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1 June 2022

Question: I was doing research on par sheet related to cascading reels slots. I wonder how the cascading math is designed? Could you give me some idea?

Answer: I haven't been able to get my hands on a PAR sheet for a cascading reels slot machine, but this is my understanding of how they did the math. Once I tell you the secret, you'll say, "Of course."

The secret is that the reel layouts are fixed, just like on a slot that does not have cascading reels. When a winning combination causes the reels to cascade, the symbols on the virtual reel above the displayed symbols fall into the empty spaces.

Let's say we have a winning combination and we have two empty positions to fill on reel 1. First, any symbols already displayed on reel 1 fall down to fill empty positions below them. Now we have to fill the two empty positions on the top of reel 1.

One way to do that is to choose the symbols from a weighted pool of symbols. Some symbols are more likely to be chosen than others. Instead of knowing the symbol in the position, we have probabilities that a symbol is in that position. We could then go through all the probabilities of each replacement position's having each symbol. That's many computations.

Another method is that the replacement symbols are determined by getting another number from the RNG and using the symbols at that virtual stop. This procedure isn't any better than choosing from a weighted pool and is probably worse.

Let's try this method. We get numbers from the RNG that tell us where to stop each reel. We also remember the stop that is displayed at the top of each reel. Let's call it the High Water Mark (HWM).

When we need to fill two positions on reel 1, the symbols in the two positions above the current HWM for reel 1 fall into place and we raise reel 1's HWM by two.

You can imagine it this way. When the reels stop, all of the symbols on the reels above the displayed symbols are stacked above the reels, ready to fall into place if needed.

With this method, there's no random element after the initial stopping places are chosen. We know exactly which symbols will fall into the empty positions for every cascade.

Now all we have to do is iterate through each combination of stopping positions for the initial starting screen. Granted, there could be quite a few of them (e.g., 10,000,000,000 if each reel has 100 stops), but we have plenty of time.

We find any winning combinations displayed, note how much they're worth, remove the symbols in the winning combinations, drop the symbols already displayed and fill the empty positions with the symbols stacked above the reels. Now we go back to see if we have winning combinations again.

When we hit a dead end -- a cascade without a winning combination, that is -- we move on to the next combination of stopping points.

Keep in mind that a large percentage of the starting screens do not have a winning combination. We're done with them after evaluating the the starting screen. Maybe this won't take as long as we thought.

How do we handle free spins? We'll have to figure out the Expected Value of a spin. The free spins usually use special bonus reel layouts. It might be simpler to work with these reels.

If anyone can confirm the procedure I described above or correct it, please write.

Here are the latest Covid data. There is a difference in the hospitalization data for US versus NV. The CDC reports total number of Covid hospital admissions. Nevada reports current hospitalizations, not admissions.

All data comes from the CDC (https://www.cdc.gov/covid-data-tracker/#cases_totalcases, https://covid.cdc.gov/covid-data-tracker/#new-hospital-admissions), except for Nevada current hospitalizations (https://nvhealthresponse.nv.gov/).

Totals Weekly Changes
US NV US NV
Date Cases  Hosp. Adm.  Deaths  Cases  Curr. Hosp.  Deaths  Cases  Hosp. Adm.  Deaths  Cases  Curr. Hosp.  Deaths
05/31/22   83,949,036   4,736,322   1,002,070   737,374   218   10,883   670,041   24,555   2,285   5,738   16   41
05/24/22   83,278,995   4,711,767   999,785   731,636   202   10,842   756,047   25,027   2,317   5,324   37   28
05/17/22   82,522,948   4,686,740   997,488   726,312   165   10,814   680,888   21,183   2,097   3,693   35   26
05/10/22   81,842,060   4,665,557   995,371   722,619   130   10,788   534,465   17,934   3,932   2,802   32   11
05/03/22   81,307,595   4,647,623   991,439   719,817   98   10,777   432,666   14,755   2,448   2,213   (6)   49
04/26/22   80,874,929   4,632,868   988,991   717,604   104   10,728   348,507   12,597   2,446   1,447   (22)   46
04/19/22   80,526,422   4,620,271   986,545   716,157   126   10,682   266,330   10,525   3,308   1,232   (37)   497
04/12/22   80,260,092   4,609,746   983,237   714,925   163   10,185   228,465   10,343   3,627   1,228   23   53
04/05/22   80,031,627   4,599,403   979,610   713,697   140   10,132   203,709   10,312   4,105   1,059   (52)   76
03/29/22   79,827,918   4,589,091   975,505   712,638   192   10,056   206,914   9,572   4,083   1,187   5   40
03/22/22   79,621,004   4,579,519   971,422   711,451   187   10,016   200,627   8,502   8,178   742   (71)   83
03/15/22   79,418,377   4,571,017   963,244   710,709   258   9,933   253,069   17,393   5,492   24,844   (20)   113
03/08/22   79,165,308   4,553,624   957,752   685,865   278   9,820   310,308   24,471   9,870   2,099   (101)   116
03/01/22   78,855,000   4,529,153   947,882   683,766   379   9,704   465,845   33,261   14,988   6,421   (189)   196
02/22/22   78,389,155   4,495,892   932,894   677,345   568   9,508   551,462   46,726   12,797   4,023   (297)   173
02/15/22   77,837,693   4,449,166   920,097   673,322   865   9,335   1,005,691   67,963   17,059   5,280   (375)   149
02/08/22   76,782,002   4,381,203   903,038   668,042   1,240   9,186   1,769,556   92,750   18,185   11,144   (387)   235
02/01/22   75,012,446   4,288,453   884,853   656,898   1,627   8,951   3,193,570   121,420   17,885   26,555   (344)   191
01/25/22   71,818,876   4,167,033   866,968   630,343   1,971   8,760   5,102,999   143,329   16,393   57,574   99   133
01/18/22   66,715,937   4,023,704   850,575   572,769   1,872   8,627   4,983,654   147,105   13,301   25,927   (1,787)   99
01/11/22   61,732,283   3,876,599   837,274   546,842   3,659   8,528   5,421,565   139,572   12,168   32,486   2,594   100
01/05/22   56,310,718   3,737,027   825,106   514,344   1,065   8,428   3,501,427   104,272   8,867   20,966   276   61
12/28/21   52,809,291   3,362,755   816,239   493,378   789   8,367   1,693,987   63,185   11,127   7,977   89   56
12/21   51,115,304   3,569,570   805,112   485,401   700   8,311   1,063,296   55,734   9,102   5,639   9   110
12/14   50,052,008   3,513,836   796,010   479,762   691   8,201   853,262   57,885   8,946   5,008   17   91
12/07   49,198,746   3,455,951   787,064   474,754   674   8,110   821,215   66,191   8,575   5,011   23   125
11/30   48,377,531   3,389,760   778,489   469,743   651   7,985   563,746   28,426   6,309   3,343   (50)   55
11/23   47,813,785   3,361,334   772,180   466,400   701   7,930   667,924   41,145   10,754   4,784   (24)   86
11/16   47,145,861   3,320,189   761,426   461,616   725   7,844   604,748   63,164   7,862   4,577   22   85
11/09   46,541,113   3,257,025   753,564   457,039   703   7,759   516,764   33,219   8,290   5,539   34   103
11/02   46,024,349   3,223,806   745,274   451,500   669   7,656   555,915   38,028   9,226   4,327   45   109
10/26   45,468,434   3,185,778   736,048   447,173   624   7,547   488,829   42,265   9,842   4,753   (87)   96
10/19   44,979,605   3,143,513   726,206   442,420   711   7,451   578,396   11,963   4,381   136
10/12   44,401,209   714,243   438,039   7,315   795,586   14,067   15,710   149
10/05   43,605,623   700,176   422,329   7,166   554,194   10,642   3,852   121
09/28   43,051,429   689,534   418,477   7,045   817,218   14,463   6,160   165
09/21   42,234,211   675,071   412,317   6,880   971,637   14,691   7,466   152
09/14   41,262,574   660,380   404,851   6,728   1,176,763   12,919   10,256   145
09/07   40,085,811   647,461   394,595   6,583   975,725   10,076   5,237   104
08/31   39,110,086   637,385   389,358   6,479   1,113,414   9,385   7,592   173
08/24   37,996,672   628,000   381,649   6,306   1,045,491   7,507   8,117   116
08/17   36,951,181   620,493   373,649   6,190   959,978   4,715   7,065   150
08/10   35,991,203   615,778   366,584   6,040   819,524   3,987   7,652   122
08/03   35,171,679   611,791   358,932   5,918   622,832   2,279   7,489   81
07/27   34,548,847   609,012   351,443   5,837   652,251   2,394   8,347   79
07/20   33,896,296   606,618   341,096   5,758   169,933   1,478   3,351   28
07/13   33,726,363   605,140   339,745   5,730   181,047   1,959   4,982   33
07/06   33,545,316   603,181   334,763   5,697   75,104   1,373   2,234   27
06/29   33,470,212   601,808   332,529   5,670   87,507   2,057   3,020   24
06/22   33,382,705   599,751   329,509   5,646   75,420   2,157   1,930   22
06/15   33,303,285   597,594   327,579   5,624   95,797   2,293   1,560   17
06/08   33,207,488   595,301   326,019   5,607   114,250   3,762   2,271   21
06/01   33,039,238   591,539   323,748   5,586   123,333   3,709   991   27
05/25   32,969,905   587,830   322,757   5,559   174,125   4,234   1,676   26
05/18   32,795,780   583,596   321,081   5,533   223,966   4,230   2,301   27
05/11   32,571,814   579,366   318,780   5,506   303,856   4,687   2,541   33
05/04   32,267,958   574,679   316,239   5,473   343,348   4,908   2,559   40
04/27   31,924,610   569,771   313,680   5,433   383,163   4,958   2,747   65
04/20   31,541,447   564,813   310,933   5,368   464,556   5,072   2,590   36
04/13   31,076,891   559,741   308,343   5,332   480,061   5,321   2,986   57
04/06   30,596,830   554,420   305,357   5,275   448,935   7,124   3,084   38
03/30   30,147,895   547,296   302,273   5,237   439,510   6,793   939   63
03/23   29,708,385   540,503   301,334   5,174   388,928   7,446   1,863   53
03/16   29,319,457   533,057   299,471   5,121   381,695   8,362   3,078   81
03/09   28,937,762   524,695   296,393   5,040   480,902   11,573   2,413   83
03/02   28,456,860   513,122   293,980   4,957   463,356   14,129   2,835   75
02/23   27,993,504   498,993   291,145   4,882   451,083   13,923   2,406   162
02/16   27,542,421   485,070   288,739   4,720   602,906   21,411   4,149   198
02/09   26,939,515   463,659   284,590   4,522   779,305   21,828   5,444   244
02/02   26,160,210   441,831   279,146   4,278   1,007,777   22,004   7,249   249
01/26   25,152,433   419,827   271,897   4,029   1,312,565   23,385   10,324   250
01/19   23,839,868   396,442   261,573   3,779   1,317,119   21,318   11,324   279
01/12   22,522,749   375,124   250,249   3,500   1,790,345   22,660   17,217   294
01/05   20,732,404   352,464   233,032   3,206   1,499,561   18,435   14,655   233
12/29/20   19,232,843   334,029   218,377   2,973   1,258,540   15,460   12,493   186
12/22   17,974,303   318,569   205,884   2,787   1,656,411   18,537   16,472   239
John Robison

John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

#### Books by John Robison:

The Slot Expert's Guide to Playing Slots
John Robison
John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

#### Books by John Robison:

The Slot Expert's Guide to Playing Slots