The integers 1,2,3 and 4 add up to 10, which was considered perfect and contained in itself the whole nature of numbers. This number was graphically represented by the figure known as the Tetratys, which became a sacred symbol for the Pythagorean's.

Numbers was responsible for "harmony", the divine principle that governed the structure of the whole world.

For the Pythagorean's the numbers had and retained a mystical significance, an independent reality. Phenomena were secondary for the only significant thing about phenomena was the way in which they reflected numbers. We have here an attitude that is utterly different from that of a mathematician of today. Mathematics had for them and also later for Plato a metaphysical as well as a purely mathematical significance.

The model for the seal configuration is twin tetractys of ten points each. (The Pythagoreans named the number 10 tetractys because it was the sum of the first four numbers.) The 13 symbolism in the seal could bear looking at again in light of this finding. 13 can be understood as the sun and 12 zodiac signs, 13 lunar months a year, or nine celestial spheres and the four directions as in Metatron's Cube. you can see there are four markers for the cardinal directions which along with the first nine circles figure to us a 3D Cartesian coordinate grid and the six directions in space , what we call the 'cube of space', or Metatron's Cube (below) which should now be seen in a new light as 13 celestial spheres. If the earth is not the center as in the nine worlds image we need new points as direction markers. And we find them in Ezekiel. A man, a lion, ox and an eagle. And Rev 4 lion calf man eagle which is Aq Leo Taurus and Scorpio. the fixed signs in the zodiac. In the context of the solar system, these represent the four corners.

8,9,16,17,24,28,30,40,41,52,8,9,16,17,24,28,30,40,41,52, etc., yet I've never seen a definitive answer on any website.


I counted 1616 whole squares (44 going down on left side, 44 going down on right side, 88 small squares in the centre). With this method the answer could also be

Geometric shape: Tetrahedron
Piece configuration: 3×3×3

Main article: Pyraminx

Tetrahedral-shaped puzzle with axes on the corners and trivial tips. It was invented in 1970 by Uwe Mèffert.

Commercial Name: Pyramorphix
Geometric shape: Tetrahedron
Piece configuration: 2×2×2

Main article: Pyramorphix


Example of a fold puzzle, created by Vesa Timonen (2002)

Example of a fold puzzle, created by Vesa Timonen (2002)

It is four squares

The aim in this particular genre of puzzles is to fold a printed piece of paper in such a way as to obtain a target picture. In principle, Rubik's Magic could be counted in this category. A better example is shown in the picture. The task is to fold the square piece of paper so that the four squares with the numbers lie next to each other without any gaps and form a square.

Impossible objects are objects which at first sight do not seem possible. The most well known impossible object is the ship in a bottle. The goal is to discover how these objects are made. Another well known puzzle is one consisting of a cube made of two pieces interlocked in 4 places by seemingly inseparable links (example). The solutions to these are to be found in different places. There are all kinds of objects which fit this description – bottles in which there are objects that are far too large (see impossible bottles), Japanese hole coins with wooden arrows and rings through them, wooden spheres in a wooden frame with far too small openings and many more.


The puzzle consists of thirteen polycubic pieces: twelve pentacubes and one tetracube. The objective is to assemble these pieces into a 4 x 4 x 4 cube. There are 19,186 distinct ways of doing so, up to rotations and reflections.

The need to find better foundries obliged him to shuttle every week between Paris and Verona; he decided to settle definitively in the latter city in 1967. After remaining there for a while, he moved to Negrar, where he would live until 2004. His sculpture became increasingly complex. The search for a “fourth dimension” became evident in Adamo Secundus and David, small works with complex possibilities of disassembly. The sculptures of this period were so successful that in the United States they acquired the status of “conversation pieces”.[citation needed


Tantrix is a hexagonal tile-based abstract game invented by Mike McManaway from New Zealand. Each of the 56 different tiles in the set contains three lines, going from one edge of the tile to another. No two lines on a tile have the same colour. There are four colours in the set: red, yellow, blue, and green. No two tiles are identical, and each is individually numbered from 1 through 56.

In Latin, tessella is a small cubical piece of claystone or glass used to make mosaics.[12] The word "tessella" means "small square" (from tessera, square, which in turn is from the Greek word τέσσερα for four). It corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay.

A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are based on it,[2] explaining its name.[1] It is commonly used as a pattern for floor tiles. When used for this, it is also known as a hopscotch pattern[3] or pinwheel pattern,[4] but it should not be confused with the mathematical pinwheel tiling, an unrelated pattern.[5]

This tiling has four-way rotational symmetry around each of its squares. When the ratio of the side lengths of the two squares is an irrational number such as the golden ratio, its cross-sections form aperiodic sequences with a similar recursive structure to the Fibonacci word. Generalizations of this tiling to three dimensions have also been studied.


Isometries of the Euclidean plane fall into four categories (see the article Euclidean plane isometry for more information).

  • Translations, denoted by Tv, where v is a vector in R2. This has the effect of shifting the plane applying displacement vector v.

  • Rotations, denoted by Rc,θ, where c is a point in the plane (the centre of rotation), and θ is the angle of rotation.

  • Reflections, or mirror isometries, denoted by FL, where L is a line in R2. (F is for "flip"). This has the effect of reflecting the plane in the line L, called the reflection axis or the associated mirror.

  • Glide reflections, denoted by GL,d, where L is a line in R2 and d is a distance. This is a combination of a reflection in the line L and a translation along Lby a distance d.


The Slothouber–Graatsma puzzle is a packing problem that calls for packing six 1 × 2 × 2 blocks and three 1 × 1 × 1 blocks into a 3 × 3 × 3 box. The solution to this puzzle is unique (up to mirror reflections and rotations). It was named after its inventors Jan Slothouber and William Graatsma.

Conway's puzzle, or Blocks-in-a-Box, is a packing problem using rectangular blocks, named after its inventor, mathematician John Conway. It calls for packing thirteen 1 × 2 × 4 blocks, one 2 × 2 × 2 block, one 1 × 2 × 2 block, and three 1 × 1 × 3 blocks into a 5 × 5 × 5 box.


The pieces of the Soma cube consist of all possible combinations of three or four unit cubes, joined at their faces, such that at least one inside corner is formed. There is one combination of three cubes that satisfies this condition, and six combinations of four cubes that satisfy this condition, of which two are mirror images of each other (see Chirality). Thus, 3 + (6 × 4) is 27, which is exactly the number of cells in a 3×3×3 cube.

Both types of tromino can be dissected into n2 smaller trominos of the same type, for any integer n > 1. That is, they are rep-tiles.[4] Continuing this dissection recursively leads to a tiling of the plane, which in many cases is an aperiodic tiling. In this context, the L-tromino is called a chair, and its tiling by recursive subdivision into four smaller L-trominos is called the chair tiling.


Both types of tromino can be dissected into n2 smaller trominos of the same type, for any integer n > 1. That is, they are rep-tiles.[4] Continuing this dissection recursively leads to a tiling of the plane, which in many cases is an aperiodic tiling. In this context, the L-tromino is called a chair, and its tiling by recursive subdivision into four smaller L-trominos is called the chair tiling.[5]


The snake cube is a mechanical puzzle, a chain of 27 or 64 cubelets, connected by an elastic band running through them. The cubelets can rotate freely. The aim of the puzzle is to arrange the chain in such a way that they will form 3 x 3 x 3 or 4 x 4 x 4 cube.

quadrant looking

Squaring the square is the problem of tiling an integral square using only other integral squares. (An integral square is a square whose sides have integer length.) The name was coined in a humorous analogy with squaring the circle. Squaring the square is an easy task unless additional conditions are set. The most studied restriction is that the squaring be perfect, meaning that the sizes of the smaller squares are all different. A related problem is squaring the plane, which can be done even with the restriction that each natural number occurs exactly once as a size of a square in the tiling. The order of a squared square is its number of constituent squares.

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Since 1999, four further probably prime repunits have been found, but it is unlikely that any of them will be proven prime in the foreseeable future because of their huge size.

There are many palindromic perfect powers nk, where n is a natural number and k is 2, 3 or 4.

  • Palindromic squares: 0, 1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944, ... (sequence A002779 in the OEIS)

  • Palindromic cubes: 0, 1, 8, 343, 1331, 1030301, 1367631, 1003003001, ... (sequence A002781 in the OEIS)

  • Palindromic fourth powers: 0, 1, 14641, 104060401, 1004006004001, ... (sequence A186080 in the OEIS)

The first nine terms of the sequence 12, 112, 1112, 11112, ... form the palindromes 1, 121, 12321, 1234321, ... (sequence A002477 in the OEIS)

The only known non-palindromic number whose cube is a palindrome is 2201, and it is a conjecture the fourth root of all the palindrome fourth powers are a palindrome with 100000...000001 (10n + 1).