Essay · December 2025
Escaping the Killer by Always Getting Closer
A rule that improves things at every step can still never finish. The rule here: get closer to the target, every single step.
The setup
You stand at point $B$. A killer starts at point $A$ and will kill you the moment he reaches $B$. He is dumb but perfectly obedient, and you get to program him with one rule: after every step, he must be closer to you than he was before. You also choose how much closer he gets each step, and in which direction he moves.
The natural belief is that this rule guarantees your death: if he keeps getting closer, how can he not arrive? But that belief hides an assumption: that getting closer also limits how much useless movement is possible. So the real question is: can you force him to walk an enormous total distance, even an infinite one, while he gets closer at every step?
The exploit
“Distance to $B$ must decrease” only says that each step lands inside the circle around $B$ through his current position. It limits his radial motion and says nothing about tangential motion. He can circle almost forever while the radius shrinks by tiny amounts. Displacement is the straight line from $A$ to $B$; total distance is the sum of every step, and a spiral can make the second as large as you like compared with the first.
The Archimedean spiral
Take $r(\theta)=R-\dfrac{m\theta}{2\pi}$. As $\theta$ increases the radius strictly decreases, so the rule is never broken. He needs $N=R/m$ full turns to reach the centre, and each turn adds about one circumference of walking for only $m$ of real progress.
The arc length from $\theta=0$ to $\theta=2\pi R/m$ is
Substitute $u=R-\frac{m\theta}{2\pi}$, so $d\theta=-\frac{2\pi}{m}\,du$ and the limits run from $u=R$ to $u=0$. With $a=\frac{m}{2\pi}$,
Evaluating and simplifying gives the plug-and-play formula
Taking the limit
As $m\to0^+$ the dominant term $\pi R^2/m$ grows without bound: $$\lim_{m\to0^+}\frac{\pi R^2}{m}=\infty.$$ So the killer can be made to walk an arbitrarily long distance (in the limit, an infinite one) while always getting closer. The discrete picture is the harmonic series: if step $n$ moves him $1/n$ along the path while his distance to you still falls, then $\sum 1/n$ diverges.
| R | m | m / 2πR | √(1 + (m/2πR)²) | L |
|---|---|---|---|---|
| 6 | 1 | 0.02653 | 1.000352 | 113.4811 |
| 6 | 0.8 | 0.02122 | 1.000225 | 141.6929 |
| 6 | 0.5 | 0.01326 | 1.000088 | 226.4144 |
| 6 | 0.2 | 0.00531 | 1.000014 | 565.5890 |
| 6 | 0.1 | 0.00265 | 1.000004 | 1131.0301 |
| 6 | 0.05 | 0.00133 | 1.000001 | 2261.9778 |
| 6 | 0.001 | 0.00003 | 1.000000 | 113097.3365 |
Why this matters
- A local rule is not a global result. A rule that seems to guarantee the goal (getting closer) does not guarantee reaching it in finite time or distance.
- Poorly specified objectives. An optimiser that “always moves closer” sounds safe, but unless its path complexity is bounded it can be arbitrarily inefficient. Gradient descent with a badly tuned learning rate does exactly this: it improves every step and still takes an impractically long route.
- The power of direction. Rotation allows almost unlimited movement inside a small, finite region.
The original paper
The essay as typeset in December 2025.