Deconstructing Hot Roll

Introduction

Hot Roll is a bonus feature added to various 3-reel slot machines by maker IGT. In the case of this analysis, it is based on the classic game Triple Double Diamond. If the player gets the Hot Roll icon on all three reels, then he will play a craps-based bonus game. In this game, the player will keep throwing two dice and winning money, until he rolls a seven, ending the bonus.

I played 284 spins of this game at the Golden Nugget on January 2, 2014. As I played, I recorded my play and then uploaded the video to YouTube. This page documents my results and attempts to reverse engineer the game to show you how it might have been programmed. For this type of game, with weighted reels, 284 spins is not enough to know exactly how IGT programmed it. What you see in this page is my best educated guess.

Rules

Hot Roll is a 20-line 3-reel slot machine.

Following are the rules for the base game.

1. The player must play all 20 lines.
2. The player may bet one to ten coins per line.
3. The pay table for each line is as follows:
    

   Hot Roll Pay Table

   Event	Pays
   Three triple diamonds (20th payline)	10,000
   Three triple diamonds (payline 1 to 19)	1,200
   Any three wilds	1,000
   Three red sevens	100
   Three purple sevens	80
   Mixed sevens	50
   Three 3-bar	30
   Three 2-bar	20
   Three 1-bar	10
   Three cherries	10
   Three mixed bars	5
   Any two cherries	5
   Any one cherry	2

   Three double diamonds would be scored as any three diamonds.

   The win of 10,000 is on a "for one" basis, relative to the total amount bet. The pay table itself mentions a win of 100,000 coins, but this is based on a ten-coin bet. If the player bets less than 10 credits per payline, then he will win 1,200 only for three Triple Diamonds on the 20th payline.

4. The Double Diamond and Triple Diamond symbols are wild and may substitute for any other symbol, except the Hot Roll or another wild, on the same pay line. Note that the wilds may not substitute for a cherry unless there is a natural cherry on the same line.
5. A double diamond will double any win on that pay line. Likewise, a triple diamond will triple any win on that pay line.
6. A combination of two wilds will both multiply. To be specific, two Double Diamonds will multiply a win by 4, one Double Diamond and one Triple Diamond will multiply a win by 6, and two Triple Diamonds will multiply a win by 9.
7. All wins for three diamonds do not get multiplied.
8. If the player gets three Hot Reel symbols anywhere on the screen, then he shall play the bonus game.
9. If the player gets three Triple Diamonds on the 20th pay line, and makes a maximum bet of 200 credits, then he shall be paid 10,000 for 1 on that line, instead of the usual 1,200 for 1 for three Triple Diamonds.
10. The pay lines are drawn as follows:
     

    Hot Roll Paylines

    Line	Reel 1	Reel 2	Reel 3
    1	middle	middle	middle
    2	top	top	top
    3	bottom	bottom	bottom
    4	top	middle	bottom
    5	bottom	middle	top
    6	middle	top	middle
    7	middle	bottom	middle
    8	bottom	middle	bottom
    9	top	middle	top
    10	top	middle	middle
    11	bottom	middle	middle
    12	middle	top	top
    13	middle	bottom	bottom
    14	top	top	middle
    15	bottom	bottom	middle
    16	middle	middle	top
    17	middle	middle	bottom
    18	top	bottom	top
    19	bottom	top	bottom
    20	top	top	bottom

 

Rules for the bonus game.

1. The player shall keep rolling a pair of dice until he rolls a seven.
2. Per Nevada law, the outcome of each die is independent and each side has a 1/6 probability, as with real dice.
3. If the player gets a seven on the first roll, then he shall win 7 times the total amount bet.
4. Otherwise, the player shall win the amount in the following table. Wins are based on the total amount bet. The player will keep all wins until he rolls a bonus-ending seven.
    

   Hot Roll Bonus

   Roll	Pays
   2 or 12	10
   3 or 11	6
   4 or 10	4
   5 or 9	3
   6 or 8	2

Data

Based on my 284 spins, I put together the order of the reel stripping and count how often each reel stopped on each position. The following table shows my results.

Triple Double Diamond Analysis

The way that three-reel slot machines usually work is to pick one random number for each reel and then map it to a position on the reel strips, according to how each stop on each reel is weighted. It is not unusual for the total number stops to sum to an even power of 2. I don't know the total number of stops for this game but for the sake of example I will assume it is 2⁸ = 256.

The following table is my best estimate of the actual reel weights. I combined trying to keep the proportions the same as the actual data and trying to achieve what I felt was a believable return for the game. You may recall that in my Las Vegas penny slot survey that the Golden Nugget came in 48th place out of 71, with an average return of 90.85%.

The way the game would be programmed, based on these weights, would be to choose three random integers from 0 to 255 (programmers always start counting at zero). It would then map those numbers to a specific stop on the reel according to the following ranges for each stop. The game would then stop each reel on the predestined position in the middle row.

Let's look at an example. Suppose the random numbers were as follows:

• Reel 1: 222
• Reel 2: 0
• Reel 3: 175

From the above table, you can see the 222 for reel 1 gets mapped to the blank between the red 7 and the 3-bar. The 0 for reel 2 gets mapped to the first blank, above the double diamond and below the red 7. The reels wrap around so the red 7 at the bottom is above the blank at the top of the list. The 175 for reel 3 would be mapped to the purple 7. The outcome would then look as follows.

Based on the weights above, the following table shows the number of combinations of each win by its multiplier for reel 1.

The next table shows the return combinations for each win. Each cell in the main body of the table is the product of the win, multiplier, and number of combinations from the table above. The total number of possible combinations is 256³ = 16,777,216. Dividing the total return combinations in the lower right cell of 10,717,885 by the total possible combinations of 16,777,216 we get 63.88%. So, for the one credit bet on the center payline, the player can expect to get back 0.6388 credits, not counting the bonus.

Since the reel stops are weighted, this analysis must be repeated for each payline. To prevent this page from getting too long for the other 19 paylines I will just present the return in the following table. Note the bottom right cell shows an average return for the base game of 68.69%.

The next table shows the number of combinations for each type of win over all 20 paylines.

The following table shows the expected number of each kind of win over all 20 paylines. The lower right cell shows the player can expect 1.46 wins per bet.

The following table shows the expected return from each kind of win over all 20 paylines. The lower right cell shows the player can expect 13.737592 credits from line pays per bet. Dividing that by a 20-unit bet, the return from the base game is 68.688%.

The next table shows the frequency of each win amount, after applying the multiplier, over all 20 paylines. The lower right cell shows a total win of 13.737592. Dividing this by 20, the total amount bet, results in a return for the base game of 68.688%.

Bonus Analysis

The rules for the bonus are stated in the rules section above. Let's start the analysis of the bonus by solving for the average win per roll, assuming it isn't a seven. The table below answers that question. The bottom right cell shows an average win of 3.733333, assuming no seven.

Next, what is the average number of rolls? If the probability of an event is p then it will take on average 1/p trials for it to happen. The probability of rolling a seven is 1/6, so it takes on average six rolls to happen. However, the player doesn't win anything for the actual roll of the seven, so there are five paying rolls before the seven.

There is also a consolation prize of 7 for rolling a seven on the first roll. The value of that is (1/6) × 7 = 1.166667. So, the average win per bonus is 1.166667 + 5 × 3.733333 = 19.833333.

As a reminder, the bonus is triggered if the player gets three Hot Roll symbols anywhere on the screen. To determine the probability of it occurring on each reel, we also need to examine the blank stops immediately above and below the Hot Roll symbol that touch the center payline. For reel 1 there are, 18 (blank) + 25 (Hot Roll) + 19 (blank) = 62 stops that when touching the center payline make the Hot Roll symbol appear anywhere in reel 1, yielding a probability of 62/256 = 0.242188.

The following table shows the probability of a Hot Roll symbol appearing on each of the three reels as well as the product. The lower right cell shows a bonus probability of 1.13%.

The overall return from the bonus is the probability of the bonus times the average win. This product is 0.011308 × 19.833333 = 0.224279.

Final Analysis

After all that, we have shown the return from the base game is 68.688% and the return from the bonus is 22.428% for a total return of 91.116%. If the player bets less than 200 credits, thus losing the max coin incentive, the return drops by 0.039% to 91.077%.

I would like to emphasize that I am not claiming this is the exact return. This page is more for an exercise in slot machine design than solving for the exact return of that one game. To determine the exact return I would need to know the exact reel weights, which is information I do not have.

Video

Video of the 288 spins this analysis is based on.

Acknowledgments

My thanks to Miplet and tringlomane for their help verifying the math above.

Written by: Michael Shackleford