Twenty One (by Groetchen, 1936)

Hello it's time for more mathematics!   Yes, I managed to get this lovely little restored trade stimulator from 1936:

Twenty One (21) by Groetchen circa 1936

These are devilish little machines designed to trick the player in to thinking it's a skill game.  Please be advised of a little operator secret:  these are NOT skill games.  The fix is in, and it's designed for you to lose, and to hopefully make you think it's your fault you lost.

We'll dive in to the mathematics of that a bit later, but first a gameplay overview:  place a nickel in, pull the handle down, and the reels spin.  The left-most windows (5, 4, 3) will all have a shutter close in front of them.  You have your two cards (windows labeled 2 & 1).  Want another card?  You can press the button under window 3 to reveal it.  Want another card?  Do the same for button 4.  When you are satisfied with your hand, press button 5 to reveal the dealer's hand.  You can't press button 4 until 3 is press, forcing an order.  Once you press button 5, the other 2 window buttons are disabled.  If you stand on 3 cards and reveal the dealer's hand, there is no way for the played to know what window 4 had.

When it arrived it was super sluggish and not really spinning at all.  I went inside:

I quickly realized it was not spinning because the pump was not working, causing the machine to slam to a stop, and be unable to finish a full cycle

the pump

The pump is what provides the slow constant resistance on a slot machine handle so that it slows the return to it's initial position.  If it's not oiled, no resistance, and so the mech is pulled by it's springs and instantly slams back into position.  The lack of timing meant the wheels weren't stopping at the right spot, the coin validator wasn't returning into place, and a whole other mess of things weren't moving properly.

A few drops of 3-in-1 and it started playing properly again.  It is not perfect --I do want to adjust it so the first wheel consistently spins for freely-- but it's playing mostly great.  Every once in a while that first wheel just gets stopped too early for my liking.

from the Mills repair book in the 50s

It's time for some math.

OK let's get in to the design of this machine and why I love it so much.

hullo

So the game is a bit odd in that the reels are numbered right to left.  For the sake of my math I am going to rearrange them.  In the above photo, we see reels 5, 4, 3, 2, then 1.

Each reel has 12 entries on it, and here is what is on the strips:

"1 or 11" is straightforward, it lets the card be a 1 or 11.

The 5th wheel shows the number the player has to BEAT.  That's hugely important to understand, because it greatly limits the avenues for a win.   Your mind sees an 18, but you need a 19.  Tie goes to the house.  So if the dealer window says you have to "beat 19", you have to have a 20 or 21 to win.

The "prize wins" column is how many "points" you win if you beat that hand.  Points!  Innocuous, fun, pure!

The 5th wheel has one entry where you can only win through blackjack.  On the first reel, there is a "blank" or zero.  If you get that, and then a 10, and then the "1 or 11" on the 3rd reel, that is blackjack.  Hey, you can win 100 points!  wow!  That's certainly an impressive number of "points" to win!

But the math is bad: (1/12)^4 bad.  0.0048% bad.  1 in 20,000 bad.  The odds of hitting the 100 coin jackpot on my Mills slot machine is 0.05%, or 10x more likely.

You have to hit one specific entry on each of 4 different reels.  There is only one blank on reel one, a lone 10 on reel 2, and a lone "11" on reel 3.  And if you get that, you still need the "21 w black jack" dealer hand.

On to the gameplay math...

Twenty One isn't as simple as the Mills slot machine math.   There are cards worth one of 2 different values.  There is the player interaction of choosing which cards to reveal.  But I was able to build a system that goes through the logistics, and reveals some strategy that I at first thought counterintuitive.

So we're going to have the computer (aka my spreadsheet) play Twenty One, and we have to establish the rules.

We get the first 2 cards.  If you look at the reel data, the maximum hand you can have is just 17.  Even if you think this makes sense from playing actual blackjack, never mistake this game for actual blackjack.  Look at what 17 can beat.  The 5th reel only has 2 spots (beat 15, beat 16) that a 17 can win on.  2/12 is 1/6 odds, but the payout is only 2 points.  OK, they mean coins.  There, I've given up the illusion!  So for a 1/6 chance, you would only net 1 coin.  And likewise you're never standing on a 16.  So that means the player will ALWAYS take the 3rd card.

The 3rd reel has 5 different 10s on it, and a 9.  These will usually bust the player if they come up, except if reel 1 is blank (zero) or if you have one of the few lower combinations possible. (5+6 can survive a 10.  6+6 and 5+7 can survive a 9. 

But the player opened that window.  They are responsible for getting that third card.  If they bust, the game makes them FEEL like they were responsible.  They pushed the button!  How could they do this to themselves???

The fix was in, and (according to my simulation,) about 43% of the time the player will be bust right away.  You have to take the three cards, and the wheels are such that you only have a 57% chance of surviving when you open that first window

OK, on to the 4th reel...

We have to state one assumption our system makes:  If the 3rd reel "1 or 11" comes up, we're adding 11 unless that will bust us.  Otherwise use 1 as the value.

The only time this rule might not hold fast for a player is if the first reel is blank.  In those cases, there is more analysis to do if you want to consider it 1 or 11.  But if the first reel isn't blank, there is no mathematical scenario where you could even select 11.

So do we open the 4th reel window?  As I said about the first 2 reels, a 17 isn't a good winner.  Plus, the 4th wheel distribution shows a number of useful small cards: 1 (or 11), 2, 2, 3, 4.  
If you have a 17?  5 different cards that are good. (41.6%)
If you have an 18?  4 different cards. (1/3)
If you have a 19? 3 cards, so 25%.

So should we open the 4th reel?  The machine is really hoping you use your real life blackjack experience here.  It would love it if you would hold on a 17.  But what about an 18?  I made sure my simulation had a variable where I could set where it would hold.

The power of your hand is determined by the dealer hand distribution.  You also have to take in to account the distribution of payouts.  You can beat a 19 and get 5, or you can get 10 depending on which wheel entry it is.

So I didn't make just one machine simulation, I made 100.  100 games of Twenty One played by our defined rules.  I then took note how many coins uhhhh I mean "points" were won over those 100 games.

Then I ran that simulation of 100 machines 100 different times.

When we hold on 18, the simulation shows we earn a median of 48.5 coins.  Points.  Coinpoints.
When we hold on 19, we earn a median of 54 coinpoints.
When we hold on 20,  (and this is where I feel it's a bit counterintuitive,) we earn a median of 52 coinpoints.

I was surprised that holding on 20 was that much stronger than 18, but the reel distribution doesn't lie.  The 4th reel cards have a distribution that truly benefits the higher holds, and the dealer hand distribution is such that a higher hand is far more likely to produce a winner.

So even if you play perfectly, the game only has a 54% rate of return.  Utterly abysmal for a gambling device.  But this machine wasn't popular because of the wins, it was popular because a casual player would think it was their fault.  At least for long enough to get a bunch of their nickels.

Is there skill?

The main skill is knowing what number to hold to maximize your odds.  (19. The answer is 19.)  You wouldn't know this unless you did the math yourself, or started getting an intuitive feel for the distribution as a frequent player.  Mind you, by the time you learn the reels through intuition you've already lost a tonne of money.

But I am interested in scenarios where the player holds after the 3rd card, doesn't bust, but then loses to the dealer.  How many times could they have actually beaten the dealer had they selected that 4th card?  The odds are surprisingly low, and we can check them because we know all the reel positions.

In my simulations, starting from a fresh deal, this scenario happens:
when holding on 18, it happens about 2%
when holding on 19, it happens just under 1% of the time.
when holding on 20, it happens like 0.1% of the time

This all means you don't have to fret about making the wrong move.  You'll almost never have a chance to if you stick by your ideal hold number.  It isn't just about the 4th reel having a card that can work for you, it's a card that will work for you that will make the difference and win you the game.  The odds are small enough to not worry.

Fun note:  this isn't a payout machine, so what is on the reel strips doesn't actually matter.  You could easily replace them and make it a far more casual-friendly game.

Want to see this kind of machine in more detail?  Here is a lovely video of someone's British version