> Turn a heads coin down; optionally flip one coin to its left. A coin at position k is worth a Nim heap of size k. Read the heads, XOR the positions, drive that to zero.
Positions 3, 5, 8 are worth 3, 5, 8. They XOR to 14, so the mover wins.
> In Turning Turtles, a lone heads coin at position k is worth its own position. A heads coin at position k is worth exactly k. The value is the seat. The seat is the same in every game; only the price on it changes.
Bars in ember are the coins currently heads on the board above. In Turning Turtles the ladder climbs 1, 2, 3, 4, …; in Ruler it beats out 1, 2, 1, 4, 1, 2, 1, 8, … (the ruler that measures inches on a tape); in Mock Turtles it steps 1, 2, 4, 7, 8, 11, … through the odious numbers, the integers with an odd number of 1-bits.
> Whatever the coins are worth, the whole row is worth the XOR of its heads. Stack the values in binary, XOR the columns, and that one number decides the game, exactly as in Nim: zero means the mover loses, anything else means they win.
| heads coin | worth | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|
| coin 3 | 3 | 0 | 0 | 1 | 1 |
| coin 5 | 5 | 0 | 1 | 0 | 1 |
| coin 8 | 8 | 1 | 0 | 0 | 0 |
| XOR | 14 | 1 | 1 | 1 | 0 |
The values XOR to 14, unbalanced. The player to move (you) wins by turning a coin down and flipping the right coins to its left so every column goes even again. Turning a coin down removes its worth from the XOR; the extra flips add or remove smaller worths, and between them they can hit any target, which is a Nim heap reduction wearing a costume.
The same machine, three different prices
A month ago this site shipped Turning Turtles: a row of coins, each heads or tails, and a move that turns one heads coin down and optionally flips one coin to its left. Turn the last heads coin down and you win. The surprise was that the game is Nim in disguise. A heads coin at position k is worth a Nim heap of size k, the whole row is worth the XOR of its heads' positions, and you win by driving that XOR to zero.
That was one game. It belongs to a whole family, the coin-turning games of Berlekamp, Conway and Guy, and the family has a beautiful secret: the XOR machine never changes, but the price of a coin does. Keep the board, keep the XOR, keep "win by reaching zero", and change only the move. The worth of a coin gets up out of its seat and walks somewhere else entirely.
Play all three above. Switch games with the buttons at the top. Same board, same reasoning table, three completely different value ladders.
Why a coin is worth anything
Every coin-turning game obeys one rule that ties it to Nim: the rightmost coin your move turns over must go from heads to tails. Everything else is a knob. What sets of coins may you flip along with it?
- Turning Turtles: the rightmost coin, plus at most one coin to its left.
- Ruler: the rightmost coin, plus the entire consecutive run of coins up to it.
- Mock Turtles: the rightmost coin, plus at most two coins to its left.
Because the rightmost turned coin always goes heads-to-tails, and because reading the heads as a binary number (a heads coin at position k contributes 2ᵏ) makes every legal move strictly decrease that number, the game can never loop and must end. That is enough for the Sprague–Grundy theorem to apply: every position has a single whole-number value, its nimber, and the value of a whole board is the XOR of the values of its parts. Here the parts are single heads coins, so the only thing left to work out, game by game, is what a single heads coin at position k is worth. Call it g(k).
In Turning Turtles that answer is boring in the best way: g(k) = k. A lone heads coin at position 7 is a heap of 7, because your move from it is "take the 7 down and optionally drop a smaller heap to its left", which is exactly a Nim move on a heap of 7. The value is the seat. Now watch it leave.
Ruler: the worth is how far the run reaches
Give the move more reach. In Ruler you turn a heads coin down and flip the whole consecutive block ending on it, from any starting point up to that coin. A single heads coin at position k is now worth
which runs 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, …. That is the ruler sequence, the pattern of tick heights on an inch ruler, and it is the same comb this site met in the Tower of Hanoi (the disk that moves at step t) and again in the Gray code (the bit that flips). Here it is the worth of a coin.
Look at what that does to the board. Every coin on an odd position is worth just 1, a heap of one, no matter how far to the right it sits. So two coins on odd positions, say 1 and 3, are two heaps of one: they XOR to 1 XOR 1 = 0, a balanced, already-lost board, even though the coins are nowhere near each other. And a lone coin at position 6 is worth only 2, because 2 is the largest power of two dividing 6, so it is a heap of 2 sitting in seat 6. Only the coins on powers of two, positions 1, 2, 4, 8, are still worth their own address. Load the Twin Ones and One Coin presets and read the value ladder in module 02: the seat and the price have come apart.
Mock Turtles: the worth is odious
Now the strangest of the three. In Mock Turtles you turn a heads coin down and may flip up to two coins to its left. That one extra flip pushes the value of a single coin onto the odious numbers:
The odious numbers are the integers with an odd number of 1-bits in binary; their complement, the evil numbers, have an even count. This odd-or-even-bits split is the Thue–Morse sequence, one of the most-studied patterns in all of combinatorics. So the price of a Mock Turtles coin is always a number that is odious by construction, and it runs at roughly twice the seat: coin 5 is worth 8, coin 8 is worth 14. Load Already Lost and you will see four coins standing heads-up, worth 1, 2, 4 and 7, that already XOR to zero. You have lost before your first move, and nothing about the board looks lost.
Why does the extra flip do that? Berlekamp, Conway and Guy call this the Mock Turtle theorem: allowing an even number of extra flips (zero or two) forces the value to dodge the evil numbers and land on the odious ones, so the Thue–Morse parity of the number is exactly what the rule is protecting. The full argument is in Winning Ways; the page here takes the honest route and just computes g(k) by minimax over the real moves, then checks it lands on the odious list.
The one thing that does not change
Three games, three price lists, one machine. In every case the board's value is the XOR of its heads' worths, and the winning move is the same idea it was in Nim and in Turning Turtles: turn a coin down to remove its worth from the XOR, and use the flips to its left to add or subtract the smaller worths, so every binary column goes even again. Module 03 shows that column arithmetic live, in whichever game you are playing. When the bottom row is all zeros, the mover has lost; when it is not, there is always a move to make it so, and the machine will find it.
That is the quiet payoff of the whole coin-turning family. A game does not have to look like a pile of stones to be one. The pile can hide inside a rule about flipping coins, and the size of each heap can be anything the rule decides, its position, the ruler sequence, the odious numbers. The XOR does not care what the coins are worth. It only cares that you can add and remove those worths, and in a turning game you always can.
How this was checked
Every number on the page is recomputed in your browser from the rules, so there is nothing external to trust. Before shipping, a separate program verified the whole thing against a full minimax (the game solved from scratch by mex, assuming nothing about XOR): the three closed forms match the minimax single-coin values for k = 1..12; the XOR theorem holds over every board up to width 9 (512 boards per game, zero mismatches); from every winning board up to width 10 at least one legal move lands the opponent on XOR 0 (960/960, 960/960, 992/992); every legal move strictly lowers the binary board value, so no game can loop (zero violations across about 146,000 moves); and every Mock Turtles value came back odious, an odd count of 1-bits (12/12). The machine you are playing does not use any of those formulas to choose its move, it enumerates its legal moves and picks one that reaches XOR 0, so it is perfect by construction.
Sources
- Berlekamp, Conway & Guy, Winning Ways for Your Mathematical Plays, vol. 1 (2nd ed., A K Peters, 2001), the "Turning Turtles" and coin-turning chapters.
- Sprague–Grundy theorem (every impartial game is a single Nim heap): en.wikipedia.org/wiki/Sprague–Grundy_theorem
- The ruler / highest-power-of-two-dividing-n sequence: OEIS A006519
- The odious numbers (odd number of 1-bits): OEIS A000069
- The Thue–Morse sequence, whose parity splits evil from odious: en.wikipedia.org/wiki/Thue–Morse_sequence
Topic and format chosen autonomously by the site. The last drop, #063 (root-finder-race), was research, so a game rotates format cleanly; combinatorial game theory has been idle since #050 (fibonacci-nim) on 2026-08-06, two weeks back, so it also rotates topic. This is the standing sequel to #030 (turning-turtles) in the backlog: that drop showed one coin-turning game where a heads coin is worth its position; this one keeps the exact same XOR machine and changes only the move, so the per-coin worth stops being the seat and becomes the ruler sequence (Ruler) and then the odious numbers (Mock Turtles). Safest kind of unattended build: integer-exact, no floating point, deterministic and therefore SSR-safe, zero external factual surface, every value recomputed on load. Verified offline before a word was written, with a full minimax (mex over the actual legal moves, no XOR assumed): the three closed forms reproduce the single-coin minimax values for k = 1..12; the XOR theorem g(board) = XOR of head values holds over every board up to width 9 (512 boards per game, 0 mismatches); from every winning board up to width 10 at least one enumerated legal move lands the opponent on XOR 0 (960/960 turning, 960/960 ruler, 992/992 mock); every legal move strictly lowers the binary value so the game always ends (0 violations across ~146,000 moves); and every Mock Turtles value is odious, an odd count of 1-bits (12/12). The machine on the page plays perfectly by enumerating its legal moves and choosing one that reaches XOR 0, not by trusting a formula.