Rarity Guide
Below I'll detail the rarity of features from the 70s Pop Series One. I'll also show the number of times we'd expect a feature to show up and compare it to what actually turned up, so we can find out which things are above and below the expect rarity.
Tiles
Probably the most complex calculation (but not that tricky) along with the lines, is the first one that works out how many tiles are going to be in the image. The full calculation used is below.
let lineTile1 = Math.random()
let lineTile2 = Math.random()
let tiles = Math.floor((((lineTile1 + (lineTile2 / 2)) / 1.5) * 14) + 5)
Right at the start two random numbers are picked, with the values between 0 and 0.99999999, which are then combined.
This is very similar to rolling two six sided dice. With two dice the chances are getting two ones or two sixes (making two and twelve) is a lot less than ending up scoring seven. You'll score 2 or 12 one time every 36 throws, but 7 one is six times. By combining two numbers we skew the results towards the middle.
In our case the lowest number possible would be (((0 + (0 / 2)) / 1.5) * 14) + 5, which is
5, while the highest would be (((0.9999 + (0.99999 / 2)) / 1.5) * 14) + 5
which is
18.
Below is the expected statistical breakdown, and what we actually got. As we can see the biggest surprise is that the 15x15 tile number is so underrepresented. If the 15x15 which is supposed to be common but isn't somehow makes it more rare than something that's supposed to be rare and is I don't know, are things rare by being unusual?
| Tiles | Percent chance | Expected | Actual | Difference | OpenSea Link |
|---|---|---|---|---|---|
| 5x5 | 1.14% | 2.93 | 3 | +0.07 | OpenSea |
| 6x6 | 3.44% | 8.81 | 6 | -2.81 | OpenSea |
| 7x7 | 5.73% | 14.69 | 19 | +4.31 | OpenSea |
| 8x8 | 8.03% | 20.56 | 24 | +3.44 | OpenSea |
| 9x9 | 10.2% | 26.11 | 27 | +0.89 | OpenSea |
| 10x10 | 10.71% | 27.42 | 24 | -3.42 | OpenSea |
| 11x11 | 10.71% | 27.42 | 31 | +3.58 | OpenSea |
| 12x12 | 10.71% | 27.42 | 26 | -1.42 | OpenSea |
| 13x13 | 10.71% | 27.42 | 30 | +2.58 | OpenSea |
| 14x14 | 10.2% | 26.11 | 27 | +0.89 | OpenSea |
| 15x15 | 8.03% | 20.56 | 9 | -11.56 | OpenSea |
| 16x16 | 5.73% | 14.69 | 17 | +2.31 | OpenSea |
| 17x17 | 3.44% | 8.81 | 11 | +2.19 | OpenSea |
| 18x18 | 1.14% | 2.93 | 2 | -0.93 | OpenSea |
Lines & Line Boost
The number of lines used is reverse proportional to the number of tiles. This is done by using the same
numbers we rolled at the start lineTile1 and lineTile2 and plugging them into
this formula...
let lines = Math.floor(9 - (lineTile1 + lineTile2) / 2 * 8) + 1
This means that the number of lines isn't randomly spread across all the values, but clusered based on the number of tiles. For example, if the we had 7x7 tiles, then we could have 7-9 lines. 12x12 tiles give us 4-6 lines, while 16x16 tiles gives us 2-4.
But there's more shenanigans than that. To give us a chance to break out of those boundaries, we also have a Line Boost, with this code...
if (Math.random() < 0.1) linesBoost--
if (Math.random() < 0.1) linesBoost -= 2
if (Math.random() < 0.1) linesBoost++
if (Math.random() < 0.1) linesBoost += 2
There's a 10% chance of each thing happening, so to have a Line Boost of -3, you'd have to pass the 10% chance of getting -1, then another 10% chance of -2, then avoiding either of the +1 and +2 boosts. Meaning there's an 0.81% chance of getting either the -3 or +3 boosts.
Finally, there's a couple of checks on the extremes to make sure we haven't done anything stupid.
if (tiles >= 16) lines--
if (lines < 1) lines = 1
if (tiles > 17 && lines > 6) lines = 6
Meaning the final number of lines breakdown is as follows, but remember, the number of tiles you first had will effect the range of lines you'll fall into. The full statistical breakdown of that will have to wait for another blog post.
First the line boosts...
| Line Boost | Percent chance | Expected | Actual | Difference | OpenSea Link |
|---|---|---|---|---|---|
| -3 | 0.81% | 2.07 | 1 | -1.07 | OpenSea |
| -2 | 7.37% | 18.88 | 14 | -4.88 | OpenSea |
| -1 | 8.19% | 20.98 | 14 | -6.98 | OpenSea |
| 0 | 67.23% | 172.11 | 187 | +14.89 | OpenSea |
| 1 | 8.19% | 20.98 | 24 | +3.02 | OpenSea |
| 2 | 7.37% | 18.88 | 15 | -3.88 | OpenSea |
| 3 | 0.81% | 2.07 | 1 | -1.07 | OpenSea |
And now the lines
| Lines | Percent chance | Expected | Actual | Difference | OpenSea Link |
|---|---|---|---|---|---|
| 1 | 4.09% | 10.47 | 9 | -1.47 | OpenSea |
| 2 | 6.01% | 15.38 | 18 | +2.62 | OpenSea |
| 3 | 6.55% | 16.77 | 13 | -3.77 | OpenSea |
| 4 | 14.17% | 36.29 | 37 | +0.71 | OpenSea |
| 5 | 19.28% | 49.36 | 48 | -1.36 | OpenSea |
| 6 | 19.61% | 50.2 | 45 | -5.2 | OpenSea |
| 7 | 15.02% | 38.46 | 48 | +9.54 | OpenSea |
| 8 | 9.63% | 24.65 | 26 | +1.35 | OpenSea |
| 9 | 4.2% | 10.76 | 9 | -1.76 | OpenSea |
| 10 | 1.07% | 2.75 | 1 | -1.75 | OpenSea |
| 11 | 0.3% | 0.78 | 2 | +1.22 | OpenSea |
| 12 | 0.02% | 0.06 | 0 | -0.06 | OpenSea |
Format
The design of each image is made up of only two tiles, straight or curved. Which tile gets chosen for each
position is defined by the format. The possible formats are; Normal, Middle,
Noise, Sideways and Top-bottom.
Normal means there's a 50/50 chance of one or the other tiles being picked.
Middle means the chance of curves being picked over straight lines increases towards the
middle of the image. This format happens to be one my favourites.
Noise is using a thing called Perlin Noise, which is a pseudorandom random number generator.
This causes the layout to be slightly more structured than the pure random of Normal.
Sideways and Top-bottom, mean that the chance of picking a straight tile over a
curved one increase as you move from one edge to the other. The end effect of these two formats are
affected by the Rotation feature, but still comes into play when combined with the
Pattern feature.
Reversed
Having decided which Format to use, Reversed flips the chance of a tile
being straight or curved. So previously if our Format was Middle (having
more curves towards the center) reversing it would put the curves into the corner and straight
tiles into the middle.
There is a base chance of 40% of the design being reversed. That chance drops down to 20% if the
Middle or Noise formats are selected. Meaning there's a 60% chance of the 40%
chance of reversing, getting reduced down to 20%. Thus Middle and Noise formats
have less chance of being reversed than the others, making a reversed Middle rarer than a
reversed Sideways.
The final overall results look like this.
Pattern
Patterns are extra designs that get added to the overall image. If a pattern is defined then on certain areas of the design a mathematical function will be used to decided if a straight or curved tile is used, and more importantly which way around that tile should go.
There is only a 5% chance that a design will be given a pattern. Once it's been decided there's going to
be a pattern one is selected from the following, with a decreasing chance; scales,
feather, flow, weave, tiles, zig and
zag.
As you can see from the results below we got lucky with patterns on this run of 256 mints. Statistically we should only have 12.8 designs with a pattern, but we got 24, almost double what was probable, with an extra 11 patterned mints.
| Patten | Percent chance | Expected | Actual | Difference | OpenSea Link |
|---|---|---|---|---|---|
| None | 95% | 243.2 | 232 | -11.2 | OpenSea |
| scales | 1.19% | 3.07 | 3 | -0.07 | OpenSea |
| feather | 1.05% | 2.68 | 6 | +3.32 | OpenSea |
| flow | 0.84% | 2.17 | 1 | -1.17 | OpenSea |
| weave | 0.69% | 1.79 | 6 | +4.21 | OpenSea |
| tiles | 0.54% | 1.4 | 5 | +3.6 | OpenSea |
| zig | 0.39% | 1.02 | 2 | +0.98 | OpenSea |
| zag | 0.24% | 0.63 | 1 | +0.37 | OpenSea |
Decoration
In addition to the patterns there's a chance of a "decoration", the most noticeable of which is the "Rose" in the middle, which can sometimes be upgraded to a "Big Rose". You can also get a "Corner rose" which is a similar design but split into the four corners.
You'll notice the corner roses more on smaller number of tiles, and was also added to making tiling the design, or wrapping it around cubes (for CV art galleries) easier.
Picking a decoration isn't quite a straight chance, roses will only appear when the design has an even number of tiles, i.e. 6x6, 8x8, 10x10, 12x12 and so on. This is so they can be centered. There's an 8% chance of a "rose" if there's an even number of tiles. If a rose gets chosen, there's a 25% chance of it being upgraded to a "Big Rose".
Meanwhile, corner roses get added 7% of the time, and if there's an even number of tiles there's then a 10% chance of adding in a center rose.
Morever, if a corner rose gets chosen, it could be over written by the middle rose. The code looks like this...
let decoration = 'None'
if (Math.random() <= 0.07) {
decoration = 'cornerRose'
if (Math.random() < 0.1 && !(tiles % 2) && tiles > 6) decoration = 'CornerRoseMiddle'
}
if (!(tiles % 2)) {
if (Math.random() <= 0.08) {
decoration = 'rose'
if (Math.random() < 0.25) decoration = 'bigRose'
}
}
Because the chances are based on the odd/eveness of the tiles the final percents end up like this...
Palette
There are sixteen palettes in total, each of which contains five colours. Each palette has an equal chance of being selected, no one palette is rarer than another. Once a palette has been randomly selected two colours are picked from it. The second one will only be used as part of a gradient.
I'll do a fuller breakdown of the individual colours that got selected from the palettes, but for the moment this is what we have.
| Palette | Colours | Percent | Expected | Actual | Difference | OpenSea |
|---|---|---|---|---|---|---|
| From the nighttime neon city wasteland of the 80s music scene | 6.25% | 16 | 12 | -4 | OpenSea | |
| Winter | 6.25% | 16 | 21 | +5 | OpenSea | |
| Grove | 6.25% | 16 | 17 | +1 | OpenSea | |
| Tuscany | 6.25% | 16 | 16 | 0 | OpenSea | |
| Delta | 6.25% | 16 | 12 | -4 | OpenSea | |
| Spray | 6.25% | 16 | 18 | +2 | OpenSea | |
| 7pm | 6.25% | 16 | 12 | -4 | OpenSea | |
| Not Plum | 6.25% | 16 | 18 | +2 | OpenSea | |
| Abandoned Flip-Flop | 6.25% | 16 | 22 | +6 | OpenSea | |
| Woodland Camping | 6.25% | 16 | 18 | +2 | OpenSea | |
| 7am | 6.25% | 16 | 17 | +1 | OpenSea | |
| False Colour Nebular | 6.25% | 16 | 22 | +6 | OpenSea | |
| Twilight Field | 6.25% | 16 | 16 | 0 | OpenSea | |
| 3pm | 6.25% | 16 | 9 | -7 | OpenSea | |
| C64 | 6.25% | 16 | 14 | -2 | OpenSea | |
| In the White Room at 3am | 6.25% | 16 | 12 | -4 | OpenSea |
Fill
The fill feature is how the colour gets applied to the image. Most of the time just a single flat colour is used, but sometimes a gradent happens.
The base chance of a gradient happening is 20%, unless the "From the nighttime neon city wasteland of the 80s music scene" palette is used, in which case the chance goes up to 80%, because that's the way I roll. So if you got one of the twelve using that palette you have a much better chance of ending up with a fill of some kind.
There are two types of fills, a straight Left-right or Top-down gradient. Or a
"Radial" fill, which is normally from the middle outwards, but sometimes from one of the four
corners, top-left, top-right, bottom-left and bottom-right.
Fills, like formats and patterns are affected by rotations.
| Fill | Percent chance | Expected | Actual | Difference | OpenSea Link |
|---|---|---|---|---|---|
| Normal | 76.25% | 195.2 | 200 | +4.8 | OpenSea |
| Left-right | 9.5% | 24.32 | 18 | -6.32 | OpenSea |
| Top-down | 9.5% | 24.32 | 25 | +0.68 | OpenSea |
| Radial | 2.83% | 7.24 | 9 | +1.76 | OpenSea |
| Radial-tl | 0.48% | 1.23 | 3 | +1.77 | OpenSea |
| Radial-tr | 0.48% | 1.23 | 0 | -1.23 | OpenSea |
| Radial-bl | 0.48% | 1.23 | 0 | -1.23 | OpenSea |
| Radial-br | 0.48% | 1.23 | 1 | -0.23 | OpenSea |
Rotate
Rotation is used to do pretty much what you'd expect. However, rotations were mainly added to affect the patterns, and as such a rotation will only happen if a pattern has been selected. Therefore normally there'd be a 5% chance of a rotation happening, but because we got lucky with the patterns on this run we have slightly more than expected rotations.
24 mints ended up getting a pattern, so out of those six should be rotated 0 degrees, six aught to be 90, six 180 and finally six 270. The table below shows what we actually got.
Inverted
The inverted effect causes the colouring to be sort of inverted, but slightly more complicated than that.
The maths however is simple, there's a 7.5% chance of an inversion.
Doubled
The doubled feature caused the tiles to be randomised and drawn twice, giving somewhat of a vinyl record effect.
There's a 2.34375% chance of a mint getting doubled, a fancy number because I was hoping for around six to get doubled out of a 256 run.
Phased
Phased is the surprise feature. In this series there aren't any moving animation but there is a colour cycle. It didn't take as long as I expected for someone to spot it, but it was a delight when they did "Hey, is mine changing color?"
Not all mints have a chance of getting phased. Phased will only show up on mints with a gradient fill, or are inverted.
If a mint's fill isn't Normal then there's a 10% chance of it getting phased. In this run we
have 56 mints with a gradient of some sort, so five to six of those should be phased.
If a mint is inverted, then there's a 25% of it being phased. We ended up with 16 inversions, so around four of those should be phased.
Together the expected verses actual are shown below, but again, phased will only happen on mints that already have one of the above features.
End note
Hopefully the above helps you explore the 256 70s Pop Series One mints a little more. I've used green text when there's more than the expected number, and red text (unless you have deuteranopia) when there's less.
I do understand though that from a collectors point of view, you're probably looking for mints where there are less than there's supposed to be. In which case keep an eye out for the red numbers.
Of course some of the mints are far rarer because they combine a number of already rare features. Soon I'll add a post highlighting some of the rarest overall mints. The best way to find out when I publish that is probably to follow me on twitter @revdancatt.
Thank you for joining in with this fun artistic adventure.