You, me, you, me. It is the most obvious way in the world to take turns — and it quietly hands the win to whoever goes first, every single time. There is a fairer order. It is one of the strangest objects in mathematics, and you can play it, hear it, and watch it balance the books to the last digit.
Two of you are dividing a hoard — coins, draft picks, the good chocolates. You take turns. To be fair, you alternate: A, B, A, B, and each of you grabs the best thing left. Feels even. It isn't.
Whoever moves first takes the single best item. Then the second player takes the second best — but on the next round the first player is choosing first again, taking the third best while the second player is left the fourth. Pair by pair, the first player wins every exchange (or ties it). The lead never shrinks. Watch it happen, then switch the order.
Under alternating order the first player cannot lose: their lead never shrinks. Under the Thue–Morse order — A B B A B A A B… — the advantage keeps changing hands, so the gap stays small. The first player gets the best item, yes; but then the second player gets the next two, and the lead is handed back almost as fast as it opens. That sequence has a name, and a 175-year-old reason it works.
Write the whole numbers in binary. Count the 1s in each. If that count is even, write 0; if it's odd, write 1. That's it. That's the Thue–Morse sequence:
It can be built three completely different ways that always agree, which is the first sign you've found something real and not an accident:
Tap a tile to read its index in binary. Gold = even number of 1-bits (the “evil” numbers); teal = odd (“odious”).
The rhythm you just heard is genuinely almost-periodic: it never settles into a loop, yet it isn't random — it's the most structured non-repeating thing there is. Marston Morse stumbled onto exactly this sequence in 1921 while studying how a line can wander forever across a curved surface without ever retracing its path — work later recognised as an early seed of symbolic dynamics.3
Now the part that should not be possible. Take the numbers 0,1,2,…,2ᵐ−1 and split them into two teams by the rule above: team A = even bit-count (“evil”), team B = odd bit-count (“odious”). The two teams are not just equal in size. Their sums are equal. So are their sums of squares. And their sums of cubes. And so on — equal all the way up through the (m−1)-th powers, and then, precisely at the m-th, the spell breaks.
Eugène Prouhet proved this in 1851 — before Thue, before Morse, in four lines of Comptes Rendus with the proof left out.1 Pick how far to go and check every column yourself; these sums are computed in exact whole-number arithmetic in your browser, no rounding to hide behind.
Each row is a power. equal means team A and team B have the identical sum of that power — proof that the split is balanced at that degree. The first differs row is where the magic runs out: always at degree m.
This is why the turn order is fair. Splitting the items by Thue–Morse parity splits them so the two halves match not just in count but in every weighted total at once — exactly the balance the greedy game needs when the best items are worth wildly more than the worst. That alternation favours the first chooser, and that the Thue–Morse order corrects it, is long-standing folklore of fair division; Cooper & Dutle (2013) make a sibling fact precise — a natural greedy procedure (each player demanding the next turn the moment they fall behind) generates exactly the Thue–Morse order.5 What's reproduced live here is the engine underneath: Prouhet's balance, and the fact that under plain alternation the first mover's lead is a sum of non-negative pairwise gaps — so it can never shrink.
There is one more reason this sequence is famous, and it's the reason it was discovered in the first place. Axel Thue built it in 1906–1912 to answer a question in pure pattern: how repetitive must an infinite string of two symbols be?2 A little repetition is unavoidable — you'll always find a doubled block somewhere (00, 11, or longer). But Thue's word never lets a block repeat three times in a row, and more: it has no overlap at all — no pattern of the form XYXYX where it nearly stutters past a double. Search it yourself.
Squares exist and are easy to find. Cubes and overlaps do not exist anywhere in the sequence — the search below scans the first 2,048 letters exhaustively and comes up empty, every time.
Every claim on this page is recomputed here, live, the moment you load it — and again, identically, offline in research/thue-morse/verify.mjs (59/59 passing). If any line below reads FAIL, the page is lying and you should not trust it.
What is proved here versus cited; the choices made; and where every fact comes from. The two rules of this place: it has to be interesting on its own terms, and it must never lie about anything real — so the seams are shown, not smoothed.
Names “evil” / “odious” for the two teams: Berlekamp–Conway–Guy, Winning Ways (1982); OEIS A001969 / A000069. Bibliographic details above are triangulated from OEIS, journal records, and the Allouche–Shallit survey “The ubiquitous Prouhet–Thue–Morse sequence” (1999); every mathematical claim on the page is independently recomputed in research/thue-morse/verify.mjs.
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