I leave early in the morning, and so am turning to sleep early tonight. So I thought I would repost one of my favorite posts over the years.......

This study was first written on Warseer several years ago, and the study was probably published elsewhere, but Warseer is were I first read it. here is the original link....


http://www.warseer.com/forums/showthread.php?65531-Do-you-roll-a-lot-of-1s-A-40k-must-read 

Before any gamer begins this Holy Quest, he or she should definitely start with the following study. Enjoy......

A huge thanks to Yade for doing this for the community.
Background
I am an engineer and I teach at ASU. In the beginning of every semester there is a lot of book work that my students need to do such as circuit board design and circuit mapping. During this time I have 4 students and a giant mechanical engineering lab at my disposal. So I decided to use my students to improve my 40K game.


Setup
In the Spring semester of 2006 I decided to debunk the dice myth "I always roll more ones". So I took a box of the red and white GW dice, a cube of 36 chessex dice, 36 square corner dice with pips (pips are indentations on the side of dice to indicate the value of the face), and 36 Vegas style square dice with no pips.

I then constructed a series of plastic barriers that would be used to keep each die independent of the others. In the lab we have a table that is 4 inches thick solid slate built on hydraulic legs to keep balance and resist independent movement. On this table we put all of the dice in the rolling container and labeled each case, giving each individual die its own chamber and number. My 4 students then shook and rolled the dice 1000 times, recording each individual result. Each die was individually rolled 1000 times, so 4 sets of 36 dice (144 dice) rolled 1000 times equals 144,000 rolls. Each die was tracked on its own and kept separate from the rest.


Results
Afterwards we calculated the results and the Chessex and GW dice averaged 29% ones. Mind you that this is an average and our high was 33 and our low was 23. We removed any statistical anomalies and came up with 29%.

Game room logic, a poor source of anything, would dictate that the side with the one is heavier and would therefore be on the bottom more. Unfortunately this is just not true, take popcorn or batholiths as an example. The 6 is too light to stop the momentum of the die, the rounded corners cannot prevent the die from turning due to the weight. In the end 1s are by far the most common result. On a 6 sided die any given number should appear 16.6% of the time, the Vegas dice were dead on and the square dice with pips were pretty close, only displaying a 19% ratio for ones.

I contacted the casino Caesar's Palace in Las Vegas and accessed their research, after much duress because they wanted to make sure I was not some gambling shark, and they had results that corroborated mine.


Following up
I then proceeded to buy more GW dice and we filled in the corners of the very same dice that we used, carefully melting the new plastic on to the old dice and filing in the corners to the right size and leveling them to .001 for accuracy. The dice then rolled more accurately but still 19% rolled ones. Over 1000 rolls from 36 dice (36,000 rolls), this variance from the expected values is just not acceptable and cannot be considered truly random.

Finally we dissected all of our dice and looked for air pockets or constitutional inconsistencies. We found a few and compared those to the results of the rolls of that individual die and there was no consistent affect generated by the dice with plastic seeds but there was one with dice that had air bubbles. We dissected the dice using a hydrogen cooled diamond saw which prevents the blade from heating up and scarring the soft plastic. Which you can imagine would melt and tear at the presence of any heat or undiverted pressure.

I sent a copy of the study to Chessex and their official response was to inform me that the amount of plastic saved from rounding the corners and hollowing out the pips of 2 dice actually gave them enough left over plastic to make a 3rd die. Economics wins.

We thought about using metal dice and because they are so heavy the weight differential is going to be significant compared to plastic dice. The ratio is so much smaller that we speculated that metal dice would be a lot more likely to be average and accurate though we did not test this assumption directly.

Lower numbers rolling more frequently is not entirely logical as the heavier part of the die ends at the top. I submitted my results to a friend in the physics department. On the original outset he also agreed that the results were counterintuitive. But he proceeded to test my theory because the results were overwhelmingly in support of the opposite expectation.

There are 2 major forces that affect the dice as they roll — gravity and centrifugal force. Gravity is a constant force and the centrifugal force is generated by the weight of the dice as they are tossed. The two forces work together but one clearly takes precedence over the other. The weight differential of the pips directly affects the centrifugal force more so than gravity. In the end, without going into research, he said that this is probably the route that I should take to determine why the dice do not roll evenly.

Interestingly, the data from Caesar's Palace proved that the dice were more likely to flip once than anything else. If you held the dice in your fingers and had the 1s face up you were more likely to roll 6s because they are on the opposite side. They were using Vegas style dice when they conducted this study. There is definitely a correlation to the way that the dice are thrown. Consequently if you go to Vegas now and you do anything at the craps table the way you roll the dice is heavily monitored. If you palm the dice at any time they are removed. You have to hold the dice in the tips of your fingers and when you toss them they have to hit the table and a wall in order to be considered a valid toss!


Conclusion - Casino Dice!
My own criticism of my study is that the nature of it was observational. It is possible, though highly unlikely, to toss a coin 144,000 times and always get heads. Does that disprove the existence of tails? Not at all.

My conclusions are highly probable and very accurate, in my opinion, based upon our results. Which in the interest of science were truly unmanipulated and random.

So I advise players to use square-cornered dice with no pips and only buy clear ones, like in Vegas, so you can see if there was a problem with the making of the dice. This is not going to prevent you from having a bad dice day but it will better ensure that you have some level of consistency.

One thing to consider is that all games workshop games (and most boardgames in general) are designed with these dice and so should reflect this in the gameplay accordingly. If you see somebody using games workshop dice for leadership tests and vegas style dice for everything else, then you can be sure that they are trying to get an unfair advantage though! Try to ensure you are using the same dice as your opponent.

12 Comments:

  1. Very interesting.
    Would like to see some corroborating sources though. I would always expect there to be some small variance as it is all down to luck, and I find it unlikely that in just 1000 rolls even a casino dice would be exactly accurate at 16.6%.

    Also what type of rolling creates this result?

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    Replies
    1. I assume that he means that accuracy fits within a 95% confidence interval (the normal test for statistical validity). I can't find any publications around it though, which is annoying.

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  2. Game science dice by Zocchi
    http://youtu.be/bR2fxoNHIuU

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  3. Very interesting, and it correlates with what I've heard previously.

    "There are 2 major forces that affect the dice as they roll — gravity and centrifugal force. Gravity is a constant force and the centrifugal force is generated by the weight of the dice as they are tossed. The two forces work together but one clearly takes precedence over the other. The weight differential of the pips directly affects the centrifugal force more so than gravity. In the end, without going into research, he said that this is probably the route that I should take to determine why the dice do not roll evenly."

    I'm not entirely in agreeance with this paragraph. Firstly, I don't believe that 'centrifugal force' is really accurate; I would think torque to be a more accurate statement. Of course this person is a Physicist, so he may still be right. Nonetheless, as centrifugal force applies to an orbit (or similar), whereas this is a rotation, I don't believe it is accurate statement.

    Additionally (though I think this was the author, not the physicist), I don't think that he meant 'weight'. Weight is actually the force applied to an object through gravity (eg, My mass is 60kg, however my 'weight' is about 600N). Therefore, weight is actually the exact same thing as the gravity. Additionally, as its really torque, not centrifugal force, weight doesn't actually affect that aspect (though weight doesn't actually impact centrifugal force either, in retrospect. Oh well). The torque is simply related to the rotational momentum of the dice.

    Thus, the forces mentioned aren't actually what are acting upon the dice. In truth they are the Force due to Gravity and air resistance. The dice also possesses torque (rotational momentum), but this isn't a force; it is kinetic energy of which air resistance acts against. I don't have the energy to examine what differing effect this would have on the outcome, but overall this paragraph seems largely innacurate.

    As an afterthought on centrifugal force, Im a little bit confused on how this person was viewing the concept. In an orbit (or similar), there is only one primary force in play. This force is centrifugal force, however the centrifugal force acting on a projectile (which is really just a rapidly decaying orbit) is actually just gravity. Thus, his entire basis seems a touch confused in its suggestions.

    Ultimately, it was a good article but with a few issues in the hypothesis to explain the phenomenon.

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    Replies
    1. Undergrad and graduate degrees in physics here...the original analysis is by an engineer, so although an engineer may prefer certain terminology its probably not a big difference. I don't have a problem with his use of gravity as a major component affecting the roll of the dice. We know the force of gravity pulls downward on the dice, acting on the center of mass toward the center of the earth. Imagine if the force was quite high, the dice would not roll as any initial movement from a flat side would have a tendency to return the dice to that flat side (the center of mass creates a torque about the corner it is attempting to roll over. The dice would not roll over unless the center of mass was tipped over an edge.

      I'm in agreement that discussing the "pip effect" as a centrifugal force is clunky. Personally I'd use rotational inertia (the tendency to stay at rest or maintain a rotation once in motion) and this is basically what he means when he says "centrifugal force is generated by the weight...". Chessex dice have a large component of their rolling affected by their inertia due to rounded corners (not a precision process), air-bubbles in the dice and missing mass due to hollowed-out pips.

      Personally I use the new GW dice that came out as I like their size and style, but I don't know if they roll any better than Chessex.

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    2. Firstly, I'll clarify what I meant regarding weight. I wasn't objecting to the reference of gravity as a contributing factor, simply to the use of the term 'weight' as opposed to mass. It's a fine difference to most people, but significant enough to be very relevant in this case.

      In regards to centrifugal force, I suppose my main qualm after more careful consideration is the reference to a 'force' here at all. In that respect, my reference to Torque was also innacurate.
      As for whether the significance was in rotational inertia or rotational momentum, I suppose it would rely upon which aspect of the roll was being focused upon. If simply looking at why the die is rolling, I would say momentum (as inertia may be stationary also). Conversely, when observing why does the die stop at a certain place, inertia would be the most relevant. As the study is focussing upon the latter, I would agree that rotational inertia is the more appropriate defenition.

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  4. Based on and scared by your studies I checked my own Chessex Dice. And they rolled very well average for any side.

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  5. Casino dice don't work well either as they are designed to be thrown on a table, not rolled. Without a good surface area to throw they are not accurate either.

    If you want balanced and accurate die, buy Backgammon Precision die, very expensive but tooled and balanced that they are the most accurate out there.

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  6. So a tourney I went to gave us rounded dice with their logo on the ones side.....they are consistently low for the reasons in this article. Scatter dice win..

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  7. Did you do Chi-Square calculations? Back long ago Dragon Magazine in the 80's had a program I did in high school to determine if a die was bias of not. You rolled X times, input the results, then it showed you how balanced (or not it was). Google has tone of online calculators now to do this.

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  8. This comment has been removed by the author.

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  9. I wonder if he would had a different result if he used a device to roll the dice instead of the 4 students. Example: Having the die slide of a slope the same exact way on to the table and testing each side. I would like to see a test where the 1 was on top and with one of the side always facing forward say the 2, do this 10,000. Then turn the die with the 1 still on top with a different side facing say the 3 and repeat the test. Then continue the test until each side has been tested and each number has been on top with each of there side tested. I would be curious of the results and how they would compare by taking some of the randomness in his test. Just a thought and something I probably wouldn't test myself.

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