The Maths Behind Palindrome Ages Explained
Mathematician Kat Phillips unpacks why some pairs of people have reversible ages—and the surprisingly elegant number theory that governs when it happens.
Written by AI. Priya Sharma
Photo: AI. Sela Marin
A text message from a non-mathematician mother turns out to be a better prompt for number theory than most seminar questions. That, in essence, is the premise of Numberphile's latest video, in which mathematician Kat Phillips walks through the puzzle her mother dropped on her during a train journey—and what began as a curiosity about two reversible ages opens into something genuinely illuminating about the structure of base-10 arithmetic.
The setup is this: Kat's mother texted her "Happy palindrome year." The calendar date offered nothing reversible. After some back-and-forth, the penny dropped—both of them had recently had birthdays. Kat had just turned 26. Her mother was 62. Their ages were, digit-for-digit, each other's reverse. Hence: palindrome year, in a loose but defensible sense of the term.
The mother, as Kat describes her, is not a mathematician but "loves throwing complicated puzzles at me and seeing how I react to them." The questions she wanted answered were, as Kat puts it, "pretty normal questions that you'd have about this": Why has this happened? Will it happen again? And is there something special about them, or does it happen to everyone?
The algebra is simpler than you might expect
Kat's approach on the train—notebook only, no Google—is worth following, because the derivation is accessible and the result is satisfying in that specific way where a general rule emerges from a concrete case.
Any two-digit number can be written as (10a + b), where a is the tens digit and b is the units digit. Its reverse is (10b + a). Subtract one from the other:
(10a + b) − (10b + a) = 9a − 9b = 9(a − b)
That's it. The difference between any two-digit number and its digit-reversal is always a multiple of nine. This isn't a property of Kat and her mother specifically—it's a property of base-10 positional notation. The number 62 and 26 differ by 36. So do 73 and 37. And 84 and 48. And 95 and 59.
What this means for the palindrome-ages question: two people will have reversible ages at some point in their lives if and only if their age gap is a multiple of nine. Age gaps of 9, 18, 27, 36, 45—all qualify. Gaps of 10, 23, 37—don't. As Kat states in the video, "any two people, if their age gap is a multiple of nine, they will have some reversible ages."
This is where the presenter, Brady Haran, articulates the natural follow-up: "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." The answer, bluntly, is no. The multiple-of-nine condition is necessary. Most parent-child age gaps don't satisfy it.
The "when" question: every 11 years
Once you've established whether a palindrome-age pair can occur, the frequency question has an equally clean answer. Return to the arithmetic: if (10a + b) and (10b + a) are a valid reversible pair with difference 36, then so are (10(a+1) + (b+1)) and (10(b+1) + (a+1))—i.e., add one to each digit. Adding one to both digits of a two-digit number adds 11 to it (10 × 1 + 1 × 1 = 11). So the next qualifying pair is 11 years later.
For Kat and her mother, the sequence runs: 40 & 4, then 51 & 15, then 62 & 26, then 73 & 37, then 84 & 48, then 95 & 59. Every 11 years, once both people are in double digits. Kat notes that the 51/15 pairing had already passed—her mother had overlooked it—which made for "a fun bit of the conversation with mom," specifically the revelation that they'd already had a palindrome year, unnoticed.
The boundary case of 40 and 4 is acknowledged as contested: "I would allow 40 and 4. But I appreciate people are going to have opinions about that." Four is a single-digit number; whether you extend the palindrome-pair concept to mixed-digit cases is a matter of definitional taste, not mathematical necessity.
What the nine-times-tables actually screen for
There's a subtlety worth naming here that the video handles gracefully. The multiple-of-nine condition on the age gap is necessary but not quite sufficient on its own—you also need to confirm that the digit difference works out practically. For an age gap of 36, the relevant factor is 36 ÷ 9 = 4. So you need pairs of digits (a, b) where a − b = 4: pairs like (6,2), (7,3), (8,4), (9,5). These are the qualifying combinations. The age gap tells you what the digit difference must be; arithmetic confirms which specific ages count.
This is why Kat checks: "6 and 2 does in fact have a difference of 4, which is quite nice."
It's also why checking a specific person's age gap against the multiples of nine is the quickest filter. Haran mentions that he checked his own age gap with Kat, found it wasn't a multiple of nine, and concluded they "probably won't share reversible ages." That "probably" is technically doing no work—if the gap isn't a multiple of nine, the reversible-age property is ruled out entirely by the algebra above.
Why this matters beyond the parlour trick
It would be easy to bracket this as recreational mathematics—pleasant, clever, essentially decorative. That framing undersells what's actually demonstrated.
The palindrome-ages puzzle is a clean illustration of how positional number systems encode structure that isn't obvious until you expand terms. The "9(a − b)" result falls directly out of the place-value definition of a two-digit number. There's no trick and no special case; it's what the notation is. Students who've memorized the "cast out nines" divisibility rule without understanding why it works would find the same algebra here explaining both phenomena.
There's also something worth noting about the mode of inquiry. Kat describes herself as an applied mathematician who doesn't work in "theorem proof style maths"—and her framing of the final result as a "conjecture" rather than a proved theorem is technically honest. The argument as presented is compelling and the pattern is clear, but a formal proof would require stating the constraints precisely (two-digit integers, fixed age gap assumption, shared-birthday simplification) and verifying the induction. The video doesn't claim to do that, and the distinction matters.
None of this diminishes the central result, which is genuinely elegant: a multiple-of-nine age gap is the entry ticket, and once you're in, the reversals recur on an 11-year clock.
The video points toward extended exploration in a companion piece where Phillips examines the question computationally and considers what palindrome ages look like in number bases other than 10—a natural extension, since the 9(a − b) result is specific to base 10. In base 8, the analogous calculation would produce a multiple-of-7 condition; in base 12, a multiple of 11. The structure generalizes; the specific numbers are artifacts of the base.
Which raises the question Kat's mother probably didn't intend to ask but asked anyway: how much of what we find "special" about certain numbers is a property of the numbers themselves, and how much is a property of the counting system we happened to build?
By Priya Sharma, Science & Health Correspondent, BuzzRAG
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