Josephus problem¶
The recurrence problem of locating the survivor or elimination order when positions in a circle are removed at a fixed counting interval.
Core Idea¶
Indexing may be zero- or one-based, counting convention determines whether the current or next position starts a round, and the classical step-two closed form does not extend unchanged to arbitrary step size. After each kth position is removed, the circle is relabeled relative to the next starting position; this self-similarity yields a recurrence mapping the survivor index for n minus one positions back into the n-position circle. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Josephus problem belongs to discrete mathematics and is useful where the analyst can specify the typed discrete mathematics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the number of positions n, step size k, circular ordering and direction, initial counting position, inclusion convention, elimination and restart rule, zero- or one-based survivor recurrence, base case, closed form for special k and full elimination permutation are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the number of positions n, step size k, circular ordering and direction, initial counting position, inclusion convention, elimination and restart rule, zero- or one-based survivor recurrence, base case, closed form for special k and full elimination permutation are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Josephus problem. Josephus problem compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed discrete mathematics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the number of positions n, step size k, circular ordering and direction, initial counting position, inclusion convention, elimination and restart rule, zero- or one-based survivor recurrence, base case, closed form for special k and full elimination permutation are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of discrete mathematics because they reuse the typed discrete mathematics carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, After each kth position is removed, the circle is relabeled relative to the next starting position; this self-similarity yields a recurrence mapping the survivor index for n minus one positions back into the n-position circle., and type the carrier, state every parameter and convention in the definition, test that the number of positions n, step size k, circular ordering and direction, initial counting position, inclusion convention, elimination and restart rule, zero- or one-based survivor recurrence, base case, closed form for special k and full elimination permutation are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Josephus problem Domain-specific
Parents (1) — more general patterns this builds on
-
Josephus problem is a kind of Recursion Prime
The proposed strict upward parent is
prime:recursion.
Hierarchy path (1) — routes to 1 parentless root
- Josephus problem → Recursion
Neighborhood in Abstraction Space¶
Josephus problem sits in a moderately populated region (43rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Recurrences & Integer Sequences (5 abstractions)
Nearest neighbors
- Leonardo number — 0.91
- Recurrence relation — 0.91
- Constant-recursive sequence — 0.90
- Motzkin number — 0.88
- Generic property — 0.88
Computed from structural-signature embeddings · 2026-09-08