The Two-Dimensional Time-Bomb Conjecture

Definitions and mathematical statement

Two-dimensional Time-Bomb question. On ${\mathbb{Z}}^2$, the Angel moves at most one square per turn and the Devil names one position--time pair $(p,t)$ per turn. The Angel loses if its $t$th move lands at $p$ when that pair was previously named. Does the Angel have a strategy to survive forever? A bomb targets the specified position at its specified move time; it is not silently replaced by a permanently forbidden square.

To express the winning condition independently of the missing turn-order convention, write $x_n\in{\mathbb{Z}}^2$ for the Angel's position after its $n$th move and let $D_n$ be the set of position--time pairs announced before that move is evaluated. Survival means\[(x_n,n)\notin D_n\quad\text{for every }n\ge1.\] The intended game-theoretic assertion is $\exists\sigma_A\,\forall\sigma_D\,\forall n\ge1$, this condition holds in the play determined by the strategies. The Angel's strategy can depend on the finite observed history, not on future Devil announcements; existence of a safe path chosen with advance knowledge would be a different question. The Devil's choices may also be adaptive.

The legal step set is not fully specified by one square. In a standard king-move interpretation one has $\|x_n-x_{n-1}\|_\infty\le1$, allowing diagonals and staying put. A nearest-neighbor $\ell_1$ interpretation gives a different game and must be distinguished. The excerpt does not state the first player, the order of each move and announcement, whether a bomb may name the immediately forthcoming Angel move or must have a delay, allowed target times, or the initial position. Accordingly $D_n$ is deliberately not identified with exactly $n$ or $n-1$ announcements here. An arbitrary initial square can conventionally be translated to the origin only after the initial-state rule is fixed. These conventions are essential to a fully defined speed-one game and are not guessed.

The supplied triage checked 2026-08-21 labels the two-dimensional Time-Bomb conjecture open and reports known higher-dimensional or larger-speed variants, citing work about the angel and devil in three dimensions. The exact rules of those variants are not reproduced, so they are not treated as solutions of the stated position--time game. This dated assessment requires independent verification.

Short English statement

Can a speed-one Angel choose its moves online so that no landing square is hit by an earlier bomb scheduled for that exact move time? The step metric, announcement timing, and allowed delays are missing. Results about other dimensions, speeds, or permanent-square games cannot be transferred without matching rules.

Sources

• Local UnsolvedMath registry, permanent ID 30000386, code catalogue entry; The Two-Dimensional Time-Bomb Conjecture, Oberwolfach Reports Open Problems.
• Combinatorics (2006), Oberwolfach Reports, DOI 10.4171/owr/2006/01, pp. 5--72. https://doi.org/10.4171/owr/2006/01. Proposed year 2006; no proposer named.
• Supplied literature label The angel and the devil in three dimensions: https://doi.org/10.1016/j.jcta.2005.03.009.
• Supplied literature label original Oberwolfach report: https://doi.org/10.4171/owr/2006/01.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field reports source checking, not independently verified here.