Invariants, casework, and pigeonhole readiness

Use elementary olympiad-style methods such as invariant tracking, organized cases, and guarantee reasoning in visible puzzle settings.

Surface

This skill appears in Invariant move legal-illegal mark, Pigeonhole box guarantee choice, Casework cover board, Invariant game challenge choice, Unique-perfect-matching edge extremum proof, Orthogonal-grid removal counterexample strategy, Geometric network minimum-cost entry, Switch-bulb one-visit identification, Region-doorway trail-start choice, Card-operation move-count pair choice, Transport-crossing sequence proof.

Task formats

Card-operation move-count pair choice

Card-operation move-count pair choice asks the player to determine the exact numbers of allowed swaps and flips in an exactly-three-move solution, then choose the ordered pair (swaps, flips).

ActionDetermine the exact numbers of allowed swaps and flips in an exactly-three-move solution, then choose the ordered pair (swaps, flips).
CheckComplete the card-operation move-count pair choice action.
UseSupport / Core / Transfer
ShowsCheck only
PresentationCompact / Standard / Extended

Practice focus

Word problems are confusing

Casework cover board

Casework cover board asks the player to cover all cases with chips and mark missing, duplicate, or impossible cases.

ActionCover all cases with chips and mark missing, duplicate, or impossible cases.
CheckComplete the casework cover board action.
UseSupport / Core / Transfer
ShowsCheck only
PresentationCompact / Standard / Extended

Practice focus

Word problems are confusing

Geometric network minimum-cost entry

Geometric network minimum-cost entry asks the player to interpret an exact geometric network domain, optionally construct a candidate, and enter the globally minimal length or cost in the declared unit.

ActionInterpret an exact geometric network domain, optionally construct a candidate, and enter the globally minimal length or cost in the declared unit.
CheckComplete the geometric network minimum-cost entry action.
UseSupport / Core / Transfer
ShowsCheck only
PresentationCompact / Standard / Extended

Practice focus

Geometry, shapes, and measurement

Invariant game challenge choice

Invariant game challenge choice asks the player to predict possible or impossible final state in a constrained move game using an invariant proof card.

ActionPredict possible or impossible final state in a constrained move game using an invariant proof card.
CheckChoose the answer that matches the visible prompt.
UseSupport / Core / Transfer
ShowsCheck only
PresentationCompact / Standard / Extended

Practice focus

Word problems are confusing

Orthogonal-grid removal counterexample strategy

Orthogonal-grid removal counterexample strategy asks the player to disprove a universal score guarantee in a centrally symmetric orthogonal-grid removal game by submitting a valid value assignment and a complete legal counter-strategy.

ActionDisprove a universal score guarantee in a centrally symmetric orthogonal-grid removal game by submitting a valid value assignment and a complete legal counter-strategy.
CheckComplete the orthogonal-grid removal counterexample strategy action.
UseSupport / Core / Transfer
ShowsCheck only
PresentationCompact / Standard / Extended

Practice focus

Word problems are confusing

Pigeonhole box guarantee choice

Pigeonhole box guarantee choice asks the player to choose the guaranteed conclusion from objects and boxes or capacities.

ActionChoose the guaranteed conclusion from objects and boxes or capacities.
CheckChoose the answer that matches the visible prompt.
UseSupport / Core / Transfer
ShowsCheck only
PresentationCompact / Standard / Extended

Practice focus

Word problems are confusing

Region-doorway trail-start choice

Region-doorway trail-start choice asks the player to interpret a complete undirected regions-and-doorways graph and select the unique start region for a trail using every doorway exactly once and ending at a named region.

ActionInterpret a complete undirected regions-and-doorways graph and select the unique start region for a trail using every doorway exactly once and ending at a named region.
CheckChoose the answer that matches the visible prompt.
UseSupport / Core / Transfer
ShowsCheck only
PresentationCompact / Standard / Extended

Practice focus

Word problems are confusing

Switch-bulb one-visit identification

Switch-bulb one-visit identification asks the player to determine a legal ordered switch strategy and use the one inspection to identify the complete switch-to-lamp correspondence.

ActionDetermine a legal ordered switch strategy and use the one inspection to identify the complete switch-to-lamp correspondence.
CheckComplete the switch-bulb one-visit identification action.
UseSupport / Core / Transfer
ShowsCheck only
PresentationCompact / Standard / Extended

Practice focus

Word problems are confusing

Transport-crossing sequence proof

Transport-crossing sequence proof asks the player to construct and order a legal crossing sequence, then complete a bounded optimality-and-uniqueness certificate.

ActionConstruct and order a legal crossing sequence, then complete a bounded optimality-and-uniqueness certificate.
CheckComplete the transport-crossing sequence proof action.
UseSupport / Core / Transfer
ShowsCheck only
PresentationCompact / Standard / Extended

Practice focus

Word problems are confusing

Unique-perfect-matching edge extremum proof

Unique-perfect-matching edge extremum proof asks the player to for a balanced finite bipartite graph constrained to have a unique perfect matching, submit the maximum edge count, an upper-bound certificate, and an attaining construction.

ActionFor a balanced finite bipartite graph constrained to have a unique perfect matching, submit the maximum edge count, an upper-bound certificate, and an attaining construction.
CheckComplete the unique-perfect-matching edge extremum proof action.
UseSupport / Core / Transfer
ShowsCheck only
PresentationCompact / Standard / Extended

Practice focus

Word problems are confusing

Selection

Fresh independent work can move toward Core, Transfer, or Challenge routes; help use or confusion keeps support visible.

Help

Hints keep attention on the visible structure; reveals show the path without counting as independent evidence.

Presentation

Presentation can reduce reading and interaction load without changing the selected skill. Practice focus can bias compatible work without breaking prerequisites or story fit.

Story fit

  • story beats with visible logic state, manipulable evidence, and a bounded player action
  • episodes with visible movement, fitting, nets, layers, grids, routes, or solid viewpoints