[0:00] Uh, I want to tell you a puzzle that my
[0:02] mom showed me and some really nice maths
[0:04] behind it. If that's all right.
[0:05] >> Your mom, is she a mathematician?
[0:06] >> No. No, she's not, but she loves
[0:08] puzzles. Um, and she really in
[0:10] particular loves throwing complicated
[0:13] puzzles at me and seeing how I react to
[0:15] them. Um, so story starts. I'm on a
[0:17] train and I get a text from my mother uh
[0:20] saying, "Happy palendrome year."
[0:25] Wasn't anything particularly palendroic
[0:27] about the date itself. I think it was
[0:29] like 2023 or something. Um the day there
[0:32] was nothing reversible going on there.
[0:34] So text my mother, what are you on
[0:37] about? And after a bit of back and
[0:38] forth, I realized that both of us had
[0:39] just had birthdays quite recently. Um
[0:41] and she was 62 and I had just turned 26.
[0:46] And this is the bit where I apologize
[0:48] for telling people my mom's age, but
[0:49] she'd noticed that our ages made this
[0:51] reversible pair. She had a few questions
[0:54] that she wanted me to answer as the the
[0:56] family mathematician. Um, namely, why
[1:00] has this happened? Uh, is it going to
[1:02] happen again? And is there something
[1:03] special about me and mom or does this
[1:05] happen to everyone? Pretty normal
[1:07] questions that you'd have, I guess,
[1:09] about about this. So, I was stuck on a
[1:11] train with nothing but a notebook. Um, I
[1:13] could have Googled it, but that's
[1:15] boring. So, I decided to take up the
[1:18] challenge and and see what was going on
[1:20] here.
[1:22] Key thing to note about this is we have
[1:24] two ages and mom is older than me. I
[1:27] think that's pretty well established. I
[1:29] can say that we're definitely going to
[1:30] have one number bigger than the other.
[1:32] Otherwise, we would both be, you know,
[1:33] the same age, 11, and that solves my
[1:36] problem. Because it's our ages, there's
[1:38] a fixed distance between the two
[1:40] numbers, right? Realistically, me and
[1:41] mom, we're very fortunate that outside
[1:43] of I think one or two weeks, we are
[1:46] exactly the same age apart throughout
[1:48] the year. So, this works for most of the
[1:50] year round, which is nice. that's going
[1:52] to is that an assumption we're going to
[1:53] work with that that your kind of your
[1:54] birthdays are on the same day sort of
[1:56] thing
[1:56] >> basically. Yeah. I think we could do it
[1:58] where you know half the year we're one
[1:59] age gap apart and half the year we're
[2:01] the other but this is a nicer story and
[2:03] makes the maths easier. So assume two
[2:06] people who have fixed ages and say their
[2:08] birthday is the same. So for me and mom
[2:11] uh the difference between our ages is 36
[2:15] which I I worked out by hand and then
[2:17] realized that's the age mom was when she
[2:18] had me and I could have just asked. So
[2:21] in particular, I'm looking for when do
[2:23] twodigit numbers, we're going to assume
[2:25] we throw out the hundreds and anything
[2:26] bigger than that. When do two-digit
[2:28] numbers with a difference of 36 have
[2:31] this reversible um property,
[2:33] >> this palendrome, this 62 26?
[2:35] >> We could start doing it by writing out.
[2:37] So we're looking for digits 62 and 26.
[2:40] And I could say that six is going to be
[2:41] a and two is going to be b um and say
[2:44] that we're looking for numbers a ba. But
[2:47] as a mathematician, if I write a b next
[2:49] to each other, I think I'm going to be
[2:51] multiplying them. So, we need to be a
[2:53] bit more formal in in how we write out
[2:54] the numbers. So, 62
[2:58] is the same as if I was going to expand
[3:00] this out. 6 * 10
[3:04] + 2, right? Pretty confident with my
[3:06] maths there, Brady.
[3:07] >> Looks good.
[3:08] >> Thank you. Um, and then similarly, 26, I
[3:11] could do the same thing and I would get
[3:12] 2 * 10 + 6. Yes. So doing this
[3:16] generally, we're looking for a number n
[3:19] which I can write as uh a * 10 + b. And
[3:26] then I'm looking for the reverse of that
[3:28] number,
[3:30] which is b * 10 + a. So we've swapped
[3:36] the digits around and then I'm calling
[3:38] them n and reverse of n. I don't know if
[3:40] that's a well established mathematical
[3:41] function, but it's the one I'm going to
[3:42] use. And then in particular what I'm
[3:44] looking for is when is the difference
[3:45] between these two numbers or what values
[3:47] of a and b work for the difference of
[3:50] 36. So if I do that computation so the
[3:54] difference between these two numbers if
[3:56] I subtract them and because I'm saying
[3:58] this one is a so a is bigger than b I
[4:01] can do it this way around b * 10 + a. If
[4:04] I expand this out and solve it we get 10
[4:06] a minus a. So I get 9 a and then I get b
[4:10] minus 10 b. So minus 9 b or 9 a minus b
[4:16] to write it nicely. So what this means
[4:18] is I haven't even got to the the 36 the
[4:21] age gap yet. But for a number a
[4:23] two-digit number in base 10 and it's
[4:25] reverse the difference between those two
[4:28] numbers will always be a multiple of
[4:29] nine which is quite cool I think at the
[4:32] very least. And and what this tells me
[4:33] is that any two people if their age gap
[4:37] is a multiple of nine, they will have
[4:39] some reversible ages.
[4:42] >> 36.
[4:44] >> Yeah. Or it could be um 18 or I'm trying
[4:48] to remember my nine times tables here.
[4:49] 27, 36, 45. I'm going to stop there
[4:53] before I get it wrong. But for me in
[4:55] particular, me and mom, it was 36. I
[4:57] hope you don't mind. I did check your
[4:58] age. we do not have this property
[5:00] because I thought that would have been
[5:01] really fun but uh unfortunately we don't
[5:04] have a multiple of nine difference
[5:05] between us so we probably won't share uh
[5:08] reversible ages.
[5:09] >> When you first told me the problem I was
[5:11] thinking I wonder if this is true for
[5:12] all mothers and their children at some
[5:14] point in their life. But but no
[5:16] >> nope no only if uh your age gap with a
[5:19] person is a multiple of nine. Uh and in
[5:22] particular in my case
[5:24] this is equal to 36. So the difference
[5:28] between these two numbers is the other
[5:29] factor of 36 in this case. So for me and
[5:33] mum, we're looking for when does the
[5:35] difference in the digits of our age
[5:38] equal four, right? When this is equal to
[5:42] 36. It's 9 * 4. So we're looking for
[5:45] values of a and b that have a difference
[5:47] of four, which we can check here. 6 and
[5:50] two does in fact have a difference of
[5:52] four, which is quite nice. Am I allowed
[5:54] to get a new bit of paper? Yes, you are.
[6:06] >> So, how often does it happen? Um, or in
[6:08] particular, when is it going to happen
[6:09] again? So, we've answered the why and
[6:11] we've answered the who. But next is the
[6:13] when. Again, we're looking for numbers n
[6:15] with digits AB and
[6:18] reversible numbers n or the reverse of
[6:21] that number. And we know that 62 and 26
[6:23] work because those were our ages. We
[6:27] proof by lived experience, I guess. But
[6:29] we're looking for particular the
[6:30] difference between the digits being
[6:31] fixed, right? It has to be equal to
[6:33] four. So whatever we do to one digit, we
[6:36] just do to the other. That's quite nice.
[6:38] Um so if I plug in, say I add one to
[6:41] each of these digits and I get 73, that
[6:44] would make my age 37 just by flipping
[6:47] them. And we can check that the
[6:49] difference between these two numbers is
[6:51] in fact 36. And you can do that maths
[6:54] yourself. The exercise to the viewer.
[6:57] Um,
[6:57] >> you added 11 to both of them.
[6:59] >> I did.
[7:00] >> Yeah.
[7:01] >> So it's plus 11. So I can do it again.
[7:04] Um, and I'd get 84 and 48 and 95 and 59.
[7:10] Yeah.
[7:11] >> So it happens every 11 years
[7:13] >> once you reach double digits because
[7:15] when you were born obviously and when
[7:17] you were
[7:18] >> when you were 11.
[7:20] >> Yeah. Hang on. So when does it kick in?
[7:22] >> So we can go backwards to check. So if
[7:24] we go from 62 and we take one off each
[7:26] side. Uh I got to really This was a fun
[7:28] bit of the conversation with mom was
[7:30] being like well actually mother we've
[7:32] already had a palendrome year. You
[7:33] you've missed one out. Um because when
[7:36] she was 51 I was 15. And this is the bit
[7:39] where people might have opinions. I
[7:41] would say because we're talking about
[7:43] twodigit numbers.
[7:45] I would allow 40 and four. But I
[7:49] appreciate people are going to have
[7:51] opinions about that. But I think it's
[7:52] really nice if we include this one
[7:54] because four is precisely the difference
[7:56] between the digits that work in this
[7:59] case of me and mom.
[8:00] >> Okay.
[8:00] >> So these are all of our possible
[8:02] palendrome years that could happen. And
[8:04] as you say, it's every 11 years. So uh
[8:07] this was the closest they think I'm a
[8:08] applied mathematician so I don't do
[8:10] theorem proof style maths and this was
[8:13] my closest conjecture of if two people
[8:17] have an age gap that's a multiple of
[8:19] nine then they will have reversible ages
[8:23] every 11 years.
[8:25] >> Right. Yeah.
[8:29] >> Yeah. Check out the links below to a
[8:32] video on number file 2 where cat
[8:34] continues this conversation and cracks
[8:37] out the code and looks at other bases.
[8:39] >> This is where the powers of coding come
[8:40] in. Um, so that's exactly what I did.
[8:43] Would you like to see the results?
[8:45] >> Also down below you can find links to
[8:46] cat and her work and find out other
[8:48] stuff she's up to.
[8:53] >> Find out what the representation of that
[8:55] number is in our given number base. find
[8:59] its reverse and then see if the
[9:01] difference between those two numbers is
[9:02] 36 base 10. Okay.