Perfect Clear Probability?

[sh1218]

So I was looking at this page [https://harddrop.com/wiki/Perfect_Clear_Opener] about perfect clears recently, and I noticed how it mentioned about probability of success for making perfect clears.

For example, "...If I piece is kept on hold throughout first bag, then there's a chance of 84.6 % (711/840) to succeed..." There's even a table below labelled first PC success rates for several setups. However, it only states the statistics and nothing more.

I'm kinda curious in how they came up with these numbers, but I don't know where to start from. Can anyone please tell me how these percentages were obtained?

[Okey_Dokey]

Those percentages assume that you can see and think multiple pieces ahead and make the optimal decisions accordingly, i.e. it assumes you get the Perfect Clear whenever the sequence allows it. For a Perfect Clear after the first 4 cleared lines, you'll need 11 pieces, if you also count the piece you have on hold in the end. That's 4 pieces into the second bag. There are 7*6*5*4 = 840 possibilities how the first 4 pieces of the second bag can look like. There's at least one way to achieve a Perfect Clear for 514 out of those 840 possibilities, if you regard the Standard PC opener where you have placed 7 pieces. That's 514/840 = 61.19 %.

For example consider the sequence TLIJ. With the help of hold, softdrop and SRS kicks, there are 4 different ways (see frames 3 to 6) to achieve a PC with those 4 pieces in the specific order. So, TLIJ is one of the 514 possibilities where a PC is possible. OJZS is also one of those 514 woking possibilities but there's only one unique way which is very hard to spot (see frame 7). Sequences like OJZS are the reason why players will not achieve the 61.19 % PC rate in practice. The sequence OIZS is NOT one of those 514 working possibilities; you'll always need at least one T/L/J piece to achieve a PC.

[fumen]v115@9gzhDeR4h0hlCeR4wwg0RpglBeBtxwg0RpglCeBtww?JeAgH9gT4Dexhhlh0CexhQpglxwg0BeBtRpglxwg0CeBtQp?JeAAABhhlh0Fewwglg0GexwHewwglg0MeAAtRAQDckDloo2?AUoo2AmOCbEUBAAABhRaRLFewSAeAPGewSJeQpQaMeAAeCh?CtFeQaAeQpGeQawSHeAPxSMeAAtiAQDckDloo2AMoo2AmOC?bE0oo2AFb+YEPqeoDy3tTAylAAAChCPFeAtQpHewwQpHeww?BtMeAAtiAQDckDloo2AUoo2AmOCbE0oo2AFbmOCFbsYEwOp?TAylAAA[/fumen]

How do I know there are 514 working possibilities? There are Japanese sites that have enumerated through all of those 840 possibilites, e.g. take this one. Same site also lists the same for the cases where you have just placed 6 of the first 7 pieces. Most prominently the case where you leave the I piece on hold (and have a 1 column wider gap to fill). Here 711 out of the 840 possibilities can end in a Perfect Clear. That's 711/840 = 84.64 %. The sequence OJZS (note the I piece on hold) has more than way to achieve the PC now (see frames 3 to 5).

[fumen]v115@9gilEeR4glRpDeR4wwg0RpCeBtxwi0DeBtwwJeAg0V?AJoo2Aw+kkDloo2APG98AQcTDEEBAAA9gi0Eexhg0xwDexh?QpglxwCeBtRpilDeBtQpJeAAeAhwhg0AtR4Eewhi0FewhBt?GewhAtR4MeAAPcAvfLuClsCSASITeDt488AwR0TAS4MrDmX?yrD9gC8h0AtR4E8zhF8g0BtG8g0AtR4C8JeAAPVAvfLuCls?CSASoyCE0IKKEBP2kEFBAAA9gC8i0R4E8zhF8Rpg0G8RpR4?C8JeAAA[/fumen]

There's also a Japanese PC "solution finder" tool you can use to calculate those chances. Here's a download for an older version of said tool, and following image explains a little how to use it.

[!--ImageUrlBegin--][a href=\\\"https://media.discordapp.net/attachments/285201280782172160/457631355946270720/solfinder.png\\\" target=\\\"_new\\\"][!--ImageUrlEBegin--][img width=\\\"500\\\" class=\\\"attach\\\" src=\\\"https://media.discordapp.net/attachments/285201280782172160/457631355946270720/solfinder.png\\\" border=\\\'0\\\' alt=\\\"IPB Image\\\" /][!--ImageUrlEnd--][/a][!--ImageUrlEEnd--]

[sh1218]

Thanks for the reply!!! Wow it's quite surprising how people actually checked all of those 840 possibilities.

I just got curious on how you mentioned that in the first setup (where you use your I piece to form a 4x4 square with the J,O,L) for a perfect clear to work you would either need T, L or a J piece.

Is there some kind of a mathematical explanation for this requirement or is it because that no sequence without these three pieces gave a solution for a perfect clear according to the link you gave me?

[Okey_Dokey]

The mathematical explanation can be found on the Perfect Clear Opener article:

[!--ImageUrlBegin--][a href=\\\"https://i.imgur.com/FsRdv65.png\\\" target=\\\"_new\\\"][!--ImageUrlEBegin--][img width=\\\"500\\\" class=\\\"attach\\\" src=\\\"https://i.imgur.com/FsRdv65.png\\\" border=\\\'0\\\' alt=\\\"IPB Image\\\" /][!--ImageUrlEnd--][/a][!--ImageUrlEEnd--]

Short version: In the fumen below, filled cells in odd-numbered columns (columns 1, 3, 5, 7 and 9) are colored in blue. Each L or J piece will add an odd number of blue cells (either 1 or 3).  Same is true for each T piece which is placed vertically. Each other placement will add an even number of blue cells. As you can see there's an odd number of blue cells after 1 bag (since T piece was placed vertically along L and J). So, you'll need to place at least one L, J or T piece to make that number even.

[fumen]v115@9gg0A8g0A8Deg0A8g0A8g0A8CeA8g0A8g0A8g0A8Be?g0A8g0A8g0A8g0A8CeA8g0A8JeAgH[/fumen]