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# Pentomino|Definition & Meaning

## Definition

**Pentominoes** are figures made from a combination of five **squares **and** joined** with one or more than one joint sides. There are **twelve** methods to construct these shapes. They are respectively named as the English alphabets from letter **T** to letter **Z** and then the letters **F, I, L, P,** and** N**.

**Pentominoes** are figures that operate with five **square** blocks joined edge to edge to form different mixtures. There are twelve achievable shapes in a set of **distinctive pentominoes** that are characterized **by T, U, V, W, X, Y, Z, F, I, L, P,** and** N.** The word **FILiPiNo** and the end of the alphabet, **TUVWXYZ **is a technique that is being used by people easily remember the **pentominoes **letters set**. **

To see what counts as a **pentomino**There are some **rules****.** If you rotate a shape and then both shapes look the same, that does not count as **pentomino** also if you flip a **shape** and then both shapes look the same then it will also not count as **pentomino.**

## About Pentominoes

Take five **identical** squares. Arrange the **squares** so that at least one edge of each square is **connected** to at least one of the other four squares. Find all such positions, then remove any **arrangement** that when turned or flipped **becomes** the same as any other **arrangement.** There are only **12** separate arrangements or pieces that can be **made.**

Each **pentomino** has a **single-letter** name to it that, on one point, produces the **shape** of that piece, as shown below.

## Constructing Rectangular Dimensions

A typical pentomino puzzle is to build a **rectangle** with the pentominoes, i.e. making it without **gaps** or without **overlapping.** A **pentomino** has an area of **5** unit **squares,** and there are **12 pentominoes,** so the box must be **60** units. **3×20, 4×15, 5×12 and 6×10 **are the only possible sizes.

In **1960,** Colin Brian Haselgrove and Jenifer Haselgrove solved the first case which was the **6×10**. The **exact** number of solutions is **2339,** excluding trivial variations that are brought by **rotating** or **reflecting** of the **rectangle,** but if we include rotation and **reflection** of a subset of the **pentominoes** (which occasionally **delivers** an extra solution in an **easy** way). There are **1010 solutions** to the **5×12** box, **368** solutions to the **4×15** box, and just **2** solutions to the **3×20** box (one of the two is **displayed** in the diagram, and the other one can be **obtained **from **rotating** the whole **block** consisting of the alphabets: **L, N, F, T, W, Y, and Z** pentominoes).

A **relatively** easy (more symmetrical) to solve the puzzle, the **2×2** hole in the center of the** 8×8** rectangle, was solved by **Dana Scott** in 1958. There are a total of **65** solutions. Scott’s algorithm was one of the **foremost** applications of a backtracking **computer** program. The four holes can be **placed** in any position **thanks** to the variations of this **puzzle.** This rule is used by one of the **external** links.

More than half of such **patterns** are resolvable, but arranging all pairs of **holes** separately near the two **corners** of the board must be one of the **exceptions, ** in such a way that a **P-pentomino** could only fit **both** corners, or on the other hand, driving a **T-pentomino** or **U-pentomino** in a corner with the end goal that another hole is made.

## Symmetry

**Pentominoes** symmetry is as **follows:**

- The English alphabet, F,
**L, N, P, and Y**can be**introduced**in eight manners: 4 each by**rotation and**reflection. Their**symmetry**group**consists**only of identity**mapping.** - T and U can be
**introduced**in 4 ways by**rotation. Aligned**with the gridlines, they have an axis of**reflection.**Their**symmetry**group consists of two**elements,**the identity and the**reflection**in a line that is parallel to the sides of the**squares.** **V and W**also can be**introduced**by rotation in 4**ways.**At**45**° to the gridlines, they have an**axis**of reflection**symmetry.**Their**symmetry**group has two elements,**identity,**and a**diagonal**reflection.- Z can be
**introduced**in 4 ways: 2 by turning, and 2 more by**reflection.**It has point**symmetry,**also called**rotational**symmetry of order 2. Its symmetry group consists of two**elements,**the identity, and the**180**°**rotation.** - I can be introduced in 2 ways and both by
**rotation.**The two axes of reflection symmetry are both aligned with the**gridlines.**Its symmetry group**consists**of four components, the identity, two reflections, and the**180°**turn. It is the dihedral gathering of**order 2**, too called the**Klein four-group.** - X can be
**introduced**in only one way.**Aligned**with the gridlines, it has four axes of**reflection**symmetry, and the**diagonals,**and rotational symmetry are of order four.

The **chiral pentominoes are****, L, N, P, Y, and Z****;** causing their **reflections** (F′, J, N′, Q, Y′, S) to obtain the number of ** one-sided** pentominoes to 18. If

**rotations**are also considered separate, then the

**pentominoes**from the very first category would be

**eightfold,**the ones from the following three categories (T, U, V, W, Z) would count fourfold, I counts

**twice,**and X counts only one time. The output for

**5×8 + 5×4 + 2 + 1**is

**63**

*set*pentominoes.

For the 2D figure, there are two more types:

**Being**introduced in 2 ways by a rotation of**90°,**with the two axes of reflection symmetry lined up with the**diagonals.**At least a heptomino is required for this type of**symmetry.**- Being
**introduced**in 2 ways, both are mirror images of each**other,**for example, a**swastika.**At least an octomino is required for this type of**symmetry.**

### Three-dimensional Puzzles

If using cubes **instead** of squares for making a set of **pentominoes,** you can also try to solve many other **puzzles.** (Wooden cubes are sold at many craft stores, but try and make sure that they are of original sizes.) The **first** puzzle with these solid pentominoes should be able to **place them** into a **3x4x5** box.

From **twelve** pentominoes, ten can be called **three-dimensional** with all of the **twelve** pieces. Create the **pentomino** figures on a **2/1** scale, but only this time another **dimension** is added– their height should be three stories tall! Now, this **technique** is relatively easy as the **pentominoes** can be used in any way that works. Remember that there are two **pentominoes** that cannot be created in this way — the **W** and the **X**.

## Example of Pentomino

Form a **rectangular** formation of pentominoes shapes with 5**x12** shapes.

### Solution

**5×12** rectangular formation pentominoes can be **formed** as:

*All images/mathematical drawings were created with GeoGebra.*