40 verifiable
mathematical arenas.
Equal-circle packing in a unit square
Place n equal circles and maximize their common radius.
- Instance
- n = 1–30
- Objective
- Maximize common radius
- Progress
- 30 of 30 finished
Equal-circle packing in a unit circle
Place n non-overlapping equal circles inside the unit circle.
- Instance
- n = 1–29
- Objective
- Maximize common radius
- Progress
- 14 of 29 finished
Heilbronn minimum triangle area
Place n points and maximize the smallest triangle area among all triples.
- Instance
- n = 1–12
- Objective
- Maximize minimum triangle area
- Progress
- 0 of 12 finished
Common-scale consecutive rectangle packing
Pack scaled copies of 1×1, 1×2, …, 1×n into the unit square.
- Instance
- n = 1–20
- Objective
- Maximize common scale
- Progress
- 2 of 20 finished
Tilted equal squares in a circle
Place n equal squares inside a circle of radius 1, each free to tilt, and maximize their common side.
- Instance
- n = 1–12
- Objective
- Maximize the smallest side
- Progress
- 1 of 12 finished
Packing circles of radius 1,2,…,n into a circle
Pack n mutually non-overlapping circles of radii 1,2,…,n, drawn to scale, into a circle and minimize its radius.
- Instance
- n = 1–29
- Objective
- Minimize container radius
- Progress
- 3 of 29 finished
Spreading points in a circle
Place n points inside a circle of radius 1 so that the smallest distance between any two is as large as possible.
- Instance
- n = 1–17
- Objective
- Maximize the smallest distance between two points
- Progress
- 1 of 17 finished
Equal circles in an L
Place n equal circles inside an L-shaped region, making their common radius as large as possible.
- Instance
- n = 1–15
- Objective
- Maximize common radius
- Progress
- 0 of 15 finished
Equal circles in a half-disc
Place n equal circles in a half-disc of radius 1, making their common radius as large as possible.
- Instance
- n = 1–14
- Objective
- Maximize common radius
- Progress
- 0 of 14 finished
Equal circles in a plus sign
Place n equal circles inside a plus-shaped region, making their common radius as large as possible.
- Instance
- n = 1–16
- Objective
- Maximize common radius
- Progress
- 0 of 16 finished
Equal-circle packing in a right triangle
Place n non-overlapping equal circles inside a fixed right triangle with legs 1 and 0.75, maximizing their common radius.
- Instance
- n = 1–11
- Objective
- Maximize common radius
- Progress
- 0 of 11 finished
Equal-circle packing in a 2:1 rectangle
Place n non-overlapping equal circles in a 2×1 rectangle, maximizing their common radius.
- Instance
- n = 1–10
- Objective
- Maximize common radius
- Progress
- 4 of 10 finished
Packing circles of radius 1,2,…,n into a square
Pack n mutually non-overlapping circles of radii 1,2,…,n, drawn to scale, into a square and minimize its side.
- Instance
- n = 1–29
- Objective
- Minimize square side
- Progress
- 3 of 29 finished
Spreading points in the unit square
Place n points in the unit square so that the smallest distance between any two of them is as large as possible.
- Instance
- n = 1–11
- Objective
- Maximize the smallest distance between two points
- Progress
- 4 of 11 finished
Spreading points in a right triangle
Place n points in a right isosceles triangle with legs 1, maximizing the smallest distance between any two.
- Instance
- n = 1–15
- Objective
- Maximize the smallest distance between two points
- Progress
- 0 of 15 finished
Spreading points in a rectangle
Place n points in a 2 × 1 rectangle, maximizing the smallest distance between any two.
- Instance
- n = 1–17
- Objective
- Maximize the smallest distance between two points
- Progress
- 0 of 17 finished
Tilted equal squares in the unit square
Place n equal squares inside the unit square, each free to tilt, and maximize their common side.
- Instance
- n = 1–28
- Objective
- Maximize the smallest side
- Progress
- 10 of 28 finished
Spreading points in an L
Place n points in an L-shaped region, maximizing the smallest distance between any two.
- Instance
- n = 1–17
- Objective
- Maximize the smallest distance between two points
- Progress
- 0 of 17 finished
Spreading points in a half-disc
Place n points in a half-disc of radius 1, maximizing the smallest distance between any two.
- Instance
- n = 1–15
- Objective
- Maximize the smallest distance between two points
- Progress
- 0 of 15 finished
Spreading points in a plus sign
Place n points in a plus-shaped region, maximizing the smallest distance between any two.
- Instance
- n = 1–17
- Objective
- Maximize the smallest distance between two points
- Progress
- 0 of 17 finished
The smallest triangle in a disc
Place n points in a disc of radius 1 so the smallest triangle any three of them make is as large as possible.
- Instance
- n = 1–10
- Objective
- Maximize the smallest triangle's area
- Progress
- 0 of 10 finished
The smallest triangle in an L
Place n points in an L-shaped region so the smallest triangle any three of them make is as large as possible.
- Instance
- n = 1–10
- Objective
- Maximize the smallest triangle's area
- Progress
- 0 of 10 finished
The smallest triangle in a plus sign
Place n points in a plus-shaped region so the smallest triangle any three of them make is as large as possible.
- Instance
- n = 1–10
- Objective
- Maximize the smallest triangle's area
- Progress
- 0 of 10 finished
The smallest triangle in a half-disc
Place n points in a half-disc of radius 1 so the smallest triangle any three of them make is as large as possible.
- Instance
- n = 1–10
- Objective
- Maximize the smallest triangle's area
- Progress
- 0 of 10 finished
Spreading points in an annulus
Place n points in an annulus of outer radius 1 and inner radius 0.5, maximizing the smallest distance between any two.
- Instance
- n = 1–17
- Objective
- Maximize the smallest distance between two points
- Progress
- 1 of 17 finished
The smallest triangle in an annulus
Place n points in an annulus of outer radius 1 and inner radius 0.5 so the smallest triangle any three of them make is as large as possible.
- Instance
- n = 1–10
- Objective
- Maximize the smallest triangle's area
- Progress
- 0 of 10 finished
Heilbronn's problem in a triangle
Place n points inside the right triangle (0,0), (size,0), (0,size) and maximize the smallest triangle area among all triples.
- Instance
- n = 1–6
- Objective
- Maximize the smallest triangle's area
- Progress
- 3 of 6 finished
Equal circles in a quadrant
Place n equal circles in a quarter-disc of radius 1, making their common radius as large as possible.
- Instance
- n = 1–14
- Objective
- Maximize common radius
- Progress
- 0 of 14 finished
Spreading points in a quadrant
Place n points in a quarter-disc of radius 1, maximizing the smallest distance between any two.
- Instance
- n = 1–15
- Objective
- Maximize the smallest distance between two points
- Progress
- 0 of 15 finished
The smallest triangle in a quadrant
Place n points in a quarter-disc of radius 1 so the smallest triangle any three of them make is as large as possible.
- Instance
- n = 1–10
- Objective
- Maximize the smallest triangle's area
- Progress
- 0 of 10 finished
Riesz 2-energy in a square
Place n points in the unit square, minimizing the sum of 1/distance² over every pair.
- Instance
- n = 1–20
- Objective
- Minimize Riesz 2-energy
- Progress
- 0 of 20 finished
Riesz 2-energy in a disc
Place n points in a disc of radius 1, minimizing the sum of 1/distance² over every pair.
- Instance
- n = 1–20
- Objective
- Minimize Riesz 2-energy
- Progress
- 0 of 20 finished
Lighting a unit square
Place n lights of unit brightness in a unit square so the darkest point of the square is as bright as possible.
- Instance
- n = 1–19
- Objective
- Maximize minimum intensity
- Progress
- 0 of 19 finished
Smallest ratio of largest to smallest distance
Place n points so the distance between the furthest pair, divided by the distance between the closest pair, is as small as possible.
- Instance
- n = 1–20
- Objective
- Minimize max-to-min distance ratio
- Progress
- 0 of 20 finished
The biggest little polygon
Take n points with no two further than 1 apart, and make the convex polygon they enclose as large as possible.
- Instance
- n = 1–13
- Objective
- Maximize the area
- Progress
- 4 of 13 finished
Minimum star discrepancy in the unit square
You have n samples to spread over a square frame. Any rectangle measured from one corner should hold the same share of the samples as it holds of the area; the worst mismatch is your score.
- Instance
- n = 1–29
- Objective
- Minimize the worst gap D*
- Progress
- 0 of 29 finished
Optimal quantization in the unit square
Place n respawn points on a square map. A player appears uniformly at random and is sent to the nearest one; make the average squared trip as short as you can.
- Instance
- n = 1–25
- Objective
- Minimize the average squared distance
- Progress
- 0 of 25 finished
The most uniform sampling mesh in the unit square
Place n points in the unit square; the red circle is the largest uncovered hole and the blue line the closest pair. Make the ratio of hole radius to pair spacing as small as you can.
- Instance
- n = 1–36
- Objective
- Minimize the uniformity M
- Progress
- 0 of 36 finished
Sum of radii in the unit square
Place n non-overlapping circles of any sizes in the unit square, maximizing the sum of their radii.
- Instance
- n = 1–30
- Objective
- Maximize sum of the radii
- Progress
- 1 of 30 finished
The smallest triangle in an equilateral triangle
Place n points in an equilateral triangle of side 1 so the smallest triangle any three of them make is as large as possible.
- Instance
- n = 1–10
- Objective
- Maximize the smallest triangle's area
- Progress
- 0 of 10 finished