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Deconstructing the Atkins Diet Slot Machine

Introduction


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I designed the Atkins Diet slot machine in 2008 for a speaking engagement at a conference on slot machines in Sydney, Australia. It was used as an example of how video slots, sometimes known as pokies, are designed. I'll start with presenting the pay table and reel strips, and then I'll show the probability and return of the various ways to win.

Pay Table

The next table shows the Atkins Diet pay table. All wins must be left aligned. The Atkins symbol is wild and can substitute for any symbol except the scale.

Atkins Diet Pay Table

Symbol 5 in a row 4 in a row 3 in a row 2 in a row
Atkins 5000 500 50 5
Steak 1000 200 40 3
Ham 500 150 30 2
Buffalo Wings 300 100 25 2
Sausage 200 75 20 0
Eggs 200 75 20 0
Butter 100 50 15 0
Cheese 100 50 15 0
Bacon 50 25 10 0
Mayonnaise 50 25 10 0

Reel Strips

The following are the reel strips for Atkins Diet.

Atkins Diet Reel Strips

Stop Reel 1 Reel 2 Reel 3 Reel 4 Reel 5
1 Scale Mayonnaise Ham Ham Bacon
2 Mayonnaise Buffalo Wings Butter Cheese Scale
3 Ham Steak Eggs Atkins Steak
4 Sausage Sausage Scale Scale Ham
5 Bacon Cheese Cheese Butter Cheese
6 Eggs Mayonnaise Mayonnaise Bacon Sausage
7 Cheese Ham Butter Cheese Butter
8 Mayonnaise Butter Ham Sausage Bacon
9 Sausage Bacon Sausage Steak Buffalo Wings
10 Butter Steak Bacon Eggs Cheese
11 Buffalo Wings Sausage Steak Bacon Sausage
12 Bacon Mayonnaise Buffalo Wings Mayonnaise Ham
13 Eggs Ham Butter Sausage Butter
14 Mayonnaise Atkins Mayonnaise Cheese Steak
15 Steak Butter Cheese Butter Mayonnaise
16 Buffalo Wings Eggs Sausage Ham Eggs
17 Butter Cheese Eggs Mayonnaise Sausage
18 Cheese Bacon Bacon Bacon Ham
19 Eggs Sausage Mayonnaise Buffalo Wings Atkins
20 Atkins Buffalo Wings Buffalo Wings Sausage Butter
21 Bacon Scale Ham Cheese Buffalo Wings
22 Mayonnaise Mayonnaise Sausage Eggs Mayonnaise
23 Ham Butter Bacon Butter Eggs
24 Cheese Cheese Cheese Buffalo Wings Ham
25 Eggs Bacon Eggs Bacon Bacon
26 Scale Eggs Atkins Mayonnaise Butter
27 Butter Buffalo Wings Buffalo Wings Eggs Steak
28 Bacon Mayonnaise Bacon Ham Mayonnaise
29 Sausage Steak Butter Sausage Sausage
30 Buffalo Wings Ham Cheese Steak Eggs
31 Steak Cheese Mayonnaise Mayonnaise Cheese
32 Butter Bacon Steak Bacon Buffalo Wings

Example Spin

Let's look at the following example spin.

The way multi-line video slots work is to pick one random number for each reel and map it onto the corresponding stop on the reel strip. Let's assume that the random number chosen designates where the reel will stop on the middle row. The symbols above and below will be those directly above and below on the reel strips.

Random numbers of 27, 14, 3, 31 and 27, for reels 1 to 5 respectively, would have resulted in the example above. The following table shows the position on the reel strips where all 15 visible symbols fall on the screen.

Atkins Diet Reel Strips

Stop Reel 1 Reel 2 Reel 3 Reel 4 Reel 5
Top 26 13 2 30 26
Middle 27 14 3 31 27
Bottom 28 15 4 32 28

It should also be noted that the reel strips wrap around. So, if the number chosen for the middle position is 1, then the top position would be 32. Likewise, if the middle position is 32, then the bottom position would be 1.

Symbol Distribution

The first step to calculating the return of a video slot is to count how often each symbol appears on each reel. The following table answers that question.

Atkins Diet Symbol Distribution

Symbol Reel 1 Reel 2 Reel 3 Reel 4 Reel 5
Atkins 1 1 1 1 1
Steak 2 3 2 2 3
Ham 2 3 3 3 4
Buffalo Wings 3 3 3 2 3
Sausage 3 3 3 4 4
Eggs 4 2 3 3 3
Butter 4 3 4 3 4
Cheese 3 4 4 4 3
Bacon 4 4 4 5 3
Mayonnaise 4 5 4 4 3
Scale 2 1 1 1 1
Total 32 32 32 32 32

Line Pay Math

There are 32 × 32 × 32 × 32 × 32 = 33,554,432 possible outcomes in Atkins Diet. The following table shows how many of these combinations result in each possible win. These figures are the result of a program with five nested loops that tallied the total for each win for each possible combination.

Atkins Diet Line Pay Combinations

Symbol 5 in a row 4 in a row 3 in a row 2 in a row Total
Atkins 1 28 513 1,024 1,566
Steak 431 2,996 32,480 326,656 362,563
Ham 955 5,157 42,112 315,392 363,616
Buffalo Wings 764 5,348 58,464 430,080 494,656
Sausage 1,595 8,613 54,432 - 64,640
Eggs 956 6,692 52,864 - 60,512
Butter 1,995 10,692 88,704 - 101,391
Cheese 1,996 13,860 85,536 - 101,392
Bacon 2,976 20,832 103,168 - 126,976
Mayonnaise 2,980 20,860 128,736 - 152,576
Total 14,649 95,078 647,009 1,073,152 1,829,888

The next table shows the probability of each possible win. This is the number of winning combinations for that win, shown in the table above, divided by the total number of combinations of 33,554,432. The total in the lower right cell reflects a hit frequency per line (probability of winning anything) of 5.45%.

Atkins Line Pay Probabilities

Symbol 5 in a row 4 in a row 3 in a row 2 in a row Total
Atkins 0.00000003 0.00000083 0.00001529 0.00003052 0.00004667
Steak 0.00001284 0.00008929 0.00096798 0.00973511 0.01080522
Ham 0.00002846 0.00015369 0.00125504 0.00939941 0.01083660
Buffalo Wings 0.00002277 0.00015938 0.00174236 0.01281738 0.01474190
Sausage 0.00004753 0.00025669 0.00162220 - 0.00192642
Eggs 0.00002849 0.00019944 0.00157547 - 0.00180340
Butter 0.00005946 0.00031865 0.00264359 - 0.00302169
Cheese 0.00005949 0.00041306 0.00254917 - 0.00302172
Bacon 0.00008869 0.00062084 0.00307465 - 0.00378418
Mayonnaise 0.00008881 0.00062168 0.00383663 - 0.00454712
Total 0.00043657 0.00283355 0.01928237 0.03198242 0.05453491

The next table shows the contribution to the return for each possible win. Each cell is the product of the probability, from the table above, and the pay table. The right cell shows a total return from line pays of 63.46%.

Atkins Diet Expected Returns

Symbol 5 in a row 4 in a row 3 in a row 2 in a row Total
Atkins (wild) 0.000149 0.000417 0.000764 0.000153 0.001483
Steak 0.012845 0.017858 0.038719 0.029205 0.098627
Ham 0.014231 0.023054 0.037651 0.018799 0.093734
Buffalo Wings 0.006831 0.015938 0.043559 0.025635 0.091963
Sausage 0.009507 0.019252 0.032444 - 0.061202
Eggs 0.005698 0.014958 0.031509 - 0.052165
Butter 0.005946 0.015932 0.039654 - 0.061532
Cheese 0.005949 0.020653 0.038238 - 0.064839
Bacon 0.004435 0.015521 0.030746 - 0.050702
Mayonnaise 0.004441 0.015542 0.038366 - 0.058349
Total 0.070029 0.159124 0.331651 0.073792 0.634597

Scatter Pay

The scale serves as a scatter pay symbol. The player gets credit for a scatter pay if it appears anywhere on the screen. Unlike line pays, scatters don't need to be left aligned or on an active pay line. To be fair, scatter pay wins are based on the total amount bet. The following table shows the number of scatters per reel, total symbols per reel, and the probability of a scatter on that reel, which is 3×(column 2)/(column 3). The reason for multiplying by 3 is there are three visible positions on each reel.

Atkins Diet Scatters per Reel

Reel Scatters Total Symbols
on Reel		 Scatter
Probability
1 2 32 0.18750
2 1 32 0.09375
3 1 32 0.09375
4 1 32 0.09375
5 1 32 0.09375

The next table shows the probability and contribution to the return for 0 to 5 total scatter symbols. The lower right cell shows that the scatter pay feature returns 6.98% to the return of the game.

Atkins Diet Scatter Pay Probabilities and Returns

Scatters Pays Combinations Probability Return
5 100 486 0.000014 0.001448
4 25 20,898 0.000623 0.015570
3 5 353,916 0.010548 0.052738
Total   33,554,432 0.011185 0.069756

Bonus

The player earns the bonus if he gets at least three scattered scales. The sum of the probability of 3 to 5 scales in the table above is 0.011185. During the bonus, the player will get ten free spins, and all wins are tripled.

In this game, free spins use the same reel strips as the base game. We have already shown that the expected line pay return is 0.634597 and the expected scatter return is 0.069756. So, if the win is tripled, then the expected win per spin is 3 × (0.634597 + 0.069756) = 2.113058.

Next, there is a rule that free spins can be won in free spins, infinitely. The expected number of total spins in a bonus is 10/(1-10×0.011185) = 11.259335.

So, with 11.259335 free spins, and an average win per spin of 2.113058, the expected final win per bonus is 11.259335 × 2.113058 = 23.791632.

The probability of winning the bonus in the first place is 0.011185, so the expected return per initial spin of the bonus feature is 0.011185 × 23.791632 = 0.266105.

Summary

The following table summarizes the three ways to win in the game. The bottom right cell shows an expected return of 97.046%. In other words, for every dollar the player bets, he can expect to get back 97¢.

Atkins Diet Summary

Item Return
Line Pays 63.460%
Scatter 6.976%
Bonus 26.610%
Total 97.046%

More Deconstructions

 

I have deconstructed lots of slot machines. Here is a list of them:

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Written by: Michael Shackleford