PentoCoin
NFT PentoCoin is a diamond in the world of crypto. It is mined from the solutions of pentomino puzzle. Derived from the Greek word for '5', and domino, a pentomino (or 5-omino) is a polyomino of order 5, that is, a polygon in the plane made of 5 equal-sized squares connected edge-to-edge. There are 12 different pentominoes.
Pentomino tiling puzzles and games are popular in recreational mathematics. A standard pentomino puzzle is to tile a rectangular box with the pentominoes, i.e. cover it without overlap and without gaps. Each of the 12 pentominoes has an area of 5 unit squares, so the box must have an area of 60 units. Possible sizes are 6×10, 5×12, 4×15 and 3×20.
Contents
History
In October 2022, the NFT collection NFT PentoCoin was created and listed on OpenSea.[1] The first item from the collection is given on the Opensea platform.[2]
Overview
All 12 pentomino shapes are numbered from 1 to 12. The number of combinations from 12 unique digits is 479,001,600. It is calculated as follows:
The number of combinations from 12 unique digits can be calculated using permutations. Since each digit must be unique in each combination, we can use the permutation formula.
The formula for permutations, P(n, r), is defined as n! / (n - r)!, where n is the number of elements and r is the number of selected elements.
In this case, we have 12 digits, and we are selecting all 12 digits (r = 12). Thus, the permutation formula will look as follows:
P(12, 12) = 12! / (12 - 12)! = 12! Where "!" denotes factorial. Let's calculate it:12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 479,001,600
The 6×10 case was first solved in 1960 by Colin Brian Haselgrove and Jenifer Haselgrove.[3] There are exactly 9356 solutions, including trivial variations obtained by rotation and reflection of the whole rectangle and including rotation and reflection of a subset of pentominoes (which sometimes provides an additional solution in a simple way). The 5×12 box has 4040 solutions, the 4×15 box has 1472 solutions, and the 3×20 box has just 8 solutions. In total, there are 14,876 combinations of pentomino shapes.
However, there are certain combinations of pentomino shapes that share the same combination of numbers. These combinations have a property called "sibling," where multiple different shape combinations correspond to a single combination of numbers. The creator of the NFT PentoCoin collection, Alex Carolus Rex Attewood, calculated that there are a total of 14,153 digit combinations that yield solutions for pentomino. Therefore, the probability of a 12-digit combination being a solution for pentomino is 0.00295204% (14,153 out of 479,001,600).
| Size | Combinations of Shapes | Combination of Digits | Uniqueness, % | Sibling | % Siblings |
| 6×10 | 9356 | 8819 | 0,00183949 | 1020 | 10,90 |
| 5×12 | 4040 | 3887 | 0,00070602 | 303 | 7,50 |
| 4×15 | 1472 | 1440 | 0,00001251 | 60 | 4,07 |
| 3×20 | 8 | 8 | 0,00000167 | 8 | 0,00 |
| Total | 14876 | 14153 | 0,00295204 | 1383 | 9,30 |
Sibling Property
| Size | Solutions | combinations of shapes | Sibling, % | Combination of digits | Sibling Level | Uniqueness, % |
| 6×10 | 9356 | 1020 | 10,90 | 483 | 47,35 | 0,00010069 |
| 5×12 | 4040 | 303 | 7,50 | 150 | 49,50 | 0,00003126 |
| 4×15 | 1472 | 60 | 4,08 | 28 | 46,67 | 0,00000585 |
| 3×20 | 8 | 0 | 0,0 | 0 | 0,00 | 0,00000000 |
| Total | 14876 | 1383 | 8,36 | 660 | 47,72 | 0,00013755 |
| Size | Solutions | Sibling | ||||||||||||||
| 2 | 3 | 4 | 5 | 6 | ||||||||||||
| Qty | % | % | Qty | % | % | Qty | % | % | Qty | % | % | Qty | % | % | ||
| 6×10 | 9356 | 907 | 6,10 | 9,69 | 39 | 0,26 | 0,42 | 40 | 0,27 | 0,43 | 10 | 0,07 | 0,11 | 24 | 0,16 | 0,26 |
| 8819 | 454 | 0,00009476 | 5,15 | 13 | 0,00000271 | 0,14 | 10 | 0,00000209 | 0,11 | 2 | 0,00000042 | 0,02 | 4 | 0.00000084 | 0,05 | |
| 5×12 | 4040 | 291 | 1,95 | 7,20 | 12 | 0,08 | 0,30 | 0 | 0,00 | 0,00 | 0 | 0,00 | 0,00 | 0 | 0,00 | 0,00 |
| 3887 | 146 | 0.00003046 | 3,76 | 4 | 0,00000084 | 0,10 | 0 | 0,00000000 | 0,00 | 0 | 0,00000000 | 0,00 | 0 | 0,00000000 | 0,00 | |
| 4×15 | 1472 | 48 | 0,32 | 3,26 | 12 | 0,08 | 0,82 | 0 | 0,00 | 0,00 | 0 | 0,00 | 0,00 | 0 | 0,00 | 0,00 |
| 1440 | 24 | 0,00000500 | 1,67 | 4 | 0,00000084 | 0,28 | 0 | 0,00000000 | 0,00 | 0 | 0,00000000 | 0,00 | 0 | 0,00000000 | 0,00 | |
| 3×20 | 8 | 0 | 0,00000000 | 0,00 | 0 | 0,00000000 | 0,00 | 0 | 0,00000000 | 0,00 | 0 | 0,00000000 | 0,00 | 0 | 0,00000000 | 0,00 |
| Total | 14876 | 1246 | 8,37 | 8,36 | 63 | 0,42 | 0,42 | 40 | 0,27 | 0,43 | 10 | 0,07 | 0,11 | 24 | 0,16 | 0,26 |
| 660 | 623 | 0,00012988 | 90,39 | 21 | 0,00000438 | 3,19 | 10 | 0,000002085 | 1,51 | 2 | 0,00000042 | 0,30 | 4 | 0.00000084 | 0,61 | |
There are 660 combinations of digits that yield 1383 combinations of pentomino shapes. Therefore, subtracting 1383 combinations of shapes with the sibling property from the total number of possible combinations, which is 14876, and adding 660 unique combinations of digits that yield pentomino solutions with the sibling property, we obtain 14153 combinations of digits that give pentomino solutions with the sibling property. Among them, 9.3% have more than one shape combination for each digit combination, indicating the presence of the sibling property.
The combination of digits 7 6 2 12 11 3 8 10 5 9 4 is the only one that serves as a solution for both the 6×10 and 5×12 size combinations of shapes. The probability of such digits combinations is 0,000000209238%.
Pento Music
Associating each of the 12 pentomino shapes with a unique musical note based on their respective numbers can create a musical representation of the pentomino solutions.
Assigning note C to shape 1 and so on, according to the table, you have generated a total of 14,153 unique minimal musical compositions corresponding to the number of unique digit combinations that yield pentomino solutions.
It's fascinating to explore the intersection of mathematics, shapes, and music in this way, creating a unique auditory experience based on the properties of the pentomino puzzles.
| pentomino ID | Name Golomb | Name
Conway |
note | sound |
| 1 | L | Q | C | |
| 2 | I | O | C | sharp |
| 3 | V | V | D | |
| 4 | T | T | D | sharp |
| 5 | X | X | E | |
| 6 | F | R | F | |
| 7 | Z | Z | F | sharp |
| 8 | N | S | G | |
| 9 | W | W | G | sharp |
| 10 | U | U | A | |
| 11 | Y | Y | A | sharp |
| 12 | P | P | B |
Songs
It is possible to compose songs using combinations of sounds.
1) In total there are 35 songs
2) There are no songs for Dragon + Leo
3) For Earth + North there are two songs only from Tiger + Gemini and Rooster + Capricornus
| Size | Classical Element | Cardinal Direction | Chinese Zodiac | Zodiac | YouTube Playlist | ||
| 6×10 | 1 | 1 | Fire | East | Rat | Aries | NFT PentoCoin #1 5 4 6 8 9 2 10 7 3 12 11 |
| 6×10 | 1 | 2 | Fire | East | Ox | Taurus | To Be Created |
| 6×10 | 1 | 3 | Fire | East | Tiger | Gemini | To Be Created |
| 6×10 | 1 | 4 | Fire | East | Rabbit | Cancer | To Be Created |
| 6×10 | 1 | 6 | Fire | East | Snake | Virgo | To Be Created |
| 6×10 | 1 | 7 | Fire | East | Horse | Libra | To Be Created |
| 6×10 | 1 | 8 | Fire | East | Goat | Scorpio | To Be Created |
| 6×10 | 1 | 9 | Fire | East | Monkey | Sagittarius | To Be Created |
| 6×10 | 1 | 10 | Fire | East | Rooster | Capricornus | To Be Created |
| 6×10 | 1 | 11 | Fire | East | Dog | Aquarius | To Be Created |
| 6×10 | 1 | 12 | Fire | East | Pig | Pisces | To Be Created |
| 5×12 | 2 | 1 | Water | South | Rat | Aries | To Be Created |
| 5×12 | 2 | 2 | Water | South | Ox | Taurus | To Be Created |
| 5×12 | 2 | 3 | Water | South | Tiger | Gemini | To Be Created |
| 5×12 | 2 | 4 | Water | South | Rabbit | Cancer] | To Be Created |
| 5×12 | 2 | 6 | Water | South | Snake | Virgo | To Be Created |
| 5×12 | 2 | 7 | Water | South | Horse | https://en.wikipedia.org/wiki/Libra_(astrology) Libra] | To Be Created |
| 5×12 | 2 | 8 | Water | South | Goat | Scorpio | To Be Created |
| 5×12 | 2 | 9 | Water | South | Monkey | Sagittarius | To Be Created |
| 5×12 | 2 | 10 | Water | South | Rooster | Capricornus | To Be Created |
| 5×12 | 2 | 11 | Water | South | Dog | Aquarius | To Be Created |
| 5×12 | 2 | 12 | Water | South | Pig | Pisces | To Be Created |
| 4×15 | 3 | 1 | Air | West | Rat | Aries | To Be Created |
| 4×15 | 3 | 2 | Air | West | Ox | Taurus | To Be Created |
| 4×15 | 3 | 3 | Air | West | Tiger | Gemini | To Be Created |
| 4×15 | 3 | 4 | Air | West | Rabbit | Cancer] | To Be Created |
| 4×15 | 3 | 6 | Air | West | Snake | Virgo | To Be Created |
| 4×15 | 3 | 7 | Air | West | Horse | Libra | To Be Created |
| 4×15 | 3 | 8 | Air | West | Goat | Scorpio | To Be Created |
| 4×15 | 3 | 9 | Air | West | Monkey | Sagittarius | To Be Created |
| 4×15 | 3 | 10 | Air | West | Rooster | Capricornus | To Be Created |
| 4×15 | 3 | 11 | Air | West | Dog | Aquarius | To Be Created |
| 4×15 | 3 | 12 | Air | West | Pig | Pisces | To Be Created |
| 3×20 | 4 | 3 | Earth | North | Tiger | Gemini | To Be Created |
| 3×20 | 4 | 10 | Earth | North | Rooster | Capricornus | To Be Created |
Price Calculation
| Size | sibling | Price, ETH | Quantity | Amount, ETH |
| 6×10 | 1 | 1000 | 8336 | 8336000 |
| 6×10 | 2 | 2000 | 907 | 1814000 |
| 6×10 | 3 | 3000 | 39 | 117000 |
| 6×10 | 4 | 4000 | 40 | 160000 |
| 6×10 | 5 | 5000 | 10 | 50000 |
| 6×10 | 6 | 6000 | 24 | 144000 |
| 5×12 | 1 | 10000 | 3737 | 37370000 |
| 5×12 | 2 | 20000 | 291 | 5820000 |
| 5×12 | 3 | 30000 | 12 | 360000 |
| 4×15 | 1 | 100000 | 1412 | 141200000 |
| 4×15 | 2 | 200000 | 48 | 9600000 |
| 4×15 | 3 | 300000 | 12 | 3600000 |
| 3×20 | 1 | 1000000 | 8 | 8000000 |
| Total | 14876 | 216571000 |
See also
References
- ↑ "PentoCoin". opensea.io. https://opensea.io/collection/pentocoin. Retrieved 9 June 2023.
- ↑ "PentoCoin #1 2 3 4 5 6 7 8 9 10 11 12". opensea.io. https://opensea.io/assets/ethereum/0x495f947276749ce646f68ac8c248420045cb7b5e/23467797326275027743874371014149683281019546929197637997700847618907801911297. Retrieved 9 June 2023.
- ↑ C. B. Haselgrove; Jenifer Haselgrove (October 1960). "A Computer Program for Pentominoes". Eureka 23: 16–18. https://www.archim.org.uk/eureka/archive/Eureka-23.pdf.
