Beyond Poker reviews games which are not poker, but which scratch the same itch or exercise similar strategic muscles. In this series you’ll find games which involve hidden information and bluffing, probabilistic thinking, risk-reward estimation, or which simply apply poker concepts like the hand rankings in a novel context.
Who doesn’t love a freeroll? The idea of having a shot at a large prize with no money of one’s own on the line is instinctively appealing to just about everyone. So, when I received a press release this week, announcing a completely free app with no in-game purchases and the chance to win $10,000 or more, I was simultaneously intrigued and skeptical.
The app in question is called 33 Numbers and, just as advertised, it costs absolutely nothing to play and will pay out a progressive jackpot ($10,105.08 as of this writing, and increasing by a few dollars per day) to whoever ultimately wins it.
Of course, there’s no such thing as a total freeroll, and as is usually the case on the internet, the fact that you’re not paying anything as the user means you’re not the customer, you’re the product. The company (33Numbers LLC) claims they’ll be funding the jackpot (and presumably their own profits) via advertising, but there doesn’t appear to be any advertising in the game at the moment. There is however, space for a banner ad at the top of the main game screen, so presumably they’re hoping that the ever-growing jackpot will eventually build enough of a user base to pull in some deep-pocketed sponsors.
Because no money is ever charged to the user, the game does not qualify as gambling in most jurisdictions. There are exceptions, however, and the app uses geolocation services to prevent players in those areas from trying for the jackpot, giving them a for-fun version to play instead. Furthermore, the game’s terms specify a minimum age of 17 to be able to play legally and be entitled to the jackpot if won.
As you’d expect from a gambling game (even one that isn’t technically gambling), the mechanics are extremely simple.
There’s a track running around the game board with space for the eponymous 33 numbers. When the game starts, a single tile numbered somewhere between 400 and 600 is drawn and placed into the centermost space on the track.
The player then receives a series of tiles randomly numbered between 1 and 1000, which they must place onto the track one at a time. Once placed, a tile cannot be moved, and they must always be placed in ascending order. If a tile can’t be placed, the game ends; for example, if a 527 has been placed immediately adjacent to a 534, then drawing any tile between 528 and 533 will result in the game being over.
The objective, naturally, is to place as many tiles as possible before hitting the game over condition. The player’s score is simply the number of tiles placed, and chasing a high score can be a fun short-term goal, but of course the real goal is to fill all 33 spaces on the track and win the jackpot.
In terms of strategy, 33 Numbers falls somewhere in between slot machines and video poker. In other words, there isn’t very much of it, but there is a little bit.
The basic strategy should be obvious to anyone, namely that you should space your numbers out in proportion to their value: If the center tile is 440 and your first draw is 200, it should go somewhat closer to the left end of the track than to the center, leaving more space for future draws between 201 and 439 than for draws between 1 and 199.
The game actually helps you out in this regard, with dynamic hints showing which numbers would divide each available gap the most evenly. For instance, if you’ve placed a 581 and a 735 with three spaces between them, those spaces will show 620, 658 and 696. Therefore, if you draw a 624, you should probably place it on the first space, since it’s closer to 620 than 658.
That said, it’s clear after a few plays that simply following the hints does not maximize your expected score. It’s much less obvious, but provable by example that it won’t always maximize your odds of winning the jackpot. The proof is given at the end of this article.
The intuitive explanation for this is that the game over condition can only be reached when tiles are placed next to each other or at the ends of the track. In the example we just looked at above, placing the 624 next to the 581 means that drawing anything between 582 and 623 will now lose; it therefore increases the odds of losing on the next draw and all subsequent ones by 4.2%.
Therefore, placing the 624 on the middle space clearly maximizes your short-term expectation of increasing your score. However, it also divides the space unevenly, which might be worse for your ultimate odds of filling every space and winning the jackpot.
As a game, 33 Number is fairly brainless, but fun as a time killer. It does occasionally present you with interesting decisions given the tension mentioned above, between wanting to arrange your numbers in proportion to their value while also wanting to create as many separate gaps between tiles as possible.
If you’re thinking about playing with the serious intention of trying for the jackpot, however, that’s something you might want to reconsider. It’s impossible for me to calculate the exact odds of winning, because doing so would require understanding perfect play, which as we’ve seen is not entirely trivial. However, my best back-of-the-envelope (I’m literally scribbling on an envelope) guesstimates put your odds of winning on any given attempt somewhere in the vicinity of one in 100 million.
Put another way, a typical score is perhaps 16-18, and getting past 20 isn’t too hard, but my best game out of perhaps 100 attempts is a 25. In the late stages of the game, the odds of drawing a safe tile get exponentially smaller with every move. The highest score attained globally was 31, back in October. The exact odds of a player winning from that point depend on the size of gaps remaining, but are probably somewhere below 1%; if we then assume that 1% shots of winning come up once every 3 months, then at current play volume it could be 20 or 30 years before someone wins.
In any case, if the game finds any traction at all, the jackpot should eventually grow big enough to cause interest and playing volume to start snowballing; I would bet that if it ever goes off at all, it will be in the high six figures by that point and with a lot more people playing than there are now.
And hey, who doesn’t love a freeroll?
Some math for the nerds out there
For those interested, here’s the proof that following the hints isn’t necessarily optimal for winning the jackpot. Imagine that we’ve placed 30 tiles already and our three remaining empty spaces fall between 1 and 21. Our hints would divide the space up as follows:
1 (6) (11) (16) 21
Assume we now draw an 8. That’s closer to 6 than to 11, so naively following the hints would have us placing it in the first space.
Now, if either of our remaining two draws is 22 or higher, we’ll lose no matter what we do, so those possibilities don’t impact our strategy at all. We’re only concerned about our relative odds when we’re already lucky enough to draw two more tiles within the 2-20 range.
When we put the 8 into the first space, we’ll need to draw a 9 through 20 on our next attempt: That’s a 12 in 18 shot, given that we’re ignoring all those times we draw anything higher than 22. The exact number we draw affects our remaining “outs” for the final draw, however. In the best case scenario, we pull a 9 or a 20 and leave ourselves with 11 possible winning numbers. In the worst case, we pull a 14 or 15, which splits the space in half and leaves us only 6 winning numbers. On average, we’ll have 8.5 winning numbers.
So, our total combined odds of winning by placing 8 in the first slot (assuming no numbers higher than 21 come up) are 12/18 x 8.5/17, which reduces to 2/3 x 1/2, or 1/3.
What happens if we put the 8 in the middle slot? Well, now we’re left with two gaps consisting of 6 and 12 possible numbers respectively, and we need to hit both to win. The order in which we hit them doesn’t matter, however; if we hit our 6/18 shot to fill the smaller gap on the first draw, we’re left with a 12/17 chance on the second and vice versa.
This makes our odds (6/18 x 12/17) + (12/18 x 6/17), which simplifies to 2 x (6 x 12) / (17 x 18). That gives us 47%, which is much better than the 33% we got by following the hints.
Alex Weldon (@benefactumgames) is a freelance writer, game designer and semipro poker player from Dartmouth, Nova Scotia, Canada.