Some books on gambling illustrated the following as a typical slot machine arrangement:
(%) Cherries 5 7 3 (O) Oranges 4 1 10 (@) Plums 6 1 4 (A) Bells 1 9 1 (C) Melons 2 1 1 (=) Bars 2 1 1
perhaps with payouts like these:
%.. 2 %%. 3 %%% 8 OOO, OO= 10 @@@, @@= 14 AAA, AA= 18 CCC, CC= 100 === 200
This is a more modern arrangement, of the general style favored in the postwar era. Lemons are completely removed from the machine; instead, the melon, a second jackpot symbol, is added. And now even a single cherry on the first reel wins. Note, however, that there were many different payout arrangements used in the postwar era; no one arrangement was nearly universal, as was the case with the prewar era.
Here, a greater effort has been taken to limit the number of times a winning combination occurs. There is only one bell on the first and third reels, so the second reel can savely have nine bells on it, and so there is only one orange, and one plum, on the second reel.
We can note that if every possible combination of the reels occurs exactly once in 8000 plays, then the machine takes in 8000 coins, it will give out the number of coins calculated as follows:
Reel Combinations Value Amount
1 2 3 Paid
5 * 13 * 20 = 1300 * 2 = 2600
5 * 7 * 17 = 595 * 3 = 1785
5 * 7 * 3 = 105 * 8 = 840
4 * 1 * 11 = 44 * 10 = 440
6 * 1 * 5 = 30 * 14 = 420
1 * 9 * 2 = 18 * 18 = 324
2 * 1 * 2 = 4 * 100 = 400
2 * 1 * 1 = 2 * 200 = 400
----
7209
for a profit of 9.8875%.
The following image illustrates some of the ways that slot machines have changed over the years.
In the top row, the first set of payouts shown are those of the first slot machine, as invented by Charles Fey. Note that these symbols, although different from those used on later machines, can be seen as corresponding to them: horseshoes become cherries, the star becomes the lemon, spades become oranges, the diamond becomes the plum, hearts become bells, and the bell becomes the bar.
The second set of payouts illustrates those offered on the earliest slot machines using fruit symbols. All the prizes are doubled, and the feature of allowing a bar on the last reel to complete most three-in-a-row combinations is added.
These early slot machines did not have payouts as large as became common later. The payout schedule shown in the diagram above for the earliest of the slot machines with fruit symbols is:
%%. 2 %%*, %%A 4 OOO, OO= 8 @@@, @@= 12 AAA, AA= 16 === 20
without a jackpot as such. These slot machines had a different arrangement of symbols on the reels from that shown in the example on the previous page, because, with the smaller prizes, the machine could still be profitable, even if the players would win more often.
One such machine had this arrangement of symbols:
(*) Lemons 1 1 3 (%) Cherries 4 4 - (O) Oranges 5 5 6 (@) Plums 3 3 5 (A) Bells 5 5 4 (=) Bars 2 2 2
With the payout scheme above, this machine's payout percentage can be calculated as follows:
Reel Combinations Value Amount
1 2 3 Paid
4 * 4 * 14 = 224 * 2 = 448
4 * 4 * 6 = 96 * 4 = 384
5 * 5 * 8 = 200 * 8 = 1600
3 * 3 * 7 = 63 * 12 = 756
5 * 5 * 6 = 150 * 16 = 2400
2 * 2 * 2 = 8 * 20 = 160
----
5748
for a percentage favoring the house of 28.15%. Although quite high, note that it is less than the 30.175% for the common design with the higher payouts given on the previous page. As well, that design offered 189 combinations paying 10 or more, while this design offers 421 combinations paying 8 or more. If players tend to put small prizes back into the machine, but quit while they are ahead after winning a larger prize, a machine with a low percentage, but which pays out most of the money in small prizes, is still very likely to take the money the player was willing to wager in the end. Of course, this depends on the setting; in some locations, players are likely to wager only one or two coins and move on, and then raw percentage is what matters.
The third set of payouts is that discussed in the example on the page on probability. This was the standard set of payouts for a slot machine; for many years prior to World War II, nearly every slot machine made used those payouts. Three bars paid 20 coins plus the jackpot, depicted here as 100 coins.
The fourth set of payouts is for the Mills Gold Award slot machine. This is one of the first machines to introduce an extra symbol which stood for an additional jackpot. Arbitrarily, the jackpot for three bars is depicted as 50 coins; winning the Gold Award provided either a larger jackpot, or a token redeemable for 100 times the amount wagered on different machines made by Mills with the new symbol.
The version of this machine with a low percentage had the following arrangement of symbols:
(*) Lemons - - 3 (%) Cherries 8 7 - (O) Oranges 3 4 5 (@) Plums 2 4 5 (A) Bells 2 3 4 (=) Bars 3 1 2 (o) Coins 2 1 1
an alternate version replaced two of the cherries, and one bar, on the first reel with lemons.
Given the payouts of
%%. 3 %%*, %%A 5 OOO, OO= 10 @@@, @@= 14 AAA, AA= 18 === 70 ooo 120
we can calculate how many coins this paid out for each 8000 coins put in, on average:
Reel Combinations Value Amount
1 2 3 Paid
8 * 7 * 13 = 728 * 3 = 2184
8 * 7 * 7 = 392 * 5 = 1960
3 * 4 * 7 = 84 * 10 = 840
2 * 4 * 7 = 56 * 14 = 784
2 * 3 * 6 = 36 * 20 = 720
3 * 1 * 2 = 6 * 70 = 420
2 * 1 * 1 = 2 * 120 = 240
----
7148
for a profit of 10.65%.
The second row continues along, showing some more modern arrangements. A single cherry symbol on the first reel could now pay out a prize, but at first cherries were still only found on the first two reels.
The first arrangement shown on the second row shows the payouts of Mills' Melon Bell machine. Here, the new symbol, a half-watermelon, simply replaced the bar, as can be seen from the fact that it can complete three oranges, three plums, and three bells.
The set of payouts in the second arrangement is also for some machines by Mills. These payouts were used both with the 21 Bell slot machine, which was described in detail in John Scarne's Complete Guide to Gambling, whose house percentage was unusually low for a slot machine, being just over 5 percent, and which had a few positions on the three reels where two symbols were present in various combinations, so that any winning combination either symbol might make would pay out, and with the 7-7-7 HiTop machine, where the extra 7 jackpot symbol was in every case overprinted on another symbol so that both would pay out, and no other symbols were combined.
(*) Lemons - - 5 (%) Cherries 3 6 - (O) Oranges 5 4 2 (@) Plums 8 3 3 (A) Bells 1 5 8 (C) Melons 1 1 1 (=) Bars 2 1 1 (7) Sevens 1 1 1
Although it does not affect the odds, the 7 on the first reel was overprinted on a plum, the one on the second reel was overprinted on an orange, and the one on the third reel was overprinted on the bar.
Again, an alternate version of the first reel was available for a less generous payout; it had three lemons on the first reel, replacing two plums (including the one overprinted with a 7) and one of the bars.
Given the payout schedule:
%.. 2 %%. 5 OOO, OO= 10 @@@, @@= 14 AAA, AA= 18 CCC, CC=, === 120 777 200
its percentages can be calculated as follows:
Reel Combinations Value Amount
1 2 3 Paid
5 * 14 * 20 = 1400 * 2 = 2800
5 * 6 * 20 = 600 * 3 = 1800
5 * 4 * 3 = 60 * 10 = 600
8 * 3 * 4 = 96 * 14 = 1344
1 * 5 * 9 = 45 * 18 = 810
1 * 1 * 2 = 2 * 120 = 240
1 * 1 * 1 = 1 * 120 = 120
1 * 1 * 1 = 1 * 200 = 200
----
7914
As this gives an even lower profit of 1.975%, which would seem to be impractical for any slot machine, even in a major casino, I can only conclude that I have misunderstood my sources of information. Cutting the jackpot in size from 100 coins to 50 changes the percentage to 2.95%, still extremely low.
The third arrangement in the bottom half of the illustration is that of the Jennings Sun Chief, a popular model in tropical resorts. The contents of its reels are as follows:
(*) Lemons - - 10 (%) Cherries 1 7 - (O) Oranges 10 1 3 (@) Plums 5 1 5 (A) Bells 1 10 1 (=) Bars 2 1 1
Making lemons fully half of the symbols on the third reel seems something of an uninspired expedient to reduce the amount paid, but the standard measure of making symbols common on one reel, and rare on another, seems to have been pushed to its limits here as well. However, in addition to allowing bars to freely replace cherries on the first two reels, this machine also offered additional payouts for three bars in a row elsewhere than on the payline.
With its payout schedule of:
%.., =.. 3 %%., %=., =%., ==. 5 OOO, OO= 11 @@@, @@= 13 AAA, AA= 18 === 150
we can work out how much it pays (possibly allowing an extra 1200 coins for the additional jackpots):
Reel Combinations Value Amount
1 2 3 Paid
3 * 12 * 20 = 720 * 3 = 2160
3 * 8 * 20 = 480 * 5 = 2400
10 * 1 * 4 = 40 * 11 = 440
5 * 1 * 6 = 30 * 13 = 390
1 * 10 * 2 = 20 * 18 = 360
2 * 1 * 1 = 2 * 150 = 300
----
5610
If we add 1200 coins to cover the extra jackpots this machine provides, we get 6810, for a percentage of 14.875%.
The fourth arrangement shown in the bottom half of the illustration finally shows an example of payouts for a more modern type of machine, on which an award is made for three cherries in a row.
The arrangement of symbols on the reels of this slot machine, also by Jennings, is:
(%) Cherries 3 7 3 (O) Oranges 7 1 10 (@) Plums 6 1 5 (A) Bells 1 10 1 (=) Bars 3 1 1
and the payment schedule it had was the following:
%.. 3 %%. 5 %%% 11 OOO, OO= 11 @@@, @@= 13 AAA, AA= 18 === 150
Once again, we can attempt to calculate this machine's percentage:
Reel Combinations Value Amount
1 2 3 Paid
3 * 13 * 20 = 780 * 3 = 2340
3 * 7 * 17 = 357 * 5 = 1785
3 * 7 * 3 = 63 * 11 = 693
7 * 1 * 11 = 77 * 11 = 847
6 * 1 * 6 = 36 * 13 = 468
1 * 10 * 2 = 20 * 18 = 360
3 * 1 * 1 = 3 * 150 = 450
----
6943
for a percentage of 13,2125%.
Some machines use an upside-down horseshoe (as is favored to retain the good luck) instead of a melon, and Bally has used a target symbol on some of their slot machines as a jackpot symbol. As well, they made an early slot machine, the Bally Double Bell, where three lemons paid out 6 coins; also, the Jennings Sweepstake Chief used the pear as a fruit that ranked below the orange; here, three pears paid 5 coins.
More recently, slot machines had been designed to have 22, 23, or 25 positions per reel, and currently electronic machines are used in casinos that simulate a slot machine with perhaps 128 positions on each reel.
An unusual slot machine, called the Multi-Bell Seven Way, was devised by Adolph Caille after he had sold the Caille Brothers slot machine business, and so he started a new company, AC Novelty, to produce and sell it.
It had seven different kinds of symbols: Strawberries, Bells, Lemons, Bars, Plums, Apples, and Oranges.
The player could choose which one of those seven symbols to wager on, and would win a jackpot when three of that symbol came up.
While it would not be necessary for all seven combinations of three symbols to be equally likely, would it be possible to arrange symbols in such a way as to achieve this, if the machine had 20 symbols on each reel, as was the standard for slot machines?
Since 20 is not a multiple of seven, it might at first seem impossible. However, that is not a reqirement. For example, there are several ways to ensure that there are eight ways to make three symbols in a row:
1 1 8 1 2 4 2 2 2
One could arrange seven different symbols like this:
Strawberries 8 1 1 Bells 1 8 1 Lemons 1 1 8 Bars 4 1 2 Plums 2 4 1 Apples 1 2 4 Oranges 2 2 2
and have 19 symbols on each reel, with three of each symbol occurring exactly eight times. Note that the first three symbols use up 10 of the positions on each reel, the next three use up 7 of the positions on each reel, and the last one uses up 2 positions. The number could be increased to 22 symbols on each reel by making the second group of three symbols like the first group, or decreased to 18 symbols on each reel by making each symbol in the second group like the seventh symbol.
It is also possible to balance things by using a less strictly symmetrical layout.
Thus, if one had four symbols with a 1 8 1 arrangement, and seven pairs of symbols with 4 1 2 and 2 1 4 arrangements, there would be eleven different symbols on three reels, each of which would have 46 symbols.
Suppose we try to apply this principle to the case where each combination of three symbols in a row can be made twelve ways. This can be done in these ways:
1 1 12 1 2 6 1 3 4 2 2 3
This suggests the following arrangement:
Strawberries 4 1 3 Bells 3 1 4 Lemons 2 3 2 Bars 2 3 2 Plums 2 3 2 Apples 2 3 2 Oranges 2 3 2
where a difference of five more symbols used on the two edge reels in the first two rows is offset by using five more symbols in the center reel on the next five rows.
Here, there are seventeen symbols on each reel; still not 20.
Another type of arrangement with this kind of symmetry is possible for seven symbols:
Strawberries 1 6 2 Bells 2 6 1 Lemons 4 1 3 Bars 3 1 4 Plums 4 1 3 Apples 3 1 4 Oranges 2 3 2
Now we once again have nineteen symbols on each reel.
And still another arrangement is possible:
Strawberries 2 1 6 Bells 6 1 2 Lemons 1 4 3 Bars 3 4 1 Plums 1 3 4 Apples 4 3 1 Oranges 2 3 2
also leading to nineteen symbols on each reel.
But if we exert ourselves to obtain a high degree of symmetry for part of the arrangement, we obtain additional freedom of choice:
Strawberries 1 3 4 Bells 4 3 1 Lemons 3 2 2 Bars 2 2 3 Plums 2 1 6 Apples 6 2 1 Oranges 1 6 2
again we have nineteen symbols on each reel, but by using a different combination for the last three symbols, we could reduce that to eighteen or seventeen, or increase it to twenty-four.
Another possible arrangement:
Strawberries 12 1 1 Bells 1 1 12 Lemons 1 6 2 Bars 2 6 1 Plums 1 3 4 Apples 4 3 1 Oranges 2 3 2
leads to twenty-three symbols on each reel.
And yet another possibility
Strawberries 1 12 1 Bells 4 1 3 Lemons 3 1 4 Bars 1 2 6 Plums 6 2 1 Apples 1 2 6 Oranges 6 2 1
still leads to twenty-two symbols on each reel.
If we increase the number of ways to make each combination to twenty, then we have the following possible ways to make each combination:
1 1 20 1 2 10 1 4 5 2 2 5
The first possibility, of course, is useless to our present objective.
One possible combination here is:
Strawberries 4 1 5 Bells 5 1 4 Lemons 1 4 5 Bars 5 4 1 Plums 1 4 5 Apples 5 4 1 Oranges 2 5 2
but this leads to twenty-three symbols on each reel.
One other number that gives the required flexibility is eighteen. If there were eighteen ways of making each jackpot, the ways to make each jackpot might be organized in these fashions:
1 1 18 1 2 9 1 3 6 2 3 3
leading to this arrangement:
Strawberries 1 3 6 Bells 6 3 1 Lemons 2 3 3 Bars 3 3 2 Plums 2 3 3 Apples 3 2 3 Oranges 3 3 2
which finally achieves the goal of twenty symbols on each reel.
However, that machine was likely not made in this fashion, because in reality there was another complicating factor not yet mentioned. As on an ordinary slot machine, there were prizes other than jackpots which could be won.
The symbol on the first reel determined which symbol one had to bet on to win, and the symbols on the third reel had a number overprinted on them which indicated the amount won, which could vary from 2 to 20 coins. (The symbols on the first reel also had a number overprinted on them, which was the number of the symbol, to emphasize that the first reel selected the winning bet.)
This seems to require that the first reel have 21 symbols on it, three of each kind, so that all seven wagers have an equal chance of winning. Since the chance of winning would be one out of seven, the average size of a prize could be either 5 or 6.
To make the odds completely equal for the seven possible bets, if winning a jackpot meant that one did not win a smaller prize at the same time, the numbers on each group of symbols on the third reel of the same kind would have to add to the same total; with a reel of 21 symbols, that total would be either 15 or 18. Fortunately, though, while this requirement could be met by putting 35 symbols on each reel, as 20+2+2+2+2 is 28, which is less than 35, this is not a necessary requirement, as in practice the mechanism would not be designed to suppress payout of the smaller prize when a jackpot is won, that being too complicated.
One problem with having three of each symbol on each wheel, however, is that then the chance of winning a jackpot is one in 343; this is high enough that a reasonably large jackpot would limit the other prizes. There does not appear to be a way to reduce the chance of a jackpot, and yet have it equal for all seven fruits, and still have 21 symbols on each of the last two reels. As it happens, however, the real AC Novelty Multi-Bell slot machine had an eighth fruit symbol, the pear, which could appear on either of the last two reels.
This makes an arrangement like the following possible:
Strawberries 3 2 3 Bells 3 3 2 Lemons 3 6 1 Bars 3 1 6 Plums 3 3 2 Apples 3 2 3 Oranges 3 3 2 Pears - 1 2
giving 3 chances out of 21 for each fruit to win a regular prize, and 18 chances instead of 27 out of 9,261 for a jackpot for each fruit, or even
Strawberries 3 1 4 Bells 3 4 1 Lemons 3 1 4 Bars 3 4 1 Plums 3 1 4 Apples 3 4 1 Oranges 3 2 2 Pears - 4 4
giving 12 chances instead of 27 for each jackpot.