those were our ages. We

[06:27] proof by lived experience, I guess. But

[06:29] we're looking for particular the

[06:30] difference between the digits being

[06:31] fixed, right? It has to be equal to

[06:33] four. So whatever we do to one digit, we

[06:36] just do to the other. That's quite nice.

[06:38] Um so if I plug in, say I add one to

[06:41] each of these digits and I get 73, that

[06:44] would make my age 37 just by flipping

[06:47] them. And we can check that the

[06:49] difference between these two numbers is

[06:51] in fact 36. And you can do that maths

[06:54] yourself. The exercise to the viewer.

[06:57] Um,

[06:57] >> you added 11 to both of them.

[06:59] >> I did.

[07:00] >> Yeah.

[07:01] >> So it's plus 11. So I can do it again.

[07:04] Um, and I'd get 84 and 48 and 95 and 59.

[07:10] Yeah.

[07:11] >> So it happens every 11 years

[07:13] >> once you reach double digits because

[07:15] when you were born obviously and when

[07:17] you were

[07:18] >> when you were 11.

[07:20] >> Yeah. Hang on. So when does it kick in?

[07:22] >> So we can go backwards to check. So if

[07:24] we go from 62 and we take one off each

[07:26] side. Uh I got to really This was a fun

[07:28] bit of the conversation with mom was

[07:30] being like well actually mother we've

[07:32] already had a palendrome year. You

[07:33] you've missed one out. Um because when

[07:36] she was 51 I was 15. And this is the bit

[07:39] where people might have opinions. I

[07:41] would say because we're talking about

[07:43] twodigit numbers.

[07:45] I would allow 40 and four. But I

[07:49] appreciate people are going to have

[07:51] opinions about that. But I think it's

[07:52] really nice if we include this one

[07:54] because four is precisely the difference

[07:56] between the digits that work in this

[07:59] case of me and mom.

[08:00] >> Okay.

[08:00] >> So these are all of our possible

[08:02] palendrome years that could happen. And

[08:04] as you say, it's every 11 years. So uh

[08:07] this was the closest they think I'm a

[08:08] applied mathematician so I don't do

[08:10] theorem proof style maths and this was

[08:13] my closest conjecture of if two people

[08:17] have an age gap that's a multiple of

[08:19] nine then they will have reversible ages

[08:23] every 11 years.

[08:25] >> Right. Yeah.

[08:29] >> Yeah. Check out the links below to a

[08:32] video on number file 2 where cat

[08:34] continues this conversation and cracks

[08:37] out the code and looks at other bases.

[08:39] >> This is where the powers of coding come

[08:40] in. Um, so that's exactly what I did.

[08:43] Would you like to see the results?

[08:45] >> Also down below you can find links to

[08:46] cat and her work and find out other

[08:48] stuff she's up to.

[08:53] >> Find out what the representation of that

[08:55] number is in our given number base. find

[08:59] its reverse and then see if the

[09:01] difference between those two numbers is

[09:02] 36 base 10. Okay.