The Comma Sequence, Landmines, and the Immortal Kangaroo

Neil Sloane, founder of the On-Line Encyclopedia of Integer Sequences, recently appeared on Numberphile to describe what he called "one of Eric Angelini's really greatest sequences of all." It has since acquired a zoological nickname: the immortal kangaroo sequence. The name is absurd and, as it turns out, load-bearing. It encodes the entire mathematical situation: a creature hopping along the number line in small, fixed jumps, through a landscape seeded with mines, whose survival depends on choices no one knows how to make.

A Rule Written in the Commas

Start with any sequence of numbers. Between consecutive terms sits a comma. Angelini's move was to define the "comma separator": read the digit immediately before the comma and the digit immediately after it, and concatenate them. In the squares 1, 4, 9, 16, the separators are 14, 49, 91.

His proposed sequence inverts that logic. Take the comma sequence, formally defined in the paper by Angelini, Sloane, Michael Branicky, Giovanni Resta, and David Wilson: the difference between consecutive terms must equal the two digits flanking the comma between them. Start at 1. The next term is 12, because the difference is 11, matching the two ones around the comma. From 12, the separator must be 2-something; adding 23 lands on 35, and sure enough, the digits around that comma read 23. Then 94 (difference 59), then 135, then 186.

The mechanism that keeps the sequence sane is the one Brady Haran kept probing in the video: the separator ignores everything except the boundary digits. Because the difference between terms is always a two-digit number, each hop moves the sequence forward by between 1 and 99. When the terms grow into eleven-digit monsters, the jumps become vanishingly small relative to the terms. The kangaroo takes tiny steps.

Landmines

Can the puzzle become unsolvable? Sloane's answer in the video was unhesitating: "Oh, yes, absolutely." A dead end occurs at what he calls a landmine: any number whose final two digits sum to 9, preceded by any number of 9s, including none. The two-digit cases number just eight: 18, 27, 36, 45, 54, 63, 72, 81. Land on one and there is no legal next term. The sequence stops.

The examples are instructive in their range. Starting from 3, the sequence reaches 36 and dies in two steps. Starting from 1, it survives 2,137,453 terms before hitting 6945. Starting from 2, it runs for roughly 2 × 10^14 terms before hitting a number with fourteen 9s followed by 18. Starting early is no protection; starting at some twenty-digit number does not help either, as I will get to.

Sloane offered a probabilistic sketch of why the lifetimes vary so wildly. In each block between consecutive powers of 10, the eight landmines cluster near the top of the block. A kangaroo passing through faces them only in the last hundred numbers of each decade. A rough model puts the per-block death chance at about 12 in 100, giving an expected lifetime around 10^8.3 steps for a random start. Sloane flagged this as "not rigorous at all," but noted it comes out "pretty close" to what computation shows. I appreciate that he drew the line himself; the estimate explains the data without pretending to prove anything.

What Has Actually Been Proved

Here is where the story gets methodologically interesting, because the strongest-sounding claim in circulation is one the researchers do not have. Haran asked whether the paper shows every starting number eventually hits a mine. Sloane's reply: "We wish. We know that we believed that was true, but we couldn't prove it" in full generality.

What the collaboration did prove, with computer-assisted arguments Sloane described as "enormously difficult," is termination in base 3. Two Rutgers graduate students then generalized the proof, establishing that every sequence dies in bases 3 through at least 643, and, crucially, in base 10 itself. So in our own numbering system, the verdict is settled: no matter where you start, the smallest-choice sequence terminates. The general conjecture, for every base larger than 2, remains open.

Base 2 is the charming exception. All binary numbers except zero begin with 1, so the separator's right digit is always 1 and the analysis collapses into something tractable. Both possible paths in base 2 are infinite. The kangaroo lives there without effort.

Immortality, with a Changed Rule

The original rule mandates choosing the smallest legal next term. Sloane calls it the firstborn: "The firstborn is the official heir to the throne." But occasionally a number has two legal successors rather than one, never more. At 14, the smallest choice gives 59; choosing instead 46 as the separator yields 60, and both satisfy the rule.

Relax the smallest-first requirement and the sequence becomes a branching tree. Every positive integer traces back through this tree to an ancestor below 100, and only 50 distinct starting points exist, because a couple of numbers under 100 have two ancestors. Sloane's counting argument is the kind I find most satisfying: the graph contains every positive integer, so it is infinite; 50 trees cover it; if all 50 were finite, the graph would be finite. Therefore at least one tree is infinite, and by König's lemma, a finitely branching infinite tree contains an infinite path. Immortality exists.

The catch is that König's lemma is a consequence of Zermelo-Fraenkel set theory plus the axiom of choice, the principle, as the Stanford Encyclopedia of Philosophy lays out, that from any collection of nonempty sets one can simultaneously select one element from each, without a rule specifying how. Sloane was blunt about what this buys: "We know there's at least one path that goes to infinity," but the proof "is not constructive. We don't know how to do it." Asked for an algorithm to pick correctly at each fork, he joked that invoking the axiom is itself the algorithm: "You just call on, you know, the magician to give you the next term. That's the algorithm, but it's not effective."

The correct choices form a binary sequence, zero for smaller successor, one for larger, catalogued in the OEIS as A399179. Sloane knows about 54 terms of it. Whether it is unique is unknown. Whether it follows a simple rule is unknown, though he allows it could: some threshold rule on multiples of 101, say, has not been ruled out.

What computation did settle is the starting point. Of the 50 trees, 49 were followed until every branch died. The survivor begins at 20. "20's the magic number," Sloane said, and its path runs 20, 22, and onward indefinitely, provided the right choices are made at each fork. We know those choices exist. We can write down roughly the first fifty of them. We cannot say what the fifty-first is, or whether a formula produces all the rest.

The Open Terrain

Three questions remain, in Sloane's own accounting. Given a starting value, is there a formula for the length of its path, or must one simply compute until death? Does termination hold in every base above 2, beyond the verified range through 643? And can the infinite path be constructed without the axiom of choice, by an effective procedure a computer could run?

Each question is a different flavor of unfinished. The first is computational bookkeeping. The second is a conjecture with strong evidence but no proof. The third sits at the border between combinatorics and logic, where existence and constructibility part ways. That a digits-and-differences puzzle, the sort of thing that looks like a coffee-break diversion, should run aground on the axiom of choice is the reason recreational mathematics earns the word mathematics. Somewhere out on the number line, a kangaroo is making choices no one can write down.

Priya Sharma covers science and research for BuzzRAG.