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 number seven.

demonstrations.wolfram.com

Many Demonstrations use Seven

Dr. Matrix

Martin Gardner’s alter ego had many connections to the number 7.

Kobon Triangle

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.

Seven Soma Cube Pieces and Seven Tangrams

Seven Touching Cylinders

How many cylinders can cover/touch each other? https://demonstrations.wolfram.com/SevenTouchingCylindersPuzzle/
And with infinite cylinders:

Seven Sevens

The Gray codes arrange the integers so that only one binary bit changes per step. Here’s order 7:
The seven-crossing knots:
The number of dominoes is a triangular number. Double-7 has a square number of dominoes.
For some reason, sets mostly end at 6, 9, 12, 15 or 18.
For the size 7 board, the non-intersection knight tour is symmetric:
On the sides of a die, opposite faces add to 7. The most common roll of 2 dice is 7.

S(2,7,49) :

In a toroidal wrapping of 7×7 grids, we can 56 7-sets so that every pair is in one set.
In each grid the digits seen by 1 are shown.
The 74 order-7 matchstick graphs:

The Seven Standardized SI Units (November 16, 2018)

In his blog for SI Units, Michael Trott took a photograph the moment it became official.
As of 2019, the International System of Units (SI) defines its seven base quantities using fixed values of fundamental physical constants. These constants form the core of our modern metrology.
Base Quantities and Corresponding Constants:
540 THz (≈555 nm) , the most perceived frequency for the human eye. Personally, I’d have gone for 547.533 THz (≈547.533 nm) .
Or a green laser, 532 nm. When they made the color wheel, they should have put green on candela. The luminosity function, V(λ),
was decided in 1924 by the Commission Internationale de l’Éclairage (CIE). Mercury Green = 546.1nm. Chartreuse= 548. Harlequin = 552.

Fano plane and triples from 1-7

The Fano plane:
Here’s another Fano plane:
There are thirty distinct Fano planes:
These divide into two sets of 15 so that any pair of Fano planes shares exactly one triple:
Here’s a game I put together based on my favourite graph:
There are 35 triples from 1-7.
Let’s arrange the 35 ways to pick triples so that each is on a unit circle:
Combining 15 Fano planes and 35 triples gives the Hoffman-Singleton Graph
Two triples are linked if they share no values. A triple on a Fano plane is linked to that plane.
Every triple is on exactly four Fano planes. Every card connects to seven other cards.
Any two non-connected cards share a connection with exactly one card.

Seven Hexahedra

As 3-D objects:

The 7 Vertex Moser Spindle and 7-coloring the Plane
The Hadwiger–Nelson problem

The Moser Spindle seven vertices. With it, one can prove at least 4 colors are needed to color the plane so that no two points of the same color are exactly one apart.
Using hexagons, the plane can be colored with seven colors:
Here’s a more complicated method I found where only about 1/302th of the plane needs the seventh color of white.
Alex Soifer published it in the Mathematical Coloring Book.
The corners of one white square are {{0,0},{x,x},{2x,0},{x,-x}} where x~0.0453993677815601129113265166981, the first real root of
Here’s the seven-coloring, with no two points of the same color within 1 of each other.
The black bars here show some of the unit distances:

The 49 8-dot patterns -- a Grid of Grids

A complete set of 8-dot patterns. Rotations and Reflections are excluded. Proper subsets only, so All and Nothing are excluded.
This is an easy set to make and I know many that have done so. I made my set with index cards and a hole punch.

Folding a Strip to Form Regular Odd n-Gons

By Izidor Hafner

MG8: Polygon Dissections

I gave a talk on these in Episode 8: Polygon Dissections. Six of them use 7 pieces.

The Seven Cell Rubik’s Cube

If we glue a cuboid of and order 2 Rubik’s cube to the corner of a table, their are 24 positions reachable with half-twists.
The graph is equivalent to all the positions of a rolling octahedron. The set of positions gives the Nauru graph.

Gosper Flowsnake

A self-avoiding curve by Bill Gosper

Show

7-Dragon -- Root 7

More order-7 space-filling curves.

Seven Pack

From Erich's Packing Center.

Series Rectangle Dissections

Here’s the Kepler triangle, based on powers of the golden ratio:
Six Kepler triangles growing in powers of the golden ratio will make a rectangle:
Seven triangles growing in powers of the supergolden ratio:
Seven triangles growing in powers of the plastic ratio:
I also found a chi ratio triangle divided into a series of 7 triangles based on the chi constant.

Conway criteria and the 7 failing heptominoes

There are 108 heptominoes, which would be a nice 12-color set:
There are 7 heptominoes that fail the Conway criteria:

Triangle into 7 Isosceles triangles

Here’s the heptagonal triangle divided into seven acute isosceles triangles:
A polynomial relating to heptagons
If the discriminant of a cubic is 49, it relates to heptagons
The vertices of a heptagon have a simple number field form:

The Grünbaum-Rigby Configuration

Many years after the configuration was found, someone had the idea to add a point at the center.
28 lines of 4 points.

Orchard -7 to 7 -- the solution Terrence Tao missed

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.
37 lines of 3 for 16 points (one at infinity).
I wrote more on the topic at Cultivating New Solutions for the Orchard-Planting Problem.

The Seven Rules of Origami and the Beloch Fold for the Heptagon

In origami, their are seven Huzita–Hatori axioms:
1. Fold through two points (line through two points).
2. Fold a point onto another point (perpendicular bisector).
3. Fold a line onto a line (angle bisector).
4. Fold through a point perpendicular to a line (line through a point perpendicular to a line).
5. Fold a point onto a line, passing the crease through another point (tangent to a parabola).
6. Fold a point onto a line and another point onto another line (common tangent to two parabolas).
7. Fold a point onto a line, making the crease perpendicular to another line (tangent to a parabola perpendicular to a line).
The sixth axiom was found in 1936 by Margherita Beloch. This fold is now named the Beloch fold, and can be used in origimi to solve various problems that cannot be solved with ruler and compass.
We can make a fold for a heptagon:

Regular Polygons of Edge Length Two

Annulus for a Regular Polygon

A moment of Venn

I’ve long been a fan of the order 7 Venn diagrams over at combinatorics.org, so I thought I’d try my hand at them.
Here’s code for the Victoria presentation of Venn-7.
That page also shows a solution in quadrilaterals. Perhaps not as obvious, but I’ve tried to make the colors match up.
And now a Venn ghost for Halloween.

Braced Heptagon to Klein 24

Start with the Braced Heptagon, a unit distant graph which braces the heptagon:
Add edges for seven lines:
Connect 3 points to the 7/1, 7/2 and 7/3 polygons:
The graph is equivalent to the Klein 24 graph:
On the hyperbolic plane:

Trisecting Cevians for 1/7

We can try with the heptagon triangle:
And the check:

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.
Here’s the unique smallest valence 7 graceful graph.
Here’s the permutation:
And the Lehmer code. For each number in the permutation, count how many subsequent values it exceeds.

Intrinsically Knotted Graphs

Conway-Gordon: all sets of 7 points in 3D standard position (no 4 coplanar) always define at least one knot.
Exactly one knot is possible

Szilassi polyhedron

The dual of the Csaszar polyhedron is the Szilassi polyhedron. Seven hexagons that all touch each other.

Kirkman Schoolgirls

15 girls in triples for 7 days.
If a graph is made of the connection between the 7 points and 7 lines of the Fano plane, you’ll get the Heawood graph.
The Heawood Graph on the Szilassi Polyhedron, which is the dual of the Császár Polyhedron:
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.
The Room square is named after Thomas Room, even though Cayley created it earlier.
A set of 28 double-8 dominoes without blanks or doubles.
abc d35 e17 f28 g46 gives the first day.
Another reference: Room Squares and Related Designs.

The Seven Circles Theorem (demo by Claude Fabre)

Take six circles tangent to each other in pairs and tangent to the unit circle on the inside. The points of contact of the six circles with the unit circle define a hexagon. The diagonals of the hexagon are concurrent.
This concurrency is obvious when the hexagon is regular. The theorem states that it still holds when the radii and the positions of the circles vary.

The Seven Bridges of Königsberg (demo by S. M. Blinder)

In the 1730s, Leonhard Euler lived in the Prussian city of Königsberg. The Pregel River runs around the center of the city (Kneiphof) and then splits into two parts. The city was then quite prosperous and the volume of commerce justified connections between the separated land masses by seven bridges. A popular problem of the day was to find a continuous path which would cross all seven bridges once and only once.

St Ives Puzzle

As I was going to St Ives,
I met a man with seven wives,
Each wife had seven sacks,
Each sack had seven cats,
Each cat had seven kits:
Kits, cats, sacks, and wives,
How many were there going to St Ives?
The number going away from St Ives is

The Integer Heptagon

In 2006, Tobias Kreisel and Sascha Kurz found a set of seven points, all at integer distances from each other. Mathematica’s text rotation and offset capabilities let you label the 21 lines with their distances in different ways.

Ramanujan–Nagell equation

Seven is the only value with multiple integer solutions:

The Seven Basic Surfaces and Seven Tetrahexes

Kleinian Integers

Heptagon Triangle and reflections

With a few intersection points of heptagon diagonals, a heptagon with double the area can be constructed.
The reflection triangle of the heptagon triangle (in green) is another heptagon triangle (in cyan) with twice the area.
Checking the side ratios:
We can look at the two diagonals:
How about their reciprocals?
The reciprocals sum to 1.

CITE THIS NOTEBOOK

Mathematical Games: number Seven across maths - graphs, geometry, designs, knots, and units​
by Ed Pegg​
Wolfram Community, STAFF PICKS, October 24, 2025
​https://community.wolfram.com/groups/-/m/t/3564738