This is part of live presentation series called Mathematical Games in which we explore a variety of games and puzzles using Wolfram Language. In this episode, we explore some covering sets.

demonstrations.wolfram.com

Many Demonstrations cover covering.

In the news:

Can you cover a tetrahedron so it is only stable on one face?
A New Pyramid-Like Shape Always Lands the Same Side Up
Related to Heppes’s Two-Tip Tetrahedron (link):
by: Izidor Hafner
A face of a polyhedron is stable if and only if the orthogonal projection of the center of mass of the polyhedron onto the plane of the face lies inside the face or on an edge. In other words, when the polyhedron is placed on that face, the center of mass is over the face. This tetrahedron has two unstable faces. If you place it on one of the unstable faces, it will topple to the other unstable face and then topple to one of the stable faces.
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In the News: The Kobon Triangle problem.

In Wheels, Life, and Other Mathematical Amusements, Gardner discussed the Kobon Triangle problem for getting the maximum coverage of triangles with a given number of lines.
On July 23, 2025, there were new advances announced.

Covering problems in Gardner’s books - Circles

Covering a circle with 5 circles
More at Erich's Packing Center.
That value also provides a solution for the 12 point Heilbronn problem maximizing the area of the smallest triangle in a unit square.

Covering problems in Gardner’s books - Ternary System

Covering problems in Gardner’s books -- Consecutive Squares

Ponting Squares

Covering problems in Gardner’s books -- Chess Coverings

One of Gardner’s favorite problems: Place 5 white queens and 3 black queens on a 5×5 chessboard so they don’t attack each other.

https://oeis.org/A051567 2*floor(2n/3) queens on a nxn board each attacking 1 other.

{0, 5, 0, 2, 149, 49, 1, 12897, 2238}
The solution for the 9x9 board is unique.

Covering problems in Gardner’s books -- Sparse Rulers

You can download Sparse Ruler Data.
Consider a sparse ruler of length 13. Clearly, five marks wouldn’t be enough since there are at most (5; 2)=10 differences between possible marks. On the other hand, six marks is enough, but because that gives (6; 2)=15 differences, there must be two repeats. {0, 1, 6, 9, 11, 13}
Here’s the unique Leech Yardstick:
Here’s unique 1m ruler.
The Pegg 1-meter ruler has a 100 partition, which measures 214cm.
NJA Sloane called the pattern “Dark Satanic Mills on a Cloudy Day”.
Part of the solution was finding hundreds of Wichmann-like Rulers.
Larger sparse rulers can be found at Sparse Rulers.
The write-up of the problem is at Hitting all the Marks.
As seen above, the length 50 sparse ruler on the left covers all lengths up to 50.

Cyclic Sum Sets

Like sparse rulers, but covers in a circle.
Here’s Singer's 1938 paper.
On a circular wheel is used, a perfect set of measuring marks is called a difference set, a Ganymede circle, or an n-switch. All the distances between selected red points are different. The distances are indicated by arcs inside the circle.
As a difference set, all possible differences are given by the set:

Perfect Partitions

We can look for perfect cyclic partitions for any N value. Here are some solutions.
Here are some trickier partitions to find.
The lengths of these partitions is as follows:
Here’s the excess pattern as it is known so far:

Difference Sets in Configurations

The {0,1,4,14,16}_21 difference set in circles and lines. All pairs of points 0-20 are connected by a circle or line.
The {0,1,3,9}_13 difference set as a configuration of parabolas

The Springer Difference Set -- Spot-It

The game of Spot-it uses a Springer difference set. Any two cards of the 57 card set share a number:
Any two cards of the 73 card set share a number:
Any two cards of the 133 card set share a number:

MG8: Polygon Dissections

I gave a talk on these in Episode 8: Polygon Dissections.

MG16: Space-filling Curves

How can we cover space with a curve?

MG17: Combinatorial Codes and Designs

The Marcel J. E. Golay 1949 paper. “The Best Single Page in Coding” - Berlekamp

Generating the Binary Golay Code

There is a natural way to generate the binary Golay code. Just use the inverse of the icosahedron! It’s that easy!
Just join with the identity matrix:

MG19: Turing Machines

Covering all programs.
What do order-n simple programs do?

MG26: Puzzles on a Square Grid

Mondrian Art Problem

Divide a square into non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles? This is known as the Mondrian art problem. This Demonstration shows optimal solutions up to size 65 and best known solutions up to 900.
Up to the square of side 100, check the "analysis" box to see the possible improvements. If all of those possibilities can be checked in a packing program without a new solution turning up, the given solution is optimal.

Thomas Kirkman Schoolgirl Problem

Fifteen young ladies in a school walk out three abreast for seven days in succession:it is required to arrange them daily,so that no two shall walk twice abreast.
The below grid was used by Cayley to solve the Kirkman Schoolgirl’s problem:
A set of 28 double-8 dominoes without blanks or doubles. This is now called a Room square.
abc d35 e17 f28 g46 gives the first day.

Smallest Projective Space

The 35 triples of the smallest projective space all have an XOR sum of zero
15 points, each on 7 planes.
15 planes, each on 7 points. {1,2,3,4,5,6,7}, {2,4,6,9,11,13,15}, {2,4,6,8,10,12,14}
Also, 35 lines each with BitXor sum 0.
Find the fifteen projective Fano planes. One of them is 1 to 7.

Social Golfer Problem (Cayley, Kirkman, Steiner)

A group consists of 32 golfers who want to play in groups of four for several days. Can they play for ten days, with no pair of golfers in the same group twice?
In 1850, Reverend Thomas Kirkman sent a related query to the readers of a popular math magazine, Lady’s and Gentleman’s Diary: Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two will walk twice abreast. Arthur Cayley and Jakob Steiner solved the problem. For groups of three, solutions are called either a resolvable Steiner triple system (RSTS) or a Kirkman triple system (KTS).

Steiner Trees

Soap films can cover a road network. These lead to Steiner trees.
by: Ferenc Beleznay
We would like to find an optimal way of connecting a given finite set of points in the plane, in the sense that the total length of the line segments joining the points is minimal. This optimization problem is referred to as the Steiner tree problem. This Demonstration supports a manual search for such a network of line segments. It can be proved that an optimal network is achieved by adding some points (called Steiner points) to the original set of so-called regular points and finding the minimal spanning tree of the resulting complete graph, where the weights of the edges are the Euclidean distances between the endpoints.

S(2,3,9) : Lo Shu Magic Square

Steiner systems cover a range. S(2,3,9) uses values up to 9 in sets of 3 so that all sets of 2 are covered. Many more coverings at the Covering Repository.
Any pair of numbers shows up in a row or column in exactly one of the grids.
Shown as triangles:

S(2,4,16) : The 9 4 Puzzle

Fill in the second grid so that any pair of numbers shows up in a row, column or main diagonal in exactly one of the grids.
To fill out the first row, there are 3 numbers to choose from. The 6 is on a diagonal.
For balance, 6 will need to be off the diagonals in the second grid.

Only memorize the 9 and 4.

Or we can arrange them all graphically (blue+magenta, yellow+green, orange):

S(2,5,25) : Magic Hexagon

Fill in the second grid so that any pair of numbers shows up in a row, column or wrapped \ diagonal in exactly one of the grids.
This is easier to see in a hexagonal grid:
This better presented with a clipped hexagon

S(2,7,49) :

For 7×7 grids, the grids are toroidal. In each grid the digits seen by 1 are shown.

A moment of Venn

The 32 regions of Venn-5 cover all possible true-false options for five sets.
Done another way:

Graceful Graphs

A graceful graph has numbered vertices and N edges such that the vertex differences are 1 to N.
Any permutation has a corresponding graceful graph. This comes from a property of Lehmer codes.
Note that the Lehmer code for the same permutation uses {0,1,2,6}.
For valence 9, the graph complement looks better.

The de Bruijn Torus

A de Bruijn sequence cyclically covers all combinations.
https://demonstrations.wolfram.com/TheDeBruijnTorus/

Covering a 24-cell with Hypercubes

We can cover a 24-cell with 3 hypercubes.

Engel 38 Spacefiller

What’s the most complex way to cover space with a convex polyhedron? The Engel polyhedron has 38 sides.
Here’s a picture of just the polyhedron:

Describe a polyhedron with minimal values.

The vertices of an icosahedron can be covered with one coordinate:
All the Platonic and Archimedean solids are easy to represent:

Hamiltonian Tours on Polyhedra (link)

We can cover the vertices of a polyhedron with a Hamiltonian tour.
by: Ed Pegg Jr
This Demonstration shows Hamiltonian tours on various polyhedra.

No Three In A Line

Can we cover every row and column of a grid with 2 markers so that there are never 3 markers in a straight line?
104 points on a 52×52 square. No 3 points are in a line.
Are larger solutions possible? No-one knows!

Orthoschemes and other Spacefilling Tetrahedra

A cube can replicate itself with 8 copies.
A few tetrahedra, known as orthoschemes, can be divided into eight similar copies of themselves.
Are there any such polyhedra?

Covering a Rolling Octahedron

We can cover all possible edge-roll orientations of an octahedron with a graph.
This gives the Nauru graph, or the Rolling Octahedron graph.
This is the same graph as the positions of a 2×2×2 Rubik’s cube with half-twists only.
We can similarly get a rolling of the icosahedron to cover all orientations. Doing the cube is a nice puzzle. I still haven’t found a nice solution for the dodecahedron.

Magic Sums

There are 32 tuples from -8 to 8 summing to zero.
For any triple of values that sum to zero, those three points are on a straight line.
Pick three numbers from -7 to 7 that sum to 0, mod 15. Are they on a straight line here?
We can add another point at infinity for the six faint horizontal green lines.

Power Wheel Graphs

How can similar triangles cover a vertex?

Life is Omniperiodic

WFR GraphCoordinationSequence

What graphs cover a particular property?
The coordination sequence is the number of vertices at distance 0, 1, 2, ... from a vertex.
Many graphs have the coordination sequence {1,3,6,4} for all vertices.

Seven Touching Cylinders

How many cylinders can cover/touch each other?

CITE THIS NOTEBOOK

Mathematical Games: Covering Sets​
by Ed Pegg​
Wolfram Community, STAFF PICKS, July 24, 2025
​https://community.wolfram.com/groups/-/m/t/3518190