Computer Dice rollers

[Nadaka]

In one random number I created I crossed the Hash codes for the generating object with the time and last random number generated as a random seed because the increment of time was too large and there were issues that prevented sharing a single RNG between objects in different threads.


One way to seed a real random number is to use the least significant digits on the CPU temperature gauge or a sample of noise from a microphone. However, these will not be available in all environments. The randomness of both these values can theoretically be manipulated. For those that need a pure random number, you can measure the decay of a radioactive emitter. But thats really overkill.

 

[Zurai]

Well, one problem I have with recording and comparing, is that any good program should even them out. But the real question I had is wether or not the computer is likely to give you a series of the same number, as sometimes happens with the die, or wether it would give a more statistically probable answer by giving the same amount of the number, but spaced out between different rolls, making the two look like this:

Real: 1 3 4 4 4 5 2 6 3

Computer: 1 4 2 5 4 3 6 4 3

Showing that the computer would have seperated the consecutive 4's becuase it is more probable, and matches better with statistics.

I think you're suffering a misconception about how RNGs work.

It seems to me that you're thinking the computer generates a number, looks at it, and decides whether it's "random enough".

That isn't what happens. The computer just generates the number and spits it out. There's no additional processing involved in the vast, vast majority of cases. Now, how the number is generated is very complex and varies depending on how many resources you have to devote to the RNG, but once a number is generated, that's the result. It doesn't weigh results based on how random they appear.

As a matter of fact, the most common complaint I've seen with RNGs is that the numbers are too streaky - the 4 4 4 in your example - even though statistically this is hogwash.

 

[Jeff Wilder]

Instead, think of it like this: "I give you one-hundred d10s, then roll them all at the same time".

The probability of a "7" appearing on any of those die is extremely high. I don't believe it is every 100%, but it is up there.

Assuming unbiased dice, the probability of getting not a single 7 in 100 rolls of d10 is (9^100/10^100). (I'm pretty sure that comes to 0.000266 or so. That's the percentage chance. The decimal might be off by a place or two, but in practical terms, it doesn't make much difference: not rolling a 7 in 100 tries is all but impossible.)

This doesn't require any special coding. As long as each individual roll has a 10 percent chance of rolling a 7 and is completely independent of every other roll, multiple rolls will take care of themselves.

(For an easier example, consider what the odds are of not rolling a 7 on just two d10. The formula is (9^2/10^2 -- that's a 90 percent chance of non-occurrence, twice, divided by the two rolls), which works out to 81 percent. And since all possible outcomes must total 1.0 (100 percent likelihood), the chance of rolling a 7 in two rolls of a d10 is (100-81) or 19 percent.

But don't tell that to my half-ogre fighter/barbarian. In three rolls of d10 and 2 rolls of d12, I have yet to roll above a 5 or 6, respectively.

 

[Glyfair]

As a matter of fact, the most common complaint I've seen with RNGs is that the numbers are too streaky - the 4 4 4 in your example - even though statistically this is hogwash.

When bridge tournaments switched to regularly using computer generated hands there were a lot of complaints about "freak deals." The simple fact is that humans weren't sufficiently shuffling the cards to get a true shuffle (7 times). Because of the way cards are collected in bridge, the suits tended to break more evenly than expected.

 

[Arkhandus]

Here you are. Observe this set of numbers rolled in one go on InvisibleCastle, using 100d20. Thousands of virtual d20s may be more than the program is designed to handle, so I stuck with just 100. Observe the occasional repeats of numbers and the spread of the numbers. Fairly random-looking to me. More 1s than 20s too, even though statistically they're just as likely to occur on a 'balanced' d20. A larger number of rolls would probably show a greater balance in the frequency. Here's another set of rolls. And one more set for the heck of it.

And of course the algorithm it uses is probably more likely to produce varied results if you don't roll several virtual dice in the same line (since the variables it bases the algorithm on are likely to have changed in the few seconds or so inbetween). And I haven't had any problems with it while using OpenRPG for roughly the past 6 years as my primary 'tabletop' for gaming.

InvisibleCastle uses the same random number generator for virtual dice rolls as found in the OpenRPG virtual tabletop (which is built on the Python programming language). It uses some algorithm or other to devise virtually-random numbers, though I don't know the specifics of what the program looks like (though I think OpenRPG is open-source, so it may be possible to find the programming code for the virtual-dice-roller if someone bothered to look for it).

 

[Asmor]

Here's 10,000d10 from my dice roller.

Seems pretty good to me, and I'm just using the basic Math.random() which is provided by JavaScript. There are much, much better pseudo-random number generators out there.

Feel free to play with it... it's all client side, so you don't need to worry about screwing up my server by rolling 1000000d1000000... Might take a little while to process for you, though. 1000 dice come out almost instantly, 10,000 took a moment... I'd say less than a second.

[sblock=10000d10, total=54562]10000d10: 7, 4, 5, 3, 1, 8, 9, 9, 5, 10, 10, 5, 3, 9, 3, 3, 8, 3, 2, 4, 4, 7, 7, 6, 9, 4, 9, 10, 10, 8, 6, 3, 7, 10, 9, 9, 1, 1, 1, 4, 4, 7, 2, 6, 9, 10, 7, 1, 1, 10, 8, 4, 8, 1, 4, 5, 4, 5, 3, 7, 5, 6, 9, 1, 1, 10, 6, 7, 2, 8, 2, 10, 4, 4, 6, 2, 9, 6, 7, 4, 8, 2, 5, 6, 7, 6, 5, 7, 6, 5, 4, 4, 10, 3, 1, 1, 9, 4, 6, 10, 5, 9, 2, 2, 3, 7, 6, 8, 1, 6, 3, 1, 5, 9, 1, 7, 8, 4, 3, 4, 9, 6, 6, 1, 7, 3, 4, 1, 10, 10, 5, 6, 5, 1, 1, 7, 2, 3, 1, 9, 1, 1, 8, 2, 9, 10, 7, 1, 4, 9, 6, 7, 3, 10, 8, 3, 6, 3, 10, 3, 3, 9, 5, 3, 3, 1, 10, 7, 7, 1, 9, 9, 4, 8, 10, 8, 2, 7, 5, 9, 2, 9, 3, 2, 4, 9, 3, 9, 5, 2, 3, 9, 4, 6, 9, 4, 8, 3, 4, 2, 8, 4, 2, 3, 10, 5, 5, 5, 1, 1, 3, 3, 6, 5, 8, 9, 9, 4, 5, 4, 9, 2, 5, 9, 6, 9, 9, 5, 9, 9, 1, 7, 9, 7, 1, 6, 3, 4, 2, 4, 8, 8, 10, 8, 10, 4, 1, 10, 5, 9, 10, 4, 6, 5, 7, 7, 10, 5, 4, 10, 1, 8, 4, 3, 2, 6, 7, 7, 10, 1, 3, 2, 3, 3, 8, 8, 2, 4, 10, 9, 3, 3, 2, 4, 4, 4, 1, 2, 3, 9, 6, 7, 9, 8, 10, 8, 10, 2, 4, 2, 2, 7, 9, 5, 2, 10, 5, 10, 5, 8, 10, 4, 3, 7, 9, 10, 10, 7, 3, 7, 9, 9, 7, 9, 2, 8, 8, 7, 6, 4, 1, 7, 4, 3, 2, 9, 1, 8, 10, 2, 10, 4, 7, 5, 7, 8, 1, 7, 10, 7, 2, 8, 6, 1, 1, 9, 2, 1, 3, 9, 1, 10, 9, 2, 1, 3, 7, 4, 10, 1, 6, 7, 2, 2, 8, 2, 5, 2, 1, 9, 7, 6, 5, 6, 1, 2, 7, 5, 3, 6, 10, 6, 2, 10, 5, 9, 8, 4, 9, 8, 8, 8, 10, 5, 7, 7, 10, 1, 4, 8, 3, 4, 3, 9, 6, 1, 10, 6, 1, 9, 1, 8, 5, 9, 9, 8, 8, 6, 2, 1, 8, 2, 3, 4, 6, 5, 9, 7, 8, 3, 10, 2, 4, 3, 3, 7, 3, 2, 10, 6, 2, 3, 3, 6, 7, 5, 2, 8, 4, 5, 9, 2, 7, 6, 6, 7, 3, 5, 2, 8, 6, 9, 1, 1, 6, 8, 3, 6, 10, 1, 4, 3, 2, 9, 3, 5, 2, 10, 2, 8, 2, 1, 4, 1, 3, 7, 8, 6, 8, 9, 4, 1, 10, 2, 4, 3, 5, 9, 5, 2, 3, 10, 10, 3, 1, 3, 8, 1, 6, 9, 3, 3, 1, 8, 7, 7, 7, 10, 5, 2, 7, 10, 2, 3, 5, 6, 5, 9, 10, 8, 7, 7, 6, 5, 3, 9, 5, 7, 7, 10, 2, 10, 6, 8, 8, 3, 7, 7, 8, 7, 5, 1, 4, 5, 2, 5, 7, 3, 8, 4, 1, 6, 7, 8, 10, 10, 8, 8, 4, 9, 10, 3, 3, 7, 8, 9, 4, 10, 3, 2, 7, 7, 3, 8, 5, 1, 6, 9, 6, 10, 1, 7, 9, 7, 8, 9, 5, 8, 6, 4, 9, 10, 10, 2, 2, 3, 6, 6, 9, 10, 3, 3, 8, 1, 5, 8, 3, 5, 6, 4, 1, 9, 10, 7, 7, 3, 2, 4, 6, 9, 5, 10, 5, 2, 7, 5, 8, 7, 9, 7, 9, 6, 2, 10, 3, 10, 10, 2, 1, 4, 4, 5, 4, 8, 2, 4, 2, 10, 6, 5, 6, 2, 7, 10, 7, 4, 9, 10, 3, 3, 7, 5, 4, 9, 10, 1, 4, 10, 3, 5, 6, 5, 8, 4, 10, 9, 2, 4, 6, 10, 1, 10, 10, 2, 4, 10, 7, 7, 7, 5, 3, 9, 5, 1, 7, 1, 2, 10, 9, 9, 10, 9, 8, 4, 8, 8, 9, 8, 9, 4, 1, 3, 6, 3, 4, 10, 7, 9, 2, 3, 1, 5, 7, 1, 5, 2, 3, 1, 1, 9, 7, 6, 2, 1, 2, 4, 1, 8, 8, 6, 3, 3, 4, 2, 8, 1, 10, 7, 10, 5, 8, 3, 3, 10, 8, 10, 10, 9, 9, 7, 3, 6, 5, 4, 2, 5, 1, 7, 7, 6, 10, 10, 4, 10, 6, 10, 3, 1, 4, 6, 2, 1, 3, 4, 5, 7, 6, 10, 5, 4, 6, 1, 6, 10, 5, 10, 10, 2, 1, 1, 2, 5, 9, 6, 10, 9, 5, 10, 3, 2, 3, 7, 4, 2, 7, 10, 9, 6, 1, 9, 10, 8, 1, 1, 10, 4, 4, 8, 1, 10, 1, 5, 9, 7, 3, 1, 5, 8, 8, 9, 7, 7, 10, 5, 4, 1, 7, 7, 8, 5, 9, 4, 5, 6, 10, 6, 1, 5, 7, 2, 6, 8, 2, 4, 2, 8, 10, 5, 5, 1, 3, 4, 9, 9, 5, 2, 10, 10, 5, 6, 4, 2, 7, 4, 7, 2, 9, 3, 3, 2, 4, 10, 5, 10, 4, 3, 7, 5, 1, 4, 4, 8, 8, 4, 9, 9, 4, 5, 2, 7, 9, 2, 1, 7, 3, 10, 1, 3, 2, 1, 10, 10, 8, 1, 4, 3, 7, 7, 1, 2, 10, 4, 9, 9, 3, 7, 7, 8, 9, 1, 5, 10, 9, 6, 8, 4, 9, 7, 2, 3, 6, 2, 3, 5, 4, 8, 1, 10, 3, 9, 2, 3, 6, 6, 10, 8, 9, 9, 3, 7, 8, 3, 2, 10, 1, 7, 2, 1, 1, 8, 2, 2, 4, 2, 10, 2, 4, 7, 9, 1, 7, 4, 8, 2, 4, 2, 10, 9, 4, 8, 1, 8, 10, 9, 3, 4, 7, 9, 8, 8, 10, 5, 5, 4, 5, 9, 9, 5, 2, 10, 8, 6, 7, 3, 5, 4, 7, 8, 3, 6, 3, 6, 7, 6, 3, 2, 8, 6, 8, 6, 4, 9, 9, 6, 9, 2, 2, 3, 9, 9, 2, 2, 2, 2, 4, 6, 2, 8, 6, 9, 4, 8, 1, 3, 5, 2, 1, 6, 10, 2, 3, 8, 4, 9, 7, 7, 7, 10, 6, 6, 3, 7, 7, 1, 5, 2, 9, 10, 10, 3, 2, 8, 3, 5, 6, 6, 4, 10, 10, 1, 10, 3, 9, 8, 2, 6, 7, 2, 4, 4, 3, 7, 2, 7, 6, 3, 5, 7, 4, 8, 6, 8, 10, 10, 5, 6, 5, 9, 6, 8, 7, 10, 10, 8, 3, 2, 10, 8, 2, 7, 4, 7, 10, 6, 2, 10, 6, 10, 3, 5, 3, 3, 6, 6, 10, 2, 4, 1, 4, 4, 8, 5, 1, 3, 6, 8, 6, 8, 2, 8, 6, 8, 10, 6, 3, 4, 2, 7, 3, 8, 6, 6, 2, 7, 9, 10, 6, 3, 6, 4, 8, 8, 9, 9, 1, 1, 10, 2, 9, 3, 6, 3, 10, 2, 7, 5, 6, 6, 1, 9, 9, 8, 5, 6, 5, 6, 4, 4, 7, 2, 6, 1, 7, 8, 1, 10, 7, 8, 5, 3, 8, 8, 3, 10, 6, 1, 7, 1, 9, 4, 7, 2, 10, 1, 6, 5, 7, 10, 6, 5, 7, 8, 9, 5, 1, 6, 5, 5, 8, 5, 5, 2, 8, 4, 6, 4, 2, 9, 5, 9, 3, 1, 7, 9, 5, 7, 7, 7, 3, 6, 1, 2, 9, 4, 5, 2, 3, 6, 7, 1, 3, 8, 8, 2, 5, 10, 6, 3, 9, 8, 3, 8, 5, 2, 9, 10, 3, 8, 4, 5, 10, 2, 2, 1, 4, 6, 4, 10, 3, 1, 5, 6, 5, 3, 5, 9, 1, 4, 10, 10, 2, 1, 8, 10, 3, 5, 8, 10, 5, 6, 4, 7, 10, 10, 6, 10, 2, 1, 6, 8, 10, 3, 5, 3, 2, 10, 3, 3, 9, 6, 6, 3, 2, 5, 2, 5, 10, 7, 4, 4, 3, 4, 10, 9, 5, 3, 3, 4, 7, 3, 3, 5, 2, 2, 3, 9, 9, 8, 6, 6, 4, 2, 5, 1, 1, 9, 5, 5, 4, 6, 3, 3, 1, 2, 9, 3, 8, 10, 3, 4, 2, 9, 9, 8, 1, 3, 2, 5, 9, 5, 1, 2, 10, 6, 2, 8, 8, 6, 10, 10, 7, 1, 10, 8, 8, 10, 3, 3, 10, 5, 7, 2, 9, 2, 4, 10, 10, 1, 8, 4, 9, 4, 2, 5, 1, 8, 2, 10, 3, 4, 10, 1, 8, 7, 3, 3, 6, 3, 3, 6, 7, 2, 4, 3, 7, 7, 4, 2, 9, 1, 8, 1, 2, 4, 5, 9, 9, 6, 5, 6, 6, 10, 8, 9, 7, 4, 3, 7, 3, 3, 4, 5, 5, 6, 8, 7, 2, 6, 10, 6, 3, 1, 9, 1, 5, 4, 3, 7, 10, 5, 1, 5, 10, 2, 3, 1, 6, 8, 2, 1, 7, 1, 4, 7, 6, 9, 3, 8, 4, 3, 6, 2, 7, 5, 9, 1, 1, 1, 6, 5, 1, 4, 3, 2, 8, 5, 10, 10, 7, 5, 10, 1, 10, 4, 5, 7, 3, 6, 10, 9, 8, 3, 10, 5, 1, 5, 1, 2, 2, 5, 6, 7, 6, 5, 5, 8, 10, 4, 2, 2, 3, 4, 5, 5, 2, 3, 2, 9, 2, 10, 1, 6, 3, 10, 10, 9, 3, 2, 10, 9, 8, 6, 7, 8, 9, 10, 6, 9, 7, 4, 7, 8, 9, 6, 6, 10, 8, 3, 8, 5, 5, 9, 1, 7, 3, 6, 7, 5, 7, 7, 2, 5, 4, 10, 7, 5, 1, 1, 9, 7, 6, 1, 3, 2, 4, 10, 10, 8, 2, 9, 2, 2, 2, 4, 3, 4, 3, 9, 7, 6, 9, 7, 8, 9, 7, 3, 1, 6, 1, 3, 9, 10, 9, 5, 10, 5, 3, 3, 9, 9, 6, 8, 8, 6, 2, 7, 1, 4, 3, 6, 2, 8, 10, 10, 4, 4, 1, 3, 10, 8, 4, 4, 10, 9, 9, 6, 5, 8, 5, 2, 10, 5, 10, 9, 2, 10, 3, 4, 4, 8, 6, 2, 1, 5, 2, 1, 2, 7, 6, 10, 10, 10, 3, 7, 2, 3, 7, 4, 10, 3, 7, 4, 6, 5, 6, 6, 2, 9, 2, 9, 7, 8, 6, 6, 1, 10, 3, 3, 6, 5, 5, 6, 5, 8, 10, 6, 8, 7, 5, 7, 9, 6, 6, 10, 8, 10, 4, 9, 5, 2, 1, 1, 2, 4, 7, 10, 1, 6, 4, 3, 4, 1, 6, 7, 9, 6, 9, 4, 2, 1, 3, 8, 1, 9, 5, 4, 6, 3, 7, 6, 8, 5, 8, 3, 8, 5, 2, 5, 8, 8, 4, 10, 7, 4, 1, 1, 5, 5, 4, 4, 7, 5, 10, 8, 3, 5, 9, 4, 4, 5, 5, 2, 10, 6, 3, 3, 3, 9, 2, 7, 8, 10, 2, 10, 7, 8, 3, 1, 3, 6, 1, 4, 9, 6, 6, 9, 2, 5, 10, 6, 8, 3, 7, 8, 10, 5, 8, 4, 9, 4, 4, 8, 2, 6, 5, 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6, 2, 10, 7, 1, 7, 2, 6, 1, 8, 5, 10, 8, 10, 4, 4, 1, 4, 6, 2, 6, 9, 3, 8, 8, 10, 4, 10, 1, 1, 4, 3, 8, 6, 1, 5, 10, 1, 10, 2, 9, 7, 8, 8, 3, 8, 7, 2, 2, 10, 6, 1, 3, 5, 8, 4, 4, 3, 7, 4, 10, 5, 2, 1, 1, 3, 6, 1, 1, 4, 4, 6, 3, 9, 2, 2, 7, 3, 4, 2, 2, 9, 4, 8, 7, 4, 4, 1, 5, 4, 1, 3, 10, 2, 10, 6, 9, 1, 3, 1, 1, 8, 2, 7, 7, 9, 6, 9, 10, 2, 4, 9, 5, 8, 3, 3, 9, 7, 3, 8, 5, 9, 3, 7, 5, 6, 7, 6, 2, 3, 3, 1, 6, 9, 4, 8, 6, 5, 7, 10, 4, 2, 6, 2, 7, 7, 3, 4, 2, 8, 5, 5, 9, 4, 8, 3, 10, 9, 5, 3, 7, 6, 1, 8, 4, 2, 8, 4, 5, 8, 4, 3, 9, 7, 5, 3, 4, 4, 3, 5, 3, 8, 6, 2, 7, 5, 8, 4, 10, 2, 2, 4, 10, 7, 2, 6, 5, 2, 8, 3, 4, 10, 3, 6, 3, 6, 1, 6, 4, 9, 9, 1, 7, 10, 8, 8, 10, 4, 9, 5, 8, 1, 4, 4, 4, 9, 7, 4, 8, 3, 1, 9, 2, 1, 3, 3, 6, 9, 8, 4, 7, 3, 7, 7, 3, 10, 7, 3, 9, 8, 1, 2, 4, 6, 3, 10, 1, 9, 1, 8, 5, 7, 4, 7, 6, 6, 7, 7, 9, 2, 5, 2, 3, 5, 7, 9, 8, 10, 9, 5, 2, 1, 1, 5, 5, 3, 6, 10, 6, 3, 9, 2, 1, 9, 10, 5, 2, 8, 3, 8, 6, 1, 8, 9, 2, 10, 8, 10, 4, 2, 4, 9, 6, 3, 9, 3, 4, 5, 5, 2, 1, 1, 5, 4, 8, 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3, 1, 4, 2, 5, 5, 8, 5, 3, 1, 7, 5, 6, 5, 3, 1, 2, 1, 2, 6, 3, 3, 2, 2, 3, 9, 6, 1, 8, 10, 2, 1, 9, 9, 7, 8, 2, 7, 3, 10, 9, 8, 3, 7, 5, 7, 5, 1, 2, 10, 9, 6, 9, 5, 4, 4, 4, 3, 3, 10, 2, 6, 7, 1, 5, 2, 10, 2, 7, 3, 1, 1, 9, 6, 3, 10, 3, 6, 2, 10, 4, 7, 2, 9, 5, 10, 6, 2, 9, 6, 4, 9, 1, 6, 9, 9, 6, 8, 6, 10, 1, 5, 6, 3, 2, 10, 10, 10, 8, 6, 10, 3, 8, 10, 10, 3, 7, 2, 3, 4, 4, 2, 1, 2, 9, 3, 3, 3, 3, 4, 3, 3, 1, 9, 4, 5, 7, 1, 3, 7, 6, 5, 8, 3, 7, 10, 6, 8, 4, 1, 6, 2, 7, 8, 10, 3, 10, 5, 2, 4, 4, 1, 3, 5, 1, 6, 4, 4, 6, 1, 3, 3, 2, 5, 4, 4, 6, 8, 5, 2, 6, 1, 10, 6, 2, 7, 3, 4, 7, 2, 1, 3, 8, 10, 4, 8, 5, 7, 7, 6, 10, 9, 3, 7, 10, 5, 5, 4, 6, 1, 7, 9, 8, 10, 4, 1, 8, 6, 1, 3, 3, 9, 2, 1, 10, 1, 4, 5, 10, 8, 6, 3, 7, 5, 3, 1, 7, 6, 1, 4, 2, 4, 10, 1, 8, 4, 6, 6, 6, 3, 2, 6, 4, 6, 5, 9, 1, 5, 7, 1, 8, 8, 4, 7, 1, 6, 7, 9, 5, 9, 1, 10, 8, 10, 8, 3, 8, 9, 6, 3, 10, 9, 6, 8, 8, 4, 10, 9, 6, 7, 7, 8, 2, 10, 8, 10, 10, 6, 7, 4, 1, 3, 9, 5, 9, 10, 6, 10, 7, 8, 9, 2, 7, 7, 3, 8, 1, 4, 10, 5, 9, 3, 2, 6, 10, 7, 2, 5, 4, 10, 8, 10, 7, 1, 6, 2, 2, 5, 9, 3, 9, 6, 3, 9, 4, 8, 8, 2, 9, 9, 7, 10, 1, 1, 7, 9, 3, 3, 7, 5, 1, 5, 3, 7, 3, 9, 1, 8, 4, 1, 6, 6, 7, 5, 1, 2, 10, 3, 6, 5, 4, 9, 3, 10, 6, 8, 3, 3, 2, 8, 4, 7, 4, 10, 9, 10, 9, 1, 8, 7, 9, 8, 6, 5, 7, 2, 3, 9, 7, 8, 3, 4, 9, 8, 9, 8, 10, 9, 2, 6, 5, 2, 6, 7, 1, 6, 7, 3, 8, 2, 1, 9, 8, 9, 1, 6, 3, 1, 5, 2, 10, 4, 3, 7, 5, 1, 3, 5, 10, 9, 7, 9, 2, 2, 4, 10, 10, 4, 8, 7, 1, 6, 2, 5, 2, 5, 8, 8, 6, 10, 1, 2, 4, 3, 9, 1, 7, 2, 10, 4, 4, 6, 6, 8, 5, 1, 10, 7, 6, 4, 3, 10, 5, 6, 1, 9, 9, 5, 4, 9, 1, 8, 5, 4, 10, 3, 1, 7, 3, 3, 9, 9, 2, 8, 10, 1, 6, 1, 2, 4, 10, 3, 5, 4, 1, 5, 4, 6, 8, 2, 8, 3, 9, 10, 8, 5, 1, 6, 2, 2, 4, 10, 8, 6, 10, 3, 9, 2, 7, 5, 1, 2, 7, 7, 4, 7, 9, 3, 4, 2, 7, 3, 6, 7, 1, 10, 8, 3, 3, 9, 2, 8, 1, 10, 4, 2, 7, 10, 10, 10, 7, 6, 2, 8, 8, 4, 7, 5, 3, 6, 1, 8, 7, 2, 8, 1, 10, 1, 4, 1, 5, 6, 9, 9, 7, 8, 2, 4, 4, 10, 6, 7, 8, 4, 3, 6, 8, 9, 5, 2, 7, 10, 9, 8, 4, 9, 2, 3, 2, 4, 2, 4, 6, 6, 10, 10, 1, 6, 1, 5, 6, 7, 4, 2, 7, 1, 3, 9, 8, 6, 4, 1, 9, 9, 7, 6, 2, 8, 4, 5, 10, 5, 6, 2, 9, 2, 3, 3, 4, 2, 5, 4, 7, 6, 7, 9, 3, 2, 8, 4, 2, 8, 7, 2, 7, 7, 10, 10, 1, 10, 5, 7, 10, 3, 9, 5, 3, 10, 6, 1, 9, 2, 10, 7, 4, 1, 2, 3, 7, 3, 10, 5, 10, 6, 3, 5, 1, 8, 9, 8, 10, 4, 8, 7, 2, 7, 6, 5, 1, 10, 2, 3, 10, 1, 10, 6, 10, 4, 9, 5, 4, 3, 10, 7, 5, 8, 4, 10, 7, 2, 9, 10, 2, 2, 2, 5, 8, 8, 4, 6, 1, 8, 6, 1, 1, 3, 9, 4, 2, 7, 3, 7, 8, 1, 2, 3, 5, 10, 3, 2, 7, 2, 2, 6, 4, 6, 10, 10, 1, 1, 8, 2, 7, 10, 1, 1, 5, 3, 7, 3, 9, 4, 7, 6, 8, 4, 3, 9, 10, 1, 10, 1, 5, 5, 9, 8, 3, 4, 9, 7, 7, 2, 3, 7, 1, 4, 1, 2, 9, 8, 2, 3, 6, 3, 2, 7, 8, 2, 6, 4, 9, 5, 2, 10, 7, 2, 5, 7, 6, 2, 5, 7, 8, 2, 2, 4, 3, 2, 7, 5, 3, 1, 7, 10, 2, 6, 2, 4, 10, 8, 2, 7, 7, 2, 9, 10, 4, 7, 4, 3, 9, 10, 8, 4, 9, 3, 10, 6, 1, 9, 9, 6, 4, 4, 8, 7, 8, 9, 1, 1, 7, 10, 4, 2, 2, 6, 10, 1, 7, 9, 5, 1, 10, 5, 8, 7, 10, 7, 9, 2, 9, 9, 7, 9, 2, 3, 7, 5, 3, 10, 2, 7, 1, 3, 3, 6, 3, 8, 7, 5, 4, 8, 10, 4, 3, 8, 8, 8, 10, 5, 9, 4, 4, 7, 2, 9, 2, 10, 6, 4, 7, 1, 9, 6, 3, 1, 4, 4, 8, 5, 9, 3, 4, 2, 3, 7, 10, 4, 8, 8, 2, 1, 6, 3, 6, 9, 1, 2, 10, 8, 4, 8, 6, 1, 2, 6, 3, 10, 6, 10, 10, 4, 9, 6, 3, 5, 4, 4, 6, 1, 6, 4, 10, 5, 9, 8, 9, 4, 10, 7, 2, 7, 3, 4, 8, 4, 3, 3, 7, 5, 2, 4, 5, 4, 10, 7, 4, 3, 2, 4, 7, 2, 4, 8, 2, 6, 3, 1, 3, 7, 7, 4, 7, 8, 5, 9, 3, 4, 6, 4, 3, 1, 1, 9, 1, 7, 5, 1, 4, 8, 6, 4, 3, 2, 8, 7, 4, 8, 6, 4, 1, 3, 9, 5, 8, 3, 9, 6, 2, 4, 9, 9, 8, 2, 10, 3, 9, 10, 9, 8, 5, 2, 4, 5, 7, 3, 2, 9, 1, 6, 6, 10, 8, 9, 1, 1, 2, 10, 6, 10, 7, 1, 7, 5, 7, 7, 9, 4, 2, 2, 5, 8, 1, 1, 6, 4, 4, 8, 7, 10, 10, 9, 8, 10, 2, 9, 4, 9, 5, 5, 9, 9, 10, 1, 1, 8, 8, 8, 9, 6, 7, 5, 2, 4, 1, 6, 8, 7, 5, 9, 9, 1, 10, 1, 4, 4, 6, 5, 4, 1, 8, 7, 3, 2, 10, 10, 5, 8, 9, 7, 6, 8, 1, 3, 2, 7, 3, 7, 1, 6, 2, 8, 5, 1, 3, 9, 10, 5, 9, 6, 4, 5, 7, 2, 4, 7, 7, 4, 10, 6, 9, 6, 6, 4, 7, 4, 7, 5, 3, 2, 2, 9, 8, 8, 8, 1, 3, 7, 6, 7, 6, 10, 6, 2, 9, 5, 5, 6, 4, 4, 9, 8, 4, 10, 1, 2, 3, 2, 1, 4, 9, 6, 6, 8, 3, 2, 2, 2, 2, 2, 1, 8, 9, 5, 10, 3, 7, 8, 9, 5, 8, 9, 9, 3, 4, 2, 8, 2, 7, 9, 8, 1, 7, 4, 10, 8, 2, 4, 3, 1, 10, 2, 9, 7, 3, 2, 6, 8, 3, 3, 7, 3, 3, 4, 7, 3, 5, 2, 3, 5, 1, 1, 1, 3, 7, 6, 3, 7, 8, 8, 10, 4, 2, 5, 7, 7, 5, 6, 6, 9, 8, 4, 6, 4, 6, 9, 7, 6, 10, 2, 8, 8, 7, 1, 8, 10, 4, 2, 6, 10, 10, 3, 4, 4, 10, 4, 5, 7, 3, 4, 9, 2, 6, 8, 9, 2, 9, 4, 8, 3, 4, 7, 10, 9, 8, 5, 3, 10, 2, 6, 5, 9, 3, 2, 7, 1, 4, 2, 10, 2, 7, 8, 9, 10, 4, 2, 3, 10, 1, 7, 6, 2, 5, 2, 10, 4, 9, 7, 7, 2, 5, 5, 2, 4, 2, 10, 5, 6, 9, 10, 3, 1, 2, 2, 8, 6, 3, 4, 1, 8, 8, 2, 8, 3, 3, 5, 8, 9, 5, 9, 3, 3, 1, 7, 1, 5, 9, 2, 4, 10, 4, 7, 7, 8, 5, 4, 2, 1, 10, 8, 4, 9, 4, 10, 4, 9, 9, 5, 5, 5, 5, 5, 6, 3, 5, 6, 1, 4, 5, 8, 8, 10, 3, 4, 8, 1, 10, 2, 8, 6, 8, 7, 4, 9, 6, 3, 7, 9, 6, 5, 2, 10, 3, 4, 9, 9, 7, 1, 3, 3, 10, 7, 10, 4, 6, 7, 5, 10, 6, 6, 8, 8, 1, 1, 2, 9, 3, 2, 4, 1, 6, 6, 6, 8, 2, 7, 1, 8, 7, 2, 1, 1, 5, 6, 5, 3, 7, 1, 3, 8, 3, 6, 10, 4, 1, 10, 1, 4, 1, 3, 2, 8, 4, 7, 8, 4, 5, 7, 5, 3, 8, 3, 8, 8, 9, 8, 6, 8, 1, 3, 5, 3, 5, 10, 8, 8, 3, 3, 3, 3, 6, 3, 5, 2, 1, 5, 2, 10, 7, 10, 3, 10, 2, 10, 6, 8, 5, 7, 9, 10, 6, 1, 5, 7, 2, 6, 6, 2, 3, 3, 9, 10, 2, 10, 2, 10, 3, 3, 8, 8, 1, 5, 9, 2, 8, 7, 6, 1, 2, 2, 1, 9, 5, 6, 5, 4, 9, 9, 4, 6, 6, 3, 4, 3, 2, 10, 10, 3, 7, 2, 6, 3, 2, 1, 8, 10, 10, 6, 2, 10, 8, 4, 5, 2, 3, 2, 5, 7, 9, 2, 7, 4, 3, 6, 5, 4, 2, 8, 8, 4, 7, 5, 1, 6, 4, 8, 4, 10, 2, 7, 8, 5, 3, 10, 10, 8, 5, 5, 5, 10, 10, 9, 7, 2, 6, 10, 8, 1, 8, 9, 2, 2, 5, 3, 7, 7, 10, 10, 4, 1, 5, 3, 4, 1, 3, 1, 7, 6, 4, 10, 4, 5, 7, 10, 5, 7, 2, 2, 1, 3, 6, 9, 9, 9, 3, 9, 8, 9, 4, 4, 6, 7, 8, 6, 1, 2, 2, 10, 4, 7, 1, 8, 7, 5, 5, 5, 9, 1, 9, 5, 7, 2, 3, 7, 5, 3, 5, 2, 9, 9, 1, 7, 9, 10, 4, 7, 1, 2, 3, 7, 3, 10, 2, 5, 8, 8, 2, 1, 2, 9, 6, 2, 2, 2, 1, 9, 10, 5, 7, 8, 9, 9, 9, 6, 10, 6, 3, 9, 7, 5, 6, 6, 10, 7, 5, 4, 7, 6, 6, 9, 3, 8, 8, 3, 1, 7, 5, 10, 6, 8, 7, 4, 4, 4, 3, 7, 2, 1, 2, 6, 7, 8, 5, 7, 1, 2, 4, 2, 5, 9, 2, 8, 10, 4, 7, 4, 4, 1, 10, 3, 5, 1, 3, 8, 7, 4, 1, 2, 4, 10, 6, 4, 8, 9, 8, 3, 4, 4, 6, 4, 6, 5, 8, 1, 10, 7, 10, 8, 7, 9, 4, 9, 3, 1, 10, 9, 6, 9, 5, 7, 1, 10, 1, 1, 2, 5, 10, 8, 2, 9, 6, 5, 9, 5, 9, 4, 2, 3, 2, 8, 10, 2, 5, 9, 2, 2, 10, 3, 4, 9, 8, 10, 3, 3, 9, 4, 6, 6, 9, 2, 7, 10, 3, 2, 4, 1, 5, 3, 5, 3, 6, 5, 4, 9, 2, 7, 8, 4, 6, 2, 10, 10, 4, 4, 9, 2, 3, 5, 6, 2, 6, 6, 10, 1, 5, 8, 6, 5, 9, 7, 4, 1, 3, 2, 6, 10, 8, 10, 10, 4, 6, 7, 7, 8, 10, 1, 8, 4, 5, 3, 7, 8, 9, 10, 1, 7, 10, 7, 3, 7, 5, 10, 4, 10, 10, 1, 9, 5, 1, 9, 3, 3, 1, 8, 3, 6, 7, 7, 5, 1, 3, 3, 2, 5, 4, 8, 1, 6, 1, 10, 8, 6, 8, 4, 1, 3, 1, 5, 2, 6, 9, 5, 2, 1, 7, 10, 1, 8, 8, 9, 9, 7, 10, 4, 6, 8, 4, 3, 1, 9, 6, 2, 2, 4, 3, 5, 3, 10, 9, 1, 7, 4, 5, 10, 7, 6, 9, 5, 5, 9, 2, 6, 6, 8, 1, 6, 2, 5, 3, 4, 5, 9, 7, 6, 10, 5, 9, 3, 3, 3, 8, 2, 3, 10, 7, 5, 4, 9, 7, 3, 8, 3, 6, 5, 9, 10, 10, 9, 4, 4, 10, 9, 6, 7, 5, 7, 10, 5, 8, 9, 3, 2, 6, 9, 5, 10, 2, 1, 8, 6, 8, 6, 8, 9, 5, 10, 7, 5, 2, 4, 5, 6, 7, 1, 4, 6, 5, 2, 3, 2, 9, 3, 9, 4, 9, 9, 9, 5, 4, 3, 6, 7, 9, 2, 7, 3, 2, 5, 2, 8, 10, 2, 2, 1, 1, 3, 8, 7, 3, 9, 8, 8, 1, 7, 5, 6, 6, 9, 1, 2, 9, 8, 1, 10, 5, 3, 7, 4, 6, 5, 7, 10, 7, 9, 8, 5, 6, 5, 6, 9, 2, 4, 4, 1, 8, 4, 3, 1, 1, 8, 6, 6, 8, 10, 10, 5, 4, 10, 5, 2, 5, 2, 7, 6, 1, 3, 5, 5, 9, 5, 3, 9, 6, 8, 5, 5, 4, 8, 3, 5, 10, 2, 9, 6, 1, 9, 4, 5, 10, 4, 8, 5, 4, 2, 7, 4, 4, 4, 6, 9, 7, 3, 10, 6, 6, 5, 5, 7, 2, 9, 10, 4, 2, 8, 5, 8, 1, 4, 4, 10, 3, 9, 9, 5, 1, 1, 2, 5, 4, 2, 10, 6, 8, 10, 2, 3, 2, 6, 4, 1, 3, 10, 6, 10, 3, 6, 10, 5, 1, 1, 9, 10, 3, 8, 4, 2, 9, 3, 2, 3, 6, 5, 3, 9, 5, 7, 3, 7, 3, 5, 4, 3, 2, 9, 1, 3, 4, 3, 3, 5, 3, 5, 1, 7, 7, 4, 2, 7, 7, 1, 3, 8, 1, 4, 10, 6, 9, 2, 10, 9, 5, 3, 6, 2, 4, 10, 4, 10, 7, 10, 1, 6, 9, 3, 2, 8, 10, 3, 10, 8, 3, 4, 9, 10, 2, 2, 1, 4, 9, 6, 4, 3, 5, 7, 4, 9, 3, 5, 7, 4, 1, 8, 4, 5, 7, 7, 6, 10, 6, 4, 7, 10, 7, 6, 6, 1, 7, 6, 9, 1, 4, 6, 6, 5, 7, 10, 5, 1, 10, 1, 1, 6, 8, 3, 4, 7, 6, 5, 6, 9, 3, 1, 3, 10, 1, 4, 2, 8, 1, 10, 2, 3, 10, 2, 7, 1, 5, 5, 7, 9, 9, 9, 5, 2, 1, 7, 8, 1, 5, 4, 2, 10, 10, 9, 7, 1, 7, 3, 5, 9, 4, 4, 9, 6, 5, 1, 6, 10, 6, 8, 7, 4, 4, 1, 6, 6, 10, 5, 1, 6, 9, 5, 7, 6, 5, 5, 7, 9, 8, 2, 5, 4, 5, 10, 5, 8, 4, 6, 4, 6, 6, 10, 1, 9, 6, 10, 5, 3, 5, 3, 2, 5, 6, 6, 5, 1, 1, 6, 3, 9, 8, 8, 1, 3, 10, 8, 2, 9, 4, 4, 10, 8, 6, 5, 8, 6, 4, 9, 2, 2, 10, 1, 6, 1, 9, 9, 7, 2, 5, 2, 10, 1, 4, 10, 8, 10, 8, 2, 2, 6, 6, 9, 9, 6, 6, 4, 10, 6, 10, 4, 3, 8, 9, 3, 1, 10, 4, 9, 10, 10, 9, 9, 8, 4, 8, 3, 8, 10, 3, 8, 8, 8, 6, 1, 4, 3, 8, 7, 10, 9, 3, 3, 6, 1, 4, 10, 6, 1, 2, 5, 6, 6, 1, 7, 5, 4, 10, 9, 6, 8, 8, 4, 2, 5, 9, 8, 5, 1, 9, 9, 7, 3, 10, 10, 9, 4, 10, 8, 6, 4, 1, 4, 4, 5, 9, 7, 1, 6, 1, 9, 7, 9, 2, 2, 1, 10, 5, 1, 1, 7, 2, 1, 5, 6, 6, 10, 10, 9, 4, 2, 1, 3, 7, 7, 7, 9, 8, 6, 10, 6, 1, 3, 1, 10, 4, 1, 7, 3, 6, 4, 10, 10, 5, 2, 2, 9, 7, 8, 7, 8, 5, 8, 7, 3, 8, 10, 3, 7, 4, 8, 7, 9, 3, 7, 1, 10, 9, 1, 1, 5, 7, 10, 9, 6, 5, 7, 8, 6, 6, 10, 3, 1, 1, 9, 4, 2, 1, 4, 3, 10, 5, 6, 9, 2, 6, 10, 4, 4, 1, 2, 8, 10, 8, 9, 8, 10, 4, 10, 4, 8, 4, 7, 6, 9, 10, 5, 4, 10, 2, 8, 9, 9, 2, 4, 6, 2, 10, 2, 3, 9, 4, 2, 3, 6, 10, 9, 4, 3, 10, 3, 9, 3, 7, 8, 5, 1, 9, 10, 10, 3, 5, 7, 10, 4, 4, 4, 10, 3, 6, 5, 9, 3, 6, 9, 7, 7, 6, 6, 2, 2, 10, 8, 1, 2, 5, 4, 5, 3, 1, 3, 1, 3, 1, 2, 1, 2, 8, 10, 4, 6, 2, 4, 7, 3, 10, 10, 10, 9, 4, 3, 9, 9, 2, 4, 10, 4, 2, 10, 8, 3, 4, 8, 9, 6, 8, 10, 4, 2, 2, 2, 10, 3, 8, 5, 10, 4, 2, 3, 7, 3, 10, 4, 9, 10, 5, 6, 6, 6, 10, 9, 3, 3, 6, 5, 5, 7, 8, 5, 5, 1, 5, 8, 5, 1, 9, 4, 6, 8, 2, 1, 8, 9, 3, 9, 4, 7, 1, 1, 7, 6, 6, 8, 5, 8, 9, 10, 6, 8, 1, 2, 7, 7, 3, 7, 1, 7, 3, 9, 3, 10, 3, 3, 4, 8, 2, 10, 5, 9, 4, 8, 6, 8, 9, 6, 9, 3, 10, 3, 9, 2, 7, 8, 4, 7, 5, 6, 7, 6, 3, 5, 8, 4, 7, 7, 3, 4, 10, 3, 5, 5, 1, 7, 1, 4, 4, 4, 8, 1, 6, 5, 7, 7, 6, 1, 10, 3, 4, 6, 3, 8, 6, 1, 4, 8, 3, 2, 4, 1, 2, 1, 6, 4, 3, 1, 5, 2, 9, 9, 8, 1, 5, 9, 9, 10, 2, 5, 2, 2, 5, 7, 8, 4, 9, 2, 9, 10, 2, 5, 6, 10, 5, 8, 10, 8, 10, 4, 8, 1, 2, 1, 5, 1, 3, 10, 6, 10, 10, 7, 1, 10, 8, 10, 10, 9, 7, 6, 1, 7, 5, 10, 7, 7, 3, 7, 3, 3, 10, 8, 7, 6, 9, 10, 2, 2, 8, 5, 8, 10, 5, 7, 7, 7, 4, 8, 9, 10, 4, 2, 10, 3, 2, 1, 7, 3, 9, 1, 1, 10, 8, 3, 8, 9, 9, 7, 1, 8, 1, 8, 1, 4, 6, 4, 5, 6, 4, 3, 8, 3, 3, 9, 6, 10, 2, 4, 4, 2, 6, 6, 7, 1, 7, 2, 6, 6, 1, 7, 9, 1, 3, 7, 9, 4, 8, 9, 8, 1, 5, 1, 4, 3, 9, 10, 3, 4, 10, 6, 2, 2, 8, 4, 2, 10, 6, 3, 7, 7, 6, 8, 10, 7, 6, 7, 2, 2, 7, 4, 10, 1, 9, 7, 7, 9, 4, 8, 3, 3, 1, 2, 9, 7, 7, 4, 7, 2, 4, 10, 5, 6, 4, 3, 6, 2, 2, 4, 8, 5, 5, 4, 2, 8, 5, 10, 10, 8, 8, 3, 1, 9, 4, 5, 5, 9, 10, 7, 3, 3, 3, 3, 4, 10, 2, 8, 10, 5, 8, 8, 10, 7, 4, 6, 9, 10, 4, 7, 7, 4, 8, 3, 6, 1, 6, 8, 1, 10, 1, 8, 2, 4, 10, 1, 1, 8, 10, 1, 8, 9, 3, 7, 7, 10, 7, 1, 6, 5, 6, 5, 7, 8, 8, 7, 4, 2, 7, 1, 1, 7, 10, 1, 5, 10, 10, 7, 8, 3, 3, 8, 2, 7, 4, 1, 9, 7, 1, 7, 5, 2, 9, 9, 6, 4, 10, 8, 6, 5, 5, 2, 9, 6, 7, 2, 10, 8, 2, 6, 10, 5, 6, 3, 6, 1, 2, 5, 5, 8, 9, 2, 5, 9, 2, 2, 8, 9, 1, 5, 1, 8, 4, 8, 9, 8, 5, 9, 9, 7, 9, 6, 10, 3, 7, 10, 1, 4, 3, 5, 7, 7, 8, 4, 3, 8, 9, 7, 10, 9, 9, 4, 5, 2, 4, 4, 10, 6, 4, 5, 8, 4, 5, 2, 8, 4, 7, 1, 3, 7, 4, 5, 8, 10, 5, 4, 6, 6, 6, 8, 1, 2, 2, 2, 7, 6, 5, 3, 1, 9, 10, 2, 4, 10, 1, 10, 1, 1, 3, 9, 4, 3, 3, 3, 3, 1, 7, 6, 4, 10, 1, 2, 6, 6, 2, 5, 9, 2, 7, 6, 1, 4, 4, 8, 3, 1, 3, 5, 4, 10, 10, 9, 7, 1, 1, 9, 4, 5, 4, 10, 4, 10, 3, 5, 6, 4, 1, 9, 1, 2, 6, 1, 7, 3, 2, 10, 2, 7, 3, 3, 6, 10, 6, 7, 3, 3, 4, 4, 10, 5, 4, 1, 4, 7, 3, 2, 8, 1, 4, 7, 9, 6, 5, 8, 5, 1, 4, 6, 5, 5, 9, 3, 9, 4, 2, 6, 8, 4, 3, 9, 5, 6, 10, 5, 5, 6, 4, 6, 1, 4, 4, 1, 5, 2, 4, 7, 5, 1, 6, 5, 1, 1, 9, 2, 8, 10, 3, 7, 4, 4, 8, 6, 8, 1, 2, 3, 4, 5, 7, 3, 1, 8, 8, 3, 9, 1, 3, 5, 1, 2, 2, 6, 1, 5, 7, 6, 7, 4, 2, 1, 4, 4, 3, 10, 4, 3, 7, 5, 7, 3, 7, 9, 6, 2, 1, 7, 6, 3, 7, 6, 5, 9, 10, 5, 5, 2, 9, 6, 8, 8, 5, 1, 8, 2, 7, 7, 6, 8, 4, 2, 2, 10, 1, 4, 9, 1, 3, 2, 9, 4, 10, 4, 1, 2, 10, 9, 1, 9, 7, 4, 3, 10, 1, 1, 3, 2, 6, 3, 6, 7, 4, 3, 6, 10, 5, 8, 7, 6, 7, 6, 5, 8, 4, 8, 6, 7, 3, 3, 6, 2, 2, 10, 4, 5, 10, 7, 10, 10, 5, 3, 10, 6, 2, 8, 2, 2, 2, 2, 7, 8, 7, 1, 3, 3, 7, 8, 4, 6, 6, 3, 6, 10, 4, 5, 5, 1, 10, 1, 3, 5, 4, 6, 10, 7, 9, 8, 4, 3, 6, 4, 6, 1, 5, 5, 1, 9, 1, 4, 7, 1, 5, 5, 7, 4, 2, 7, 2, 3, 5, 3, 3, 6, 10, 3, 3, 9, 1, 1, 3, 1, 6, 8, 2, 7, 7, 5, 1, 9, 9, 4, 2, 5, 7, 5, 9, 3, 10, 1, 5, 7, 6, 7, 4, 5, 4, 1, 8, 2, 8, 6, 9, 2, 10, 6, 8, 5, 2, 8, 5, 6, 10, 2, 8, 10, 6, 8, 5, 10, 5, 4, 8, 6, 8, 3, 10, 7, 9, 10, 7, 7, 4, 1, 3, 1, 9, 8, 8, 10, 8, 1, 9, 1, 5, 10, 7, 4, 1, 4, 2, 2, 10, 6, 7, 1, 5, 10, 3, 6, 5, 10, 5, 5, 4, 1, 6, 4, 7, 1, 8, 10, 3, 1, 5, 6, 1, 6, 2, 5, 3, 8, 8, 8, 2, 10, 3, 3, 9, 10, 6, 1, 6, 6, 4, 9, 7, 10, 4, 1, 5, 9, 4, 4, 4, 4, 6, 1, 7, 7, 3, 3, 5, 4, 4, 9, 7, 9, 2, 4, 7, 9, 2, 6, 5, 6, 10, 10, 10, 7, 10, 3, 10, 9, 6, 5, 8, 3, 7, 9, 9, 4, 2, 5, 10, 7, 5, 1, 6, 2, 10, 1, 5, 8, 6, 4, 5, 3, 3, 5, 5, 8, 10, 6, 2, 7, 9, 7, 10, 4, 2, 9, 3, 9, 3, 8, 5, 7, 2, 3, 8, 4, 1, 5, 7, 2, 2, 8, 6, 8, 7, 6, 7, 8, 8, 6, 7, 7, 6, 9, 5, 4, 9, 7, 7, 6, 1, 5, 8, 2, 9, 2, 5, 8, 1, 6, 7, 3, 10, 3, 5, 7, 3, 2, 7, 2, 9, 6, 3, 5, 3, 4, 9, 4, 9, 3, 4, 1, 7, 7, 1, 6, 9, 5, 1, 1, 2, 5, 10, 1, 5, 2, 6, 1, 2, 7, 7, 10, 1, 1, 5, 9, 3, 2, 1, 8, 10, 3, 5, 9, 6, 6, 7, 1, 7, 7, 9, 8, 5, 1, 4, 1, 4, 2, 7, 1, 4, 4, 2, 10, 8, 4, 6, 1, 10, 9, 5, 10, 9, 10, 9, 4, 1, 10, 8, 7, 10, 8, 6, 7, 7, 1, 1, 2, 8, 4, 1, 10, 9, 7, 7, 3, 3, 6, 8, 1, 7, 3, 8, 6, 5, 4, 1, 2, 7, 10, 4, 2, 7, 7. Total: 54562.[/sblock]

Incidentally, that's an uncommonly good set, I think. When I tried 1000d1000 twice, I got something like an average roll of 480 the first time and 490 the second time, compared to an average roll here of 5.46.

 

[Pinotage]

[sblock=10000d10, total=54562]10000d10: 7, 4, 5, 3, 1, 8, 9, 9, 5, 10, 10, 5, 3, 9, 3, 3, 8, 3, 2, 4, 4, 7, 7, 6, 9, 4, 9, 10, 10, 8, 6, 3, 7, 10, 9, 9, 1, 1, 1, 4, 4, 7, 2, 6, 9, 10, 7, 1, 1, 10, 8, 4, 8, 1, 4, 5, 4, 5, 3, 7, 5, 6, 9, 1, 1, 10, 6, 7, 2, 8, 2, 10, 4, 4, 6, 2, 9, 6, 7, 4, 8, 2, 5, 6, 7, 6, 5, 7, 6, 5, 4, 4, 10, 3, 1, 1, 9, 4, 6, 10, 5, 9, 2, 2, 3, 7, 6, 8, 1, 6, 3, 1, 5, 9, 1, 7, 8, 4, 3, 4, 9, 6, 6, 1, 7, 3, 4, 1, 10, 10, 5, 6, 5, 1, 1, 7, 2, 3, 1, 9, 1, 1, 8, 2, 9, 10, 7, 1, 4, 9, 6, 7, 3, 10, 8, 3, 6, 3, 10, 3, 3, 9, 5, 3, 3, 1, 10, 7, 7, 1, 9, 9, 4, 8, 10, 8, 2, 7, 5, 9, 2, 9, 3, 2, 4, 9, 3, 9, 5, 2, 3, 9, 4, 6, 9, 4, 8, 3, 4, 2, 8, 4, 2, 3, 10, 5, 5, 5, 1, 1, 3, 3, 6, 5, 8, 9, 9, 4, 5, 4, 9, 2, 5, 9, 6, 9, 9, 5, 9, 9, 1, 7, 9, 7, 1, 6, 3, 4, 2, 4, 8, 8, 10, 8, 10, 4, 1, 10, 5, 9, 10, 4, 6, 5, 7, 7, 10, 5, 4, 10, 1, 8, 4, 3, 2, 6, 7, 7, 10, 1, 3, 2, 3, 3, 8, 8, 2, 4, 10, 9, 3, 3, 2, 4, 4, 4, 1, 2, 3, 9, 6, 7, 9, 8, 10, 8, 10, 2, 4, 2, 2, 7, 9, 5, 2, 10, 5, 10, 5, 8, 10, 4, 3, 7, 9, 10, 10, 7, 3, 7, 9, 9, 7, 9, 2, 8, 8, 7, 6, 4, 1, 7, 4, 3, 2, 9, 1, 8, 10, 2, 10, 4, 7, 5, 7, 8, 1, 7, 10, 7, 2, 8, 6, 1, 1, 9, 2, 1, 3, 9, 1, 10, 9, 2, 1, 3, 7, 4, 10, 1, 6, 7, 2, 2, 8, 2, 5, 2, 1, 9, 7, 6, 5, 6, 1, 2, 7, 5, 3, 6, 10, 6, 2, 10, 5, 9, 8, 4, 9, 8, 8, 8, 10, 5, 7, 7, 10, 1, 4, 8, 3, 4, 3, 9, 6, 1, 10, 6, 1, 9, 1, 8, 5, 9, 9, 8, 8, 6, 2, 1, 8, 2, 3, 4, 6, 5, 9, 7, 8, 3, 10, 2, 4, 3, 3, 7, 3, 2, 10, 6, 2, 3, 3, 6, 7, 5, 2, 8, 4, 5, 9, 2, 7, 6, 6, 7, 3, 5, 2, 8, 6, 9, 1, 1, 6, 8, 3, 6, 10, 1, 4, 3, 2, 9, 3, 5, 2, 10, 2, 8, 2, 1, 4, 1, 3, 7, 8, 6, 8, 9, 4, 1, 10, 2, 4, 3, 5, 9, 5, 2, 3, 10, 10, 3, 1, 3, 8, 1, 6, 9, 3, 3, 1, 8, 7, 7, 7, 10, 5, 2, 7, 10, 2, 3, 5, 6, 5, 9, 10, 8, 7, 7, 6, 5, 3, 9, 5, 7, 7, 10, 2, 10, 6, 8, 8, 3, 7, 7, 8, 7, 5, 1, 4, 5, 2, 5, 7, 3, 8, 4, 1, 6, 7, 8, 10, 10, 8, 8, 4, 9, 10, 3, 3, 7, 8, 9, 4, 10, 3, 2, 7, 7, 3, 8, 5, 1, 6, 9, 6, 10, 1, 7, 9, 7, 8, 9, 5, 8, 6, 4, 9, 10, 10, 2, 2, 3, 6, 6, 9, 10, 3, 3, 8, 1, 5, 8, 3, 5, 6, 4, 1, 9, 10, 7, 7, 3, 2, 4, 6, 9, 5, 10, 5, 2, 7, 5, 8, 7, 9, 7, 9, 6, 2, 10, 3, 10, 10, 2, 1, 4, 4, 5, 4, 8, 2, 4, 2, 10, 6, 5, 6, 2, 7, 10, 7, 4, 9, 10, 3, 3, 7, 5, 4, 9, 10, 1, 4, 10, 3, 5, 6, 5, 8, 4, 10, 9, 2, 4, 6, 10, 1, 10, 10, 2, 4, 10, 7, 7, 7, 5, 3, 9, 5, 1, 7, 1, 2, 10, 9, 9, 10, 9, 8, 4, 8, 8, 9, 8, 9, 4, 1, 3, 6, 3, 4, 10, 7, 9, 2, 3, 1, 5, 7, 1, 5, 2, 3, 1, 1, 9, 7, 6, 2, 1, 2, 4, 1, 8, 8, 6, 3, 3, 4, 2, 8, 1, 10, 7, 10, 5, 8, 3, 3, 10, 8, 10, 10, 9, 9, 7, 3, 6, 5, 4, 2, 5, 1, 7, 7, 6, 10, 10, 4, 10, 6, 10, 3, 1, 4, 6, 2, 1, 3, 4, 5, 7, 6, 10, 5, 4, 6, 1, 6, 10, 5, 10, 10, 2, 1, 1, 2, 5, 9, 6, 10, 9, 5, 10, 3, 2, 3, 7, 4, 2, 7, 10, 9, 6, 1, 9, 10, 8, 1, 1, 10, 4, 4, 8, 1, 10, 1, 5, 9, 7, 3, 1, 5, 8, 8, 9, 7, 7, 10, 5, 4, 1, 7, 7, 8, 5, 9, 4, 5, 6, 10, 6, 1, 5, 7, 2, 6, 8, 2, 4, 2, 8, 10, 5, 5, 1, 3, 4, 9, 9, 5, 2, 10, 10, 5, 6, 4, 2, 7, 4, 7, 2, 9, 3, 3, 2, 4, 10, 5, 10, 4, 3, 7, 5, 1, 4, 4, 8, 8, 4, 9, 9, 4, 5, 2, 7, 9, 2, 1, 7, 3, 10, 1, 3, 2, 1, 10, 10, 8, 1, 4, 3, 7, 7, 1, 2, 10, 4, 9, 9, 3, 7, 7, 8, 9, 1, 5, 10, 9, 6, 8, 4, 9, 7, 2, 3, 6, 2, 3, 5, 4, 8, 1, 10, 3, 9, 2, 3, 6, 6, 10, 8, 9, 9, 3, 7, 8, 3, 2, 10, 1, 7, 2, 1, 1, 8, 2, 2, 4, 2, 10, 2, 4, 7, 9, 1, 7, 4, 8, 2, 4, 2, 10, 9, 4, 8, 1, 8, 10, 9, 3, 4, 7, 9, 8, 8, 10, 5, 5, 4, 5, 9, 9, 5, 2, 10, 8, 6, 7, 3, 5, 4, 7, 8, 3, 6, 3, 6, 7, 6, 3, 2, 8, 6, 8, 6, 4, 9, 9, 6, 9, 2, 2, 3, 9, 9, 2, 2, 2, 2, 4, 6, 2, 8, 6, 9, 4, 8, 1, 3, 5, 2, 1, 6, 10, 2, 3, 8, 4, 9, 7, 7, 7, 10, 6, 6, 3, 7, 7, 1, 5, 2, 9, 10, 10, 3, 2, 8, 3, 5, 6, 6, 4, 10, 10, 1, 10, 3, 9, 8, 2, 6, 7, 2, 4, 4, 3, 7, 2, 7, 6, 3, 5, 7, 4, 8, 6, 8, 10, 10, 5, 6, 5, 9, 6, 8, 7, 10, 10, 8, 3, 2, 10, 8, 2, 7, 4, 7, 10, 6, 2, 10, 6, 10, 3, 5, 3, 3, 6, 6, 10, 2, 4, 1, 4, 4, 8, 5, 1, 3, 6, 8, 6, 8, 2, 8, 6, 8, 10, 6, 3, 4, 2, 7, 3, 8, 6, 6, 2, 7, 9, 10, 6, 3, 6, 4, 8, 8, 9, 9, 1, 1, 10, 2, 9, 3, 6, 3, 10, 2, 7, 5, 6, 6, 1, 9, 9, 8, 5, 6, 5, 6, 4, 4, 7, 2, 6, 1, 7, 8, 1, 10, 7, 8, 5, 3, 8, 8, 3, 10, 6, 1, 7, 1, 9, 4, 7, 2, 10, 1, 6, 5, 7, 10, 6, 5, 7, 8, 9, 5, 1, 6, 5, 5, 8, 5, 5, 2, 8, 4, 6, 4, 2, 9, 5, 9, 3, 1, 7, 9, 5, 7, 7, 7, 3, 6, 1, 2, 9, 4, 5, 2, 3, 6, 7, 1, 3, 8, 8, 2, 5, 10, 6, 3, 9, 8, 3, 8, 5, 2, 9, 10, 3, 8, 4, 5, 10, 2, 2, 1, 4, 6, 4, 10, 3, 1, 5, 6, 5, 3, 5, 9, 1, 4, 10, 10, 2, 1, 8, 10, 3, 5, 8, 10, 5, 6, 4, 7, 10, 10, 6, 10, 2, 1, 6, 8, 10, 3, 5, 3, 2, 10, 3, 3, 9, 6, 6, 3, 2, 5, 2, 5, 10, 7, 4, 4, 3, 4, 10, 9, 5, 3, 3, 4, 7, 3, 3, 5, 2, 2, 3, 9, 9, 8, 6, 6, 4, 2, 5, 1, 1, 9, 5, 5, 4, 6, 3, 3, 1, 2, 9, 3, 8, 10, 3, 4, 2, 9, 9, 8, 1, 3, 2, 5, 9, 5, 1, 2, 10, 6, 2, 8, 8, 6, 10, 10, 7, 1, 10, 8, 8, 10, 3, 3, 10, 5, 7, 2, 9, 2, 4, 10, 10, 1, 8, 4, 9, 4, 2, 5, 1, 8, 2, 10, 3, 4, 10, 1, 8, 7, 3, 3, 6, 3, 3, 6, 7, 2, 4, 3, 7, 7, 4, 2, 9, 1, 8, 1, 2, 4, 5, 9, 9, 6, 5, 6, 6, 10, 8, 9, 7, 4, 3, 7, 3, 3, 4, 5, 5, 6, 8, 7, 2, 6, 10, 6, 3, 1, 9, 1, 5, 4, 3, 7, 10, 5, 1, 5, 10, 2, 3, 1, 6, 8, 2, 1, 7, 1, 4, 7, 6, 9, 3, 8, 4, 3, 6, 2, 7, 5, 9, 1, 1, 1, 6, 5, 1, 4, 3, 2, 8, 5, 10, 10, 7, 5, 10, 1, 10, 4, 5, 7, 3, 6, 10, 9, 8, 3, 10, 5, 1, 5, 1, 2, 2, 5, 6, 7, 6, 5, 5, 8, 10, 4, 2, 2, 3, 4, 5, 5, 2, 3, 2, 9, 2, 10, 1, 6, 3, 10, 10, 9, 3, 2, 10, 9, 8, 6, 7, 8, 9, 10, 6, 9, 7, 4, 7, 8, 9, 6, 6, 10, 8, 3, 8, 5, 5, 9, 1, 7, 3, 6, 7, 5, 7, 7, 2, 5, 4, 10, 7, 5, 1, 1, 9, 7, 6, 1, 3, 2, 4, 10, 10, 8, 2, 9, 2, 2, 2, 4, 3, 4, 3, 9, 7, 6, 9, 7, 8, 9, 7, 3, 1, 6, 1, 3, 9, 10, 9, 5, 10, 5, 3, 3, 9, 9, 6, 8, 8, 6, 2, 7, 1, 4, 3, 6, 2, 8, 10, 10, 4, 4, 1, 3, 10, 8, 4, 4, 10, 9, 9, 6, 5, 8, 5, 2, 10, 5, 10, 9, 2, 10, 3, 4, 4, 8, 6, 2, 1, 5, 2, 1, 2, 7, 6, 10, 10, 10, 3, 7, 2, 3, 7, 4, 10, 3, 7, 4, 6, 5, 6, 6, 2, 9, 2, 9, 7, 8, 6, 6, 1, 10, 3, 3, 6, 5, 5, 6, 5, 8, 10, 6, 8, 7, 5, 7, 9, 6, 6, 10, 8, 10, 4, 9, 5, 2, 1, 1, 2, 4, 7, 10, 1, 6, 4, 3, 4, 1, 6, 7, 9, 6, 9, 4, 2, 1, 3, 8, 1, 9, 5, 4, 6, 3, 7, 6, 8, 5, 8, 3, 8, 5, 2, 5, 8, 8, 4, 10, 7, 4, 1, 1, 5, 5, 4, 4, 7, 5, 10, 8, 3, 5, 9, 4, 4, 5, 5, 2, 10, 6, 3, 3, 3, 9, 2, 7, 8, 10, 2, 10, 7, 8, 3, 1, 3, 6, 1, 4, 9, 6, 6, 9, 2, 5, 10, 6, 8, 3, 7, 8, 10, 5, 8, 4, 9, 4, 4, 8, 2, 6, 5, 9, 5, 8, 8, 2, 6, 2, 7, 9, 2, 10, 6, 6, 4, 8, 10, 1, 10, 9, 7, 3, 1, 1, 5, 2, 1, 6, 1, 8, 9, 6, 7, 1, 8, 5, 5, 8, 9, 5, 6, 1, 3, 6, 1, 4, 9, 4, 1, 4, 5, 3, 1, 5, 9, 9, 5, 3, 6, 7, 8, 10, 3, 5, 7, 2, 2, 8, 10, 10, 9, 7, 8, 2, 4, 5, 3, 1, 4, 8, 7, 5, 2, 1, 9, 10, 6, 2, 5, 1, 7, 6, 6, 6, 7, 5, 10, 7, 5, 1, 5, 4, 5, 2, 5, 4, 5, 10, 3, 3, 6, 8, 8, 10, 8, 8, 7, 4, 5, 2, 10, 6, 7, 2, 2, 9, 4, 9, 5, 6, 8, 1, 5, 2, 10, 5, 6, 5, 6, 1, 10, 6, 10, 6, 2, 2, 2, 5, 5, 9, 10, 7, 3, 5, 8, 10, 2, 2, 10, 5, 2, 7, 3, 1, 1, 2, 3, 8, 2, 10, 7, 7, 4, 6, 10, 8, 9, 8, 5, 1, 10, 10, 7, 10, 7, 9, 8, 9, 10, 8, 10, 5, 10, 9, 5, 4, 5, 7, 9, 1, 9, 8, 7, 4, 4, 6, 2, 1, 10, 9, 7, 10, 3, 2, 5, 8, 8, 8, 3, 4, 9, 1, 9, 1, 8, 8, 8, 8, 6, 1, 7, 10, 4, 7, 5, 4, 7, 8, 8, 9, 10, 7, 3, 2, 8, 10, 3, 4, 10, 9, 7, 9, 8, 1, 1, 10, 10, 3, 6, 6, 9, 9, 3, 4, 2, 2, 9, 1, 10, 7, 4, 4, 3, 6, 8, 6, 7, 6, 9, 6, 4, 4, 4, 10, 1, 1, 5, 9, 3, 2, 2, 8, 9, 1, 3, 10, 4, 9, 6, 5, 2, 4, 1, 9, 7, 6, 4, 8, 3, 1, 10, 1, 8, 8, 3, 9, 5, 10, 8, 7, 9, 3, 6, 2, 10, 7, 1, 7, 2, 6, 1, 8, 5, 10, 8, 10, 4, 4, 1, 4, 6, 2, 6, 9, 3, 8, 8, 10, 4, 10, 1, 1, 4, 3, 8, 6, 1, 5, 10, 1, 10, 2, 9, 7, 8, 8, 3, 8, 7, 2, 2, 10, 6, 1, 3, 5, 8, 4, 4, 3, 7, 4, 10, 5, 2, 1, 1, 3, 6, 1, 1, 4, 4, 6, 3, 9, 2, 2, 7, 3, 4, 2, 2, 9, 4, 8, 7, 4, 4, 1, 5, 4, 1, 3, 10, 2, 10, 6, 9, 1, 3, 1, 1, 8, 2, 7, 7, 9, 6, 9, 10, 2, 4, 9, 5, 8, 3, 3, 9, 7, 3, 8, 5, 9, 3, 7, 5, 6, 7, 6, 2, 3, 3, 1, 6, 9, 4, 8, 6, 5, 7, 10, 4, 2, 6, 2, 7, 7, 3, 4, 2, 8, 5, 5, 9, 4, 8, 3, 10, 9, 5, 3, 7, 6, 1, 8, 4, 2, 8, 4, 5, 8, 4, 3, 9, 7, 5, 3, 4, 4, 3, 5, 3, 8, 6, 2, 7, 5, 8, 4, 10, 2, 2, 4, 10, 7, 2, 6, 5, 2, 8, 3, 4, 10, 3, 6, 3, 6, 1, 6, 4, 9, 9, 1, 7, 10, 8, 8, 10, 4, 9, 5, 8, 1, 4, 4, 4, 9, 7, 4, 8, 3, 1, 9, 2, 1, 3, 3, 6, 9, 8, 4, 7, 3, 7, 7, 3, 10, 7, 3, 9, 8, 1, 2, 4, 6, 3, 10, 1, 9, 1, 8, 5, 7, 4, 7, 6, 6, 7, 7, 9, 2, 5, 2, 3, 5, 7, 9, 8, 10, 9, 5, 2, 1, 1, 5, 5, 3, 6, 10, 6, 3, 9, 2, 1, 9, 10, 5, 2, 8, 3, 8, 6, 1, 8, 9, 2, 10, 8, 10, 4, 2, 4, 9, 6, 3, 9, 3, 4, 5, 5, 2, 1, 1, 5, 4, 8, 2, 8, 7, 9, 8, 3, 7, 10, 3, 10, 7, 7, 8, 9, 5, 8, 4, 7, 1, 5, 4, 7, 10, 4, 4, 4, 8, 4, 8, 4, 3, 10, 5, 10, 7, 10, 1, 6, 7, 9, 1, 6, 5, 5, 7, 10, 1, 7, 2, 6, 9, 3, 10, 3, 8, 8, 1, 1, 4, 4, 7, 6, 2, 6, 4, 1, 2, 8, 5, 9, 3, 3, 6, 4, 9, 9, 10, 2, 8, 9, 10, 6, 3, 3, 10, 7, 4, 3, 8, 3, 8, 7, 5, 5, 8, 2, 1, 1, 3, 9, 5, 3, 3, 1, 7, 2, 3, 6, 1, 1, 5, 10, 7, 8, 4, 3, 1, 1, 9, 5, 10, 3, 9, 5, 5, 4, 8, 6, 9, 4, 6, 1, 3, 4, 6, 9, 9, 3, 10, 8, 5, 2, 2, 2, 4, 2, 5, 9, 2, 7, 4, 10, 10, 6, 5, 9, 1, 3, 9, 7, 8, 2, 9, 5, 1, 10, 3, 1, 4, 7, 7, 2, 3, 9, 7, 5, 9, 6, 5, 6, 8, 7, 1, 5, 5, 4, 8, 4, 2, 7, 5, 5, 9, 9, 8, 9, 6, 6, 4, 1, 1, 7, 10, 3, 4, 4, 8, 7, 7, 9, 8, 7, 9, 1, 2, 1, 4, 7, 3, 2, 7, 8, 5, 1, 8, 10, 7, 9, 7, 1, 3, 5, 7, 8, 2, 3, 4, 1, 7, 4, 9, 7, 1, 2, 1, 3, 6, 10, 8, 4, 1, 5, 4, 1, 1, 8, 8, 2, 3, 5, 4, 8, 4, 5, 4, 8, 6, 7, 10, 1, 8, 4, 1, 3, 6, 5, 1, 8, 9, 2, 10, 8, 1, 5, 4, 7, 4, 5, 9, 5, 4, 7, 7, 6, 2, 3, 5, 10, 10, 7, 6, 1, 2, 4, 3, 5, 4, 10, 6, 8, 7, 2, 7, 7, 10, 8, 2, 10, 7, 6, 4, 1, 4, 5, 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2, 10, 6, 8, 5, 7, 9, 10, 6, 1, 5, 7, 2, 6, 6, 2, 3, 3, 9, 10, 2, 10, 2, 10, 3, 3, 8, 8, 1, 5, 9, 2, 8, 7, 6, 1, 2, 2, 1, 9, 5, 6, 5, 4, 9, 9, 4, 6, 6, 3, 4, 3, 2, 10, 10, 3, 7, 2, 6, 3, 2, 1, 8, 10, 10, 6, 2, 10, 8, 4, 5, 2, 3, 2, 5, 7, 9, 2, 7, 4, 3, 6, 5, 4, 2, 8, 8, 4, 7, 5, 1, 6, 4, 8, 4, 10, 2, 7, 8, 5, 3, 10, 10, 8, 5, 5, 5, 10, 10, 9, 7, 2, 6, 10, 8, 1, 8, 9, 2, 2, 5, 3, 7, 7, 10, 10, 4, 1, 5, 3, 4, 1, 3, 1, 7, 6, 4, 10, 4, 5, 7, 10, 5, 7, 2, 2, 1, 3, 6, 9, 9, 9, 3, 9, 8, 9, 4, 4, 6, 7, 8, 6, 1, 2, 2, 10, 4, 7, 1, 8, 7, 5, 5, 5, 9, 1, 9, 5, 7, 2, 3, 7, 5, 3, 5, 2, 9, 9, 1, 7, 9, 10, 4, 7, 1, 2, 3, 7, 3, 10, 2, 5, 8, 8, 2, 1, 2, 9, 6, 2, 2, 2, 1, 9, 10, 5, 7, 8, 9, 9, 9, 6, 10, 6, 3, 9, 7, 5, 6, 6, 10, 7, 5, 4, 7, 6, 6, 9, 3, 8, 8, 3, 1, 7, 5, 10, 6, 8, 7, 4, 4, 4, 3, 7, 2, 1, 2, 6, 7, 8, 5, 7, 1, 2, 4, 2, 5, 9, 2, 8, 10, 4, 7, 4, 4, 1, 10, 3, 5, 1, 3, 8, 7, 4, 1, 2, 4, 10, 6, 4, 8, 9, 8, 3, 4, 4, 6, 4, 6, 5, 8, 1, 10, 7, 10, 8, 7, 9, 4, 9, 3, 1, 10, 9, 6, 9, 5, 7, 1, 10, 1, 1, 2, 5, 10, 8, 2, 9, 6, 5, 9, 5, 9, 4, 2, 3, 2, 8, 10, 2, 5, 9, 2, 2, 10, 3, 4, 9, 8, 10, 3, 3, 9, 4, 6, 6, 9, 2, 7, 10, 3, 2, 4, 1, 5, 3, 5, 3, 6, 5, 4, 9, 2, 7, 8, 4, 6, 2, 10, 10, 4, 4, 9, 2, 3, 5, 6, 2, 6, 6, 10, 1, 5, 8, 6, 5, 9, 7, 4, 1, 3, 2, 6, 10, 8, 10, 10, 4, 6, 7, 7, 8, 10, 1, 8, 4, 5, 3, 7, 8, 9, 10, 1, 7, 10, 7, 3, 7, 5, 10, 4, 10, 10, 1, 9, 5, 1, 9, 3, 3, 1, 8, 3, 6, 7, 7, 5, 1, 3, 3, 2, 5, 4, 8, 1, 6, 1, 10, 8, 6, 8, 4, 1, 3, 1, 5, 2, 6, 9, 5, 2, 1, 7, 10, 1, 8, 8, 9, 9, 7, 10, 4, 6, 8, 4, 3, 1, 9, 6, 2, 2, 4, 3, 5, 3, 10, 9, 1, 7, 4, 5, 10, 7, 6, 9, 5, 5, 9, 2, 6, 6, 8, 1, 6, 2, 5, 3, 4, 5, 9, 7, 6, 10, 5, 9, 3, 3, 3, 8, 2, 3, 10, 7, 5, 4, 9, 7, 3, 8, 3, 6, 5, 9, 10, 10, 9, 4, 4, 10, 9, 6, 7, 5, 7, 10, 5, 8, 9, 3, 2, 6, 9, 5, 10, 2, 1, 8, 6, 8, 6, 8, 9, 5, 10, 7, 5, 2, 4, 5, 6, 7, 1, 4, 6, 5, 2, 3, 2, 9, 3, 9, 4, 9, 9, 9, 5, 4, 3, 6, 7, 9, 2, 7, 3, 2, 5, 2, 8, 10, 2, 2, 1, 1, 3, 8, 7, 3, 9, 8, 8, 1, 7, 5, 6, 6, 9, 1, 2, 9, 8, 1, 10, 5, 3, 7, 4, 6, 5, 7, 10, 7, 9, 8, 5, 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4, 6, 6, 10, 1, 9, 6, 10, 5, 3, 5, 3, 2, 5, 6, 6, 5, 1, 1, 6, 3, 9, 8, 8, 1, 3, 10, 8, 2, 9, 4, 4, 10, 8, 6, 5, 8, 6, 4, 9, 2, 2, 10, 1, 6, 1, 9, 9, 7, 2, 5, 2, 10, 1, 4, 10, 8, 10, 8, 2, 2, 6, 6, 9, 9, 6, 6, 4, 10, 6, 10, 4, 3, 8, 9, 3, 1, 10, 4, 9, 10, 10, 9, 9, 8, 4, 8, 3, 8, 10, 3, 8, 8, 8, 6, 1, 4, 3, 8, 7, 10, 9, 3, 3, 6, 1, 4, 10, 6, 1, 2, 5, 6, 6, 1, 7, 5, 4, 10, 9, 6, 8, 8, 4, 2, 5, 9, 8, 5, 1, 9, 9, 7, 3, 10, 10, 9, 4, 10, 8, 6, 4, 1, 4, 4, 5, 9, 7, 1, 6, 1, 9, 7, 9, 2, 2, 1, 10, 5, 1, 1, 7, 2, 1, 5, 6, 6, 10, 10, 9, 4, 2, 1, 3, 7, 7, 7, 9, 8, 6, 10, 6, 1, 3, 1, 10, 4, 1, 7, 3, 6, 4, 10, 10, 5, 2, 2, 9, 7, 8, 7, 8, 5, 8, 7, 3, 8, 10, 3, 7, 4, 8, 7, 9, 3, 7, 1, 10, 9, 1, 1, 5, 7, 10, 9, 6, 5, 7, 8, 6, 6, 10, 3, 1, 1, 9, 4, 2, 1, 4, 3, 10, 5, 6, 9, 2, 6, 10, 4, 4, 1, 2, 8, 10, 8, 9, 8, 10, 4, 10, 4, 8, 4, 7, 6, 9, 10, 5, 4, 10, 2, 8, 9, 9, 2, 4, 6, 2, 10, 2, 3, 9, 4, 2, 3, 6, 10, 9, 4, 3, 10, 3, 9, 3, 7, 8, 5, 1, 9, 10, 10, 3, 5, 7, 10, 4, 4, 4, 10, 3, 6, 5, 9, 3, 6, 9, 7, 7, 6, 6, 2, 2, 10, 8, 1, 2, 5, 4, 5, 3, 1, 3, 1, 3, 1, 2, 1, 2, 8, 10, 4, 6, 2, 4, 7, 3, 10, 10, 10, 9, 4, 3, 9, 9, 2, 4, 10, 4, 2, 10, 8, 3, 4, 8, 9, 6, 8, 10, 4, 2, 2, 2, 10, 3, 8, 5, 10, 4, 2, 3, 7, 3, 10, 4, 9, 10, 5, 6, 6, 6, 10, 9, 3, 3, 6, 5, 5, 7, 8, 5, 5, 1, 5, 8, 5, 1, 9, 4, 6, 8, 2, 1, 8, 9, 3, 9, 4, 7, 1, 1, 7, 6, 6, 8, 5, 8, 9, 10, 6, 8, 1, 2, 7, 7, 3, 7, 1, 7, 3, 9, 3, 10, 3, 3, 4, 8, 2, 10, 5, 9, 4, 8, 6, 8, 9, 6, 9, 3, 10, 3, 9, 2, 7, 8, 4, 7, 5, 6, 7, 6, 3, 5, 8, 4, 7, 7, 3, 4, 10, 3, 5, 5, 1, 7, 1, 4, 4, 4, 8, 1, 6, 5, 7, 7, 6, 1, 10, 3, 4, 6, 3, 8, 6, 1, 4, 8, 3, 2, 4, 1, 2, 1, 6, 4, 3, 1, 5, 2, 9, 9, 8, 1, 5, 9, 9, 10, 2, 5, 2, 2, 5, 7, 8, 4, 9, 2, 9, 10, 2, 5, 6, 10, 5, 8, 10, 8, 10, 4, 8, 1, 2, 1, 5, 1, 3, 10, 6, 10, 10, 7, 1, 10, 8, 10, 10, 9, 7, 6, 1, 7, 5, 10, 7, 7, 3, 7, 3, 3, 10, 8, 7, 6, 9, 10, 2, 2, 8, 5, 8, 10, 5, 7, 7, 7, 4, 8, 9, 10, 4, 2, 10, 3, 2, 1, 7, 3, 9, 1, 1, 10, 8, 3, 8, 9, 9, 7, 1, 8, 1, 8, 1, 4, 6, 4, 5, 6, 4, 3, 8, 3, 3, 9, 6, 10, 2, 4, 4, 2, 6, 6, 7, 1, 7, 2, 6, 6, 1, 7, 9, 1, 3, 7, 9, 4, 8, 9, 8, 1, 5, 1, 4, 3, 9, 10, 3, 4, 10, 6, 2, 2, 8, 4, 2, 10, 6, 3, 7, 7, 6, 8, 10, 7, 6, 7, 2, 2, 7, 4, 10, 1, 9, 7, 7, 9, 4, 8, 3, 3, 1, 2, 9, 7, 7, 4, 7, 2, 4, 10, 5, 6, 4, 3, 6, 2, 2, 4, 8, 5, 5, 4, 2, 8, 5, 10, 10, 8, 8, 3, 1, 9, 4, 5, 5, 9, 10, 7, 3, 3, 3, 3, 4, 10, 2, 8, 10, 5, 8, 8, 10, 7, 4, 6, 9, 10, 4, 7, 7, 4, 8, 3, 6, 1, 6, 8, 1, 10, 1, 8, 2, 4, 10, 1, 1, 8, 10, 1, 8, 9, 3, 7, 7, 10, 7, 1, 6, 5, 6, 5, 7, 8, 8, 7, 4, 2, 7, 1, 1, 7, 10, 1, 5, 10, 10, 7, 8, 3, 3, 8, 2, 7, 4, 1, 9, 7, 1, 7, 5, 2, 9, 9, 6, 4, 10, 8, 6, 5, 5, 2, 9, 6, 7, 2, 10, 8, 2, 6, 10, 5, 6, 3, 6, 1, 2, 5, 5, 8, 9, 2, 5, 9, 2, 2, 8, 9, 1, 5, 1, 8, 4, 8, 9, 8, 5, 9, 9, 7, 9, 6, 10, 3, 7, 10, 1, 4, 3, 5, 7, 7, 8, 4, 3, 8, 9, 7, 10, 9, 9, 4, 5, 2, 4, 4, 10, 6, 4, 5, 8, 4, 5, 2, 8, 4, 7, 1, 3, 7, 4, 5, 8, 10, 5, 4, 6, 6, 6, 8, 1, 2, 2, 2, 7, 6, 5, 3, 1, 9, 10, 2, 4, 10, 1, 10, 1, 1, 3, 9, 4, 3, 3, 3, 3, 1, 7, 6, 4, 10, 1, 2, 6, 6, 2, 5, 9, 2, 7, 6, 1, 4, 4, 8, 3, 1, 3, 5, 4, 10, 10, 9, 7, 1, 1, 9, 4, 5, 4, 10, 4, 10, 3, 5, 6, 4, 1, 9, 1, 2, 6, 1, 7, 3, 2, 10, 2, 7, 3, 3, 6, 10, 6, 7, 3, 3, 4, 4, 10, 5, 4, 1, 4, 7, 3, 2, 8, 1, 4, 7, 9, 6, 5, 8, 5, 1, 4, 6, 5, 5, 9, 3, 9, 4, 2, 6, 8, 4, 3, 9, 5, 6, 10, 5, 5, 6, 4, 6, 1, 4, 4, 1, 5, 2, 4, 7, 5, 1, 6, 5, 1, 1, 9, 2, 8, 10, 3, 7, 4, 4, 8, 6, 8, 1, 2, 3, 4, 5, 7, 3, 1, 8, 8, 3, 9, 1, 3, 5, 1, 2, 2, 6, 1, 5, 7, 6, 7, 4, 2, 1, 4, 4, 3, 10, 4, 3, 7, 5, 7, 3, 7, 9, 6, 2, 1, 7, 6, 3, 7, 6, 5, 9, 10, 5, 5, 2, 9, 6, 8, 8, 5, 1, 8, 2, 7, 7, 6, 8, 4, 2, 2, 10, 1, 4, 9, 1, 3, 2, 9, 4, 10, 4, 1, 2, 10, 9, 1, 9, 7, 4, 3, 10, 1, 1, 3, 2, 6, 3, 6, 7, 4, 3, 6, 10, 5, 8, 7, 6, 7, 6, 5, 8, 4, 8, 6, 7, 3, 3, 6, 2, 2, 10, 4, 5, 10, 7, 10, 10, 5, 3, 10, 6, 2, 8, 2, 2, 2, 2, 7, 8, 7, 1, 3, 3, 7, 8, 4, 6, 6, 3, 6, 10, 4, 5, 5, 1, 10, 1, 3, 5, 4, 6, 10, 7, 9, 8, 4, 3, 6, 4, 6, 1, 5, 5, 1, 9, 1, 4, 7, 1, 5, 5, 7, 4, 2, 7, 2, 3, 5, 3, 3, 6, 10, 3, 3, 9, 1, 1, 3, 1, 6, 8, 2, 7, 7, 5, 1, 9, 9, 4, 2, 5, 7, 5, 9, 3, 10, 1, 5, 7, 6, 7, 4, 5, 4, 1, 8, 2, 8, 6, 9, 2, 10, 6, 8, 5, 2, 8, 5, 6, 10, 2, 8, 10, 6, 8, 5, 10, 5, 4, 8, 6, 8, 3, 10, 7, 9, 10, 7, 7, 4, 1, 3, 1, 9, 8, 8, 10, 8, 1, 9, 1, 5, 10, 7, 4, 1, 4, 2, 2, 10, 6, 7, 1, 5, 10, 3, 6, 5, 10, 5, 5, 4, 1, 6, 4, 7, 1, 8, 10, 3, 1, 5, 6, 1, 6, 2, 5, 3, 8, 8, 8, 2, 10, 3, 3, 9, 10, 6, 1, 6, 6, 4, 9, 7, 10, 4, 1, 5, 9, 4, 4, 4, 4, 6, 1, 7, 7, 3, 3, 5, 4, 4, 9, 7, 9, 2, 4, 7, 9, 2, 6, 5, 6, 10, 10, 10, 7, 10, 3, 10, 9, 6, 5, 8, 3, 7, 9, 9, 4, 2, 5, 10, 7, 5, 1, 6, 2, 10, 1, 5, 8, 6, 4, 5, 3, 3, 5, 5, 8, 10, 6, 2, 7, 9, 7, 10, 4, 2, 9, 3, 9, 3, 8, 5, 7, 2, 3, 8, 4, 1, 5, 7, 2, 2, 8, 6, 8, 7, 6, 7, 8, 8, 6, 7, 7, 6, 9, 5, 4, 9, 7, 7, 6, 1, 5, 8, 2, 9, 2, 5, 8, 1, 6, 7, 3, 10, 3, 5, 7, 3, 2, 7, 2, 9, 6, 3, 5, 3, 4, 9, 4, 9, 3, 4, 1, 7, 7, 1, 6, 9, 5, 1, 1, 2, 5, 10, 1, 5, 2, 6, 1, 2, 7, 7, 10, 1, 1, 5, 9, 3, 2, 1, 8, 10, 3, 5, 9, 6, 6, 7, 1, 7, 7, 9, 8, 5, 1, 4, 1, 4, 2, 7, 1, 4, 4, 2, 10, 8, 4, 6, 1, 10, 9, 5, 10, 9, 10, 9, 4, 1, 10, 8, 7, 10, 8, 6, 7, 7, 1, 1, 2, 8, 4, 1, 10, 9, 7, 7, 3, 3, 6, 8, 1, 7, 3, 8, 6, 5, 4, 1, 2, 7, 10, 4, 2, 7, 7. Total: 54562.[/sblock]

Good on average, yes. But take a look at the last few lines and see how often 7, 7 comes up. It comes up far too frequently to make a truly random generator. Good RNGs just don't do that. Instances of 3 numbers in a row seem very common as well, when really, they shouldn't be. The last ten lines has 8,8,8, 10,10,10 and even a very improbably 4,4,4,4. A quick scan through the numbers revealed four in a row of the same number on at least 3 occasions, which is too likely.

Pinotage

 

[Delta]

Well, one problem I have with recording and comparing, is that any good program should even them out. But the real question I had is wether or not the computer is likely to give you a series of the same number, as sometimes happens with the die, or wether it would give a more statistically probable answer by giving the same amount of the number, but spaced out between different rolls, making the two look like this:

Real: 1 3 4 4 4 5 2 6 3

Computer: 1 4 2 5 4 3 6 4 3

Showing that the computer would have seperated the consecutive 4's becuase it is more probable, and matches better with statistics.

There are some heavy-duty mathematical tests established to qualify good psedurandom number generators. For example, you'll find what's called the "BSI Evaluation List" at the following link. What you're describing above would only qualify as a "K1" quality generator, the poorest, weakest category in the list.

http://en.wikipedia.org/wiki/Pseudorandom_number_generator#BSI_evaluation_criteria

To get to stage "K2" quality, the generator has to pass all the following statistical tests:

(1) Monobit test (nearly equal number of one and zero bits in the sequence).
(2) Poker test (a special instance of the chi-squared test).
(3) Runs test (counts the frequency of runs of various lengths).
(4) Longruns test (checks whether there exists any run of length 34 or greater in 20 000 bits of the sequence).
(5) Autocorrelation test.

Again, that's just for K2 quality. There are K3 and K4 categories beyond that, but they're theoretically a lot more complicated than that to qualify.

 

[Asmor]

Good on average, yes. But take a look at the last few lines and see how often 7, 7 comes up. It comes up far too frequently to make a truly random generator. Good RNGs just don't do that. Instances of 3 numbers in a row seem very common as well, when really, they shouldn't be. The last ten lines has 8,8,8, 10,10,10 and even a very improbably 4,4,4,4. A quick scan through the numbers revealed four in a row of the same number on at least 3 occasions, which is too likely.

Pinotage

As I said, I'm just using Javascript's built in random number generator, which has a reputation for being pretty bad.

The chances of rolling the same number 3 times in a row is 1/100. The chance of rolling a number 4 times in a row is 1/1000. So theoretically, assuming "perfect" randomness, there should be 100 triplets and 10 quadruplets. Actually, for that matter, there should be 1 quintuplet.

 

[Asmor]

Incidentally, that does bring up a good point, that I think I implied but deserves to be stated:

Not all pseudo-random number generators are the same. There are good ones, and there are bad ones. Unfortunately, I don't know of many dice rollers which proudly proclaim a proven-effective PRNG as one of the features... Which is kind of silly, since that's probably the most important thing for a dice roller after usability.