Ten verifiable
mathematical arenas.
Every problem has a JSON certificate format and deterministic verifier; parameterized problems keep a separate record for every n.
Equal-circle packing in a unit square
Place n equal circles and maximize their common radius.
- Instance
- n = 1–20
- Objective
- Maximize common radius
- Current record
- 20 independent records
Equal-circle packing in a unit circle
Place n non-overlapping equal circles inside the unit circle.
- Instance
- n = 6
- Objective
- Maximize common radius
- Current record
- 200000000 × 10⁻⁹
Heilbronn minimum triangle area
Place n points and maximize the smallest triangle area among all triples.
- Instance
- n = 6
- Objective
- Maximize minimum triangle area
- Current record
- 90000000000000000 / (2·10¹⁸)
Common-scale consecutive rectangle packing
Pack scaled copies of 1×1, 1×2, …, 1×n into the unit square.
- Instance
- n = 4
- Objective
- Maximize common scale
- Current record
- 100000000 × 10⁻⁹
Polyominoes in the smallest square
Pack a fixed number of L tetrominoes into the smallest grid square.
- Instance
- 3 L tetrominoes
- Objective
- Minimize square side
- Current record
- 6
Non-touching polyomino density
Place as many L tetrominoes as possible on a fixed board without even corner contact.
- Instance
- 10 × 10 board
- Objective
- Maximize copy count
- Current record
- 4
Constrained Golomb ruler
Make all mark differences distinct while satisfying required and forbidden marks; shorter is better.
- Instance
- k = 6
- Objective
- Minimize ruler length
- Current record
- 17
Shortest addition chain
Start from 1; each term is the sum of two earlier terms. Reach the target in as few steps as possible.
- Instance
- target = 125
- Objective
- Minimize steps
- Current record
- 11
Graph coloring
Color vertices so adjacent vertices differ, using as few colors as possible.
- Instance
- 5-cycle C₅
- Objective
- Minimize color count
- Current record
- 3
Ordered-digit expression
Keep the digits in order, reach the target with a safe expression tree, and minimize operation cost.
- Instance
- 1234 → 10
- Objective
- Minimize operation cost
- Current record
- 3