The Collatz Zoo

What happens when we change the rules? The classical Collatz map uses $3x+1$ for odd numbers and $x/2$ for even. But we can build a whole zoo of dynamical systems by varying the multiplier $n$, the divisor $y$, and the additive constant $c$:

$$f(x) = \begin{cases} x/y & \text{if } y \mid x \\ nx + c & \text{otherwise} \end{cases}$$

The classical conjecture is the special case $(n, y, c) = (3, 2, 1)$. Studying the full family shows why 3x+1 is expected to converge — and why most of its relatives are expected not to. Everything on this page is a heuristic, an analogy, or a computation over a stated range. None of it is a proof.

Multiplier n: 3 Divisor y: 2 Constant c: 1

System: 3x+1, x/2 μ = 0.752 — Subcritical

Thermodynamics

Criticality μ	0.7521
Energy input/kick	1.585 bits
Avg drains/kick	1.996
Net per cycle	-0.411 bits
Fundamental const	log₂(6) = 2.5850

Survey (odd starts x = 3..499; an orbit is left unresolved past 10¹⁵ or after 2000 steps)

100% pass through 1

Pass through 1: 249Cycle, not through 1: 0Unresolved: 0

Sample orbit: x = 27

112 values, peak 9,232, reached 1 (stopped there)

Thermodynamic Framework

Each system in the zoo has measurable "physics." We treat the orbit as an energy transfer process:

Quantity	Formula	Physical Analogue
Potential energy	$\log_y(x)$	Height above ground state
Energy input per kick	$\log_y(n)$	Work done by the $nx+c$ step
Energy drain per step	$1$	Dissipation from the $x/y$ step
Fundamental constant	$\log_y(ny)$	The "speed of light" of the system
Criticality $\mu$	$n / y^{E[v_y]}$	Subcritical ($\mu < 1$) or supercritical ($\mu > 1$)

The Conservation Law

Every system obeys a conservation equation:

$$s \cdot \log_y(ny) = T - \log_y(x_{\text{initial} }/x_{\text{final} }) + \varepsilon$$

where $s$ is the number of kick steps (odd steps), $T$ is the total steps, and $\varepsilon \le 0$ captures the residual from the $+c$ corrections.

For classical 3x+1, on an orbit followed down to 1: $s \cdot \log_2(6) = T - \log_2(x_0) + \varepsilon$, with $\varepsilon \le 0$ always, and $\varepsilon \ge -0.33$ for every start below $10^6$ (computed; the minimum, $-0.326$, is at $x_0 = 993$).

This is the first law — energy is conserved up to the residual $\varepsilon$.

The Critical Threshold

The criticality parameter measures the average drift, and nothing more:

$$\mu = \frac{n}{y^{E[v_y]} }$$

where $E[v_y]$ is the expected $y$-adic valuation of $nx+c$ for random inputs. For $x/2$ systems with $n$ and $c$ odd, $E[v_2] = 2$ (each bit position is equally likely to end a run of zeros), giving:

$$\mu = \frac{n}{y^2} = \frac{n}{4}$$

The average drift changes sign at $\mu = 1$, i.e., $n = 4$:

System	$\mu$	Odd starts 3 to 499 (computed)
$3x+1, \; x/2$	$3/4 = 0.75$	all reach 1
$5x+1, \; x/2$	$5/4 = 1.25$	84% pass $10^{15}$; the rest reach 1 or the cycles through 13 and 17
$7x+1, \; x/2$	$7/4 = 1.75$	98% pass $10^{15}$
$9x+1, \; x/2$	$9/4 = 2.25$	97% pass $10^{15}$

Passing $10^{15}$ is not divergence: no orbit of $5x+1$ has been proved to diverge.

3 is the only odd number between 1 and 4. So among the $nx+1$, $x/2$ systems with odd $n > 1$, $3x+1$ is the only one whose average drift is downward. That is a classical heuristic, not a proof: an average says nothing certain about every orbit. The explorer above has a blunt example: for $2x+1$, $x/3$ it reports $\mu \approx 0.87$, yet every start that is 2 modulo 3 is never divided at all ($2x+1$ is again 2 modulo 3) and runs off to infinity.

The Three Laws

A thermodynamic analogy, in three "laws":

First Law: Conservation

$$s \cdot \log_2(6) = T - \log_2(x_0) + \varepsilon$$

Energy input equals energy output plus residual. This holds for all $(n, y, c)$ systems — it's a counting identity. The fundamental constant $\log_2(6)$ arises because $6 = 2 \times 3 = y \times n$.

Second Law: Dissipation

$$\mu = 3/4 < 1$$

The average energy drain exceeds the average energy input. On average, a kick-drain cycle loses 0.415 bits. This is the arrow of time — orbits trend downward on average.

For $5x+1$: $\mu = 5/4 > 1$, so the arrow points upward, and most orbits appear to grow without bound.

Third "Law" (conjectural): No Perpetual Motion

Even though the average is contraction, individual orbits could in principle avoid the average — staying in "unlucky" residue classes that produce fewer drains per kick. The finite fuel picture is a heuristic for why this might not persist: staying unlucky imposes more and more constraints on the bits of the starting number.

This "law" is not proved — it is essentially the Collatz conjecture itself — and it's where the $+1$ in $3x+1$ enters critically. The average drift doesn't depend on which odd $c$ is used, but which cycles exist does, as the next section shows.

The Role of $c$

Varying $c$ while keeping $n=3, y=2$ reveals a surprise:

$c$	$\mu$	Starts that pass through 1	Distinct cycles reached	Notes
1	0.75	100%	1	Only $1 \to 4 \to 2 \to 1$ seen
3	0.75	0%	1	$3 \mid c$: the $3x+1$ map on multiples of 3
5	0.75	13%	6	Several cycles
7	0.75	69%	2	1 is not on a cycle: $1 \to 10 \to 5 \to 22 \to \dots \to 5$
9	0.75	0%	1	$3 \mid c$: the $3x+1$ map on multiples of 9
11	0.75	19%	3
13	0.75	47%	10

(Computed for the odd starts from 3 to 499; every one ended in a cycle.)

Criticality $\mu$ is identical for all odd values of $c$ — it depends only on $n$ and $y$. Every such system with $n=3, y=2$ is subcritical, and for every odd $c$ from 1 to 19 every odd start from 3 to 999 settles into some cycle (computed). But $c$ determines the ground state landscape:

• $c \equiv 0 \pmod{3}$: after its first odd step an orbit stays among multiples of 3 for good ($3x + c$ is then a multiple of 3, and halving keeps the factor), so it never comes back to 1. Every orbit tested ends in a cycle. For $c = 3$ the map on multiples of 3, and for $c = 9$ the map on multiples of 9, is the $3x+1$ map in disguise, so "always cycles" there is the Collatz conjecture again.
• $c$ even: $3x + c$ is odd whenever $x$ is, so an odd number is never halved again and its orbit grows forever.
• $c = 1$: 1 lies on the cycle $1 \to 4 \to 2 \to 1$, and every orbit tested ends there — a single observed ground state.

In physics terms: $n$ and $y$ determine the thermodynamics (does energy dissipate on average?). The constant $c$ determines the ground state degeneracy (how many stable configurations exist?). Among the systems tabulated here, classical Collatz and its two disguises ($c = 3$ and $c = 9$) are the ones with a single observed cycle; only for $c = 1$ does that cycle pass through 1.

Related

• Eisenstein Lattice — orbits as walks on the triangular lattice
• The Transfer Operator — the spectrum of the map cut off at a modulus
• abc Conjecture — how close powers of 2 and 3 can be
• Finite Fuel — a heuristic for why individual orbits might not escape the average
• Bit Destruction — the size of each drop in bits