A Bone of Your Own - An FOFC probability game

[QuikSand]

Okay, after thinking for a few minutes on a new puzzle, I have instead come up with a new game. I'm a fan of probability exercises... so that's where this will be grounded.

Your job will be to design a die. (Or, in old-time parlance, a "bone.") In familiar D&D fashion, your die will have 20 sides... I'm sure there's a name for this perfect three-dimensional object, but it escapes me. Is a dodecahedron 12 or 20... I can never remember. That aside...

On the usual 20-sided die, you see the numbers 1 through 20. Way too simple. For this game, on your die, you may take the 210 "points" from the basic die, and you are free to distribute them however you like, as long as each side has an integer value of at least 1.



Here's how the game will be played:

Round One will see each die rolled several times, for a cumulative score:

-Roll three times, take all three
-Roll three times, drop the low
-Roll three times, drop the high
-Roll three times, drop the high and the low

That means that a total of ten rolls will count, summed together to give the round one score. The top 3 finishers will advance to Round Two.


Round Two will see the thre dice rolled several times, to eliminate the lowest-scoring of them:

-Roll three times, drop the high and low
-Roll five times, drop the high and low, take the average of the other three
-Roll seven times, drop the high and low, take the average of the other five
-Roll five times, take the lowest roll
-Roll five times, take the highest roll

The sum of these five scores determines the score for round two. The lowest total score for round two only is eliminated. In the event of a tie, the die with the lower score in Round One is eliminated.


Round Three will see the final two dice compete head-to-head to determine the champion:

-Each die gets 10 rolls, with the higher of the two gets one point per win, plus 0.1 point for each full multiple of the other's score it reaches. For example -- A rolls a 3, B rolls a 17. B scores 1 point for the win, and since 17/3=5.666, B also receives an additional 0.5 points -- earning a total of 1.5 for the roll.

The higher total score after 10 such rolls is declared the winner. In the event of a tie, extra rolls will continue until the tie is broken.



My current thinking is that we will start the contest with six entries. That means of the six, one half (three) will advance to level two. If that affects your thinking, please use that as your default.


Give some thought to the design of your die, given the rules above. I will spend a little time preparing an Excel sheet to automate the contest, and hope to have something fairly easy to run soon.

I have not yet decided whether I am going to submit an entry. I may.


When you have the design of your die completed, send it to me via PM. In the body of the PM, just include the numbers on your die, in ascending order, please -- one number per line of text. It should look like this:

1
2
3
4
5
.
.
.

I think with that setup, I ought to be able to paste the data into Excel, and set this up fairly easily.


In the future, FYI, I am considering a more rigorously automated version of this -- perhaps one that might involve rerolling the dice (or combination of rolls) maybe 100 or 1,000 times -- to better tease out the short-term randomness involved here. But for now, it will be a game based mostly on luck, with a bit of design involved. Hopefully.

Enjoy.

[sovereignstar]

hehe.. you said bone.

[QuikSand]

I like to offer a little something for everyone.

[Greyroofoo]

i'm interested.....

[paperstreetsoap]

can we use a number twice?

[cthomer5000]

I like to offer a little something for everyone.
You really crack me up sometimes.

[TredWel]

I'm game. PM sent.

[Buzzbee]

I'm game. PM sent.
Ditto.

[Desnudo]

Sent

[QuikSand]

can we use a number twice?

Absolutely. If not, you don't have many options, do you?

[albionmoonlight]

sent

[Eaglesfan27]

Sent.

[Brillig]

Icosahedron

[KWhit]

Sent.

[QuikSand]

Icosahedron

I guess the dodecahedron is a 12-sider, then. I'm pretty sure I have seen a different name for the 20-sided shape, though... this one is brand new to me.

[QuikSand]

BTW, I have 11 entries so far, and I have been thinking of one of my own. So, we will plan to run two "heats" of the contest with the submissions I have received.

Might have some time today to start setting this up in Excel.

[Alf]

2/3 as for all QS puzzles ;)

[QuikSand]

Okay, I have set up a spreadsheet (not fully automated yet, I'm working on that)... and am ready to run the first heat of six entries. I randomly selected two groups of six for our first trial runs...

[QuikSand]

Here are the first six entries:
(Sorry, I don't know how to better format these)


QuikSand's Kwhit EaglesFan27 Vince TredWel Celeval
Mean 313
1 3 1 1 1 1 1 1 1 10 1 10
2 3 2 1 2 1 2 1 2 10 2 10
3 3 3 1 3 3 3 1 3 10 3 10
4 3 4 10 4 5 4 1 4 10 4 10
5 3 5 10 5 7 5 1 5 10 5 10
6 13 6 10 6 7 6 7 6 10 6 10
7 13 7 10 7 9 7 8 7 10 7 10
8 13 8 10 8 9 8 10 8 10 8 10
9 13 9 10 9 11 9 10 9 10 9 10
10 13 10 10 10 11 10 10 10 10 10 10
11 13 11 11 11 13 11 16 11 11 11 10
12 13 12 11 12 13 12 16 12 11 12 10
13 13 13 11 13 13 13 16 13 11 13 10
14 13 14 11 14 13 14 16 14 11 14 11
15 13 15 11 15 15 15 16 15 11 15 11
16 13 16 11 16 15 16 16 16 11 16 11
17 13 17 17 17 15 17 16 17 11 17 11
18 13 18 18 18 15 18 16 18 11 18 12
19 13 19 18 19 17 19 16 19 11 19 12
20 13 20 18 20 17 20 16 20 11 20 12
210 210 210 210 210 210

[QuikSand]

ROUND ONE


QuikSand's Kwhit EaglesFan27 Vince TredWel Celeval
Mean 313
1 3 1 1 1 1 1 1 1 10 1 10
2 3 2 1 2 1 2 1 2 10 2 10
3 3 3 1 3 3 3 1 3 10 3 10
4 3 4 10 4 5 4 1 4 10 4 10
5 3 5 10 5 7 5 1 5 10 5 10
6 13 6 10 6 7 6 7 6 10 6 10
7 13 7 10 7 9 7 8 7 10 7 10
8 13 8 10 8 9 8 10 8 10 8 10
9 13 9 10 9 11 9 10 9 10 9 10
10 13 10 10 10 11 10 10 10 10 10 10
11 13 11 11 11 13 11 16 11 11 11 10
12 13 12 11 12 13 12 16 12 11 12 10
13 13 13 11 13 13 13 16 13 11 13 10
14 13 14 11 14 13 14 16 14 11 14 11
15 13 15 11 15 15 15 16 15 11 15 11
16 13 16 11 16 15 16 16 16 11 16 11
17 13 17 17 17 15 17 16 17 11 17 11
18 13 18 18 18 15 18 16 18 11 18 12
19 13 19 18 19 17 19 16 19 11 19 12
20 13 20 18 20 17 20 16 20 11 20 12
210 210 210 210 210 210

Rd 1 4 3 18 18 13 13 11 16 12 11 13 10
11 13 7 10 2 1 13 16 16 11 15 11
6 13 11 11 20 17 17 16 9 10 2 10
sub 29 39 31 48 32 31

15 13 20 18 16 15 20 16 9 10 9 10
1 3 13 11 6 7 11 16 7 10 11 10
15 13 16 11 3 3 1 1 5 10 16 11
sub 26 29 22 32 20 21

20 13 14 11 16 15 6 7 9 10 6 10
3 3 15 11 3 3 9 10 2 10 14 11
17 13 5 10 3 3 5 1 20 11 8 10
sub 16 21 6 8 20 20

1 3 11 11 6 7 12 16 9 10 20 12
18 13 14 11 15 15 5 1 6 10 16 11
13 13 15 11 2 1 20 16 6 10 14 11
sub 13 11 7 16 10 11

Rd 1 84 100 66 104 82 83

[QuikSand]

QuikSand's Kwhit Vince
Mean 313
1 3 1 1 1 1
2 3 2 1 2 1
3 3 3 1 3 1
4 3 4 10 4 1
5 3 5 10 5 1
6 13 6 10 6 7
7 13 7 10 7 8
8 13 8 10 8 10
9 13 9 10 9 10
10 13 10 10 10 10
11 13 11 11 11 16
12 13 12 11 12 16
13 13 13 11 13 16
14 13 14 11 14 16
15 13 15 11 15 16
16 13 16 11 16 16
17 13 17 17 17 16
18 13 18 18 18 16
19 13 19 18 19 16
20 13 20 18 20 16

Rd 2 3 3 11 11 19 16
18 13 3 1 12 16
5 3 9 10 3 1
sub 3 10 16

4 3 10 10 4 1
11 13 4 10 11 16
5 3 10 10 11 16
16 13 13 11 12 16
14 13 7 10 14 16
sub 9.666666667 10 16

16 13 10 10 15 16
20 13 9 10 18 16
15 13 3 1 11 16
10 13 17 17 11 16
14 13 19 18 13 16
20 13 1 1 14 16
9 13 12 11 20 16
sub 13 9.8 16

19 13 4 10 8 10
13 13 14 11 6 7
9 13 9 10 9 10
13 13 1 1 4 1
4 3 20 18 9 10
sub 13 18 10

7 13 18 18 17 16
12 13 12 11 3 1
8 13 13 11 4 1
13 13 3 1 10 10
15 13 4 10 3 1
sub 13 1 1

Rd 2 51.66666667 48.8 59

[QuikSand]

QuikSand's Vince
Mean 313
1 3 1 1
2 3 2 1
3 3 3 1
4 3 4 1
5 3 5 1
6 13 6 7
7 13 7 8
8 13 8 10
9 13 9 10
10 13 10 10
11 13 11 16
12 13 12 16
13 13 13 16
14 13 14 16
15 13 15 16
16 13 16 16
17 13 17 16
18 13 18 16
19 13 19 16
20 13 20 16

Rd 3 8 13 1 1
2.3 0

19 13 2 1
2.3 0

8 13 9 10
1.1 0

9 13 12 16
0 1.1

3 3 9 10
0 1.3

4 3 2 1
1.3 0

6 13 15 16
0 1.1

6 13 17 16
0 1.1

15 13 17 16
0 1.1

9 13 20 16
0 1.1

Rd 3 7 6.8

[QuikSand]

This was not rigged, I assure you. The only thing nonrandom about it at all was that I found two identical entries, and opted to keep the second for the second heat, rather than the first.

[albionmoonlight]

Interesting to see what others are doing with their dice.

[QuikSand]

A few thoughts on my own creation:

- I think the exact score of 84 is a fairly likely result for the first round... some weird luck either way will send me up or down by 10 points sometimes, but I'm going to sacore 84 a lot, and I think that ought to be good enough to make the top 3 out of 6 fairly often. If this were a contest with 10 or 20 entries, and only the top 3 qualified, then I'd be screwed here.

- My die just plain got lucky in the second round, rolling an awful lot of 13s. I haven;t looked to see how everyone else did in terms of pure luck, but I know I was on the good side (and still only barely qualified).

- In the last stage, any die with very low values is susceptible to getting the big score put up against it... and I got very lucky against Vince to start right away with two big scores. In the end, he won 6 out of 10 rolls, but the bonus points I got when he rolled a 1 to my 13 twice was enough to give me the slight final edge. Rolling a 1, in particular, in that last round is a real danger -- I wanted to keep 1s off my die completely -- it's tough to score a huge jackpot against a 3.

[QuikSand]

Heat two coming in a minute or two...

[Alf]

FYI : The Platonic solids http://www.shef.ac.uk/nps/courses/groups/plato.html (great page where you can rotate the objetcs)

And 20 faces is icosahedron

Each solid is related to an element (earth, wind, fire, water and the Universe)

(in french here : http://www.col-camus-soufflenheim.ac-strasbourg.fr/Page.php?IDP=330&IDD=0)

[QuikSand]

HEAT TWO

ROUND ONE


Airhog albionmoonlight Desnudo Buzzbee Daimyo paperstreetsoap

1 10 1 1 1 1 1 2 1 10 1 1
2 10 2 1 2 1 2 2 2 10 2 1
3 10 3 1 3 1 3 12 3 10 3 1
4 10 4 1 4 1 4 12 4 10 4 2
5 10 5 1 5 1 5 12 5 10 5 2
6 10 6 1 6 1 6 12 6 10 6 2
7 10 7 1 7 1 7 12 7 10 7 2
8 10 8 1 8 1 8 12 8 10 8 4
9 10 9 16 9 1 9 12 9 10 9 4
10 10 10 16 10 1 10 12 10 10 10 4
11 11 11 16 11 20 11 12 11 11 11 5
12 11 12 16 12 20 12 12 12 11 12 5
13 11 13 16 13 20 13 12 13 11 13 5
14 11 14 16 14 20 14 12 14 11 14 10
15 11 15 16 15 20 15 12 15 11 15 10
16 11 16 16 16 20 16 12 16 11 16 10
17 11 17 16 17 20 17 12 17 11 17 35
18 11 18 16 18 20 18 12 18 11 18 35
19 11 19 21 19 20 19 12 19 11 19 35
20 11 20 21 20 20 20 12 20 11 20 37
210 210 210 220 210 210

Rd 1 18 11 18 16 16 20 1 2 14 11 1 1
1 10 20 21 19 20 17 12 17 11 10 4
18 11 13 16 4 1 1 2 9 10 8 4
sub 32 53 41 16 32 9

20 11 10 16 10 1 15 12 9 10 5 2
19 11 17 16 19 20 4 12 2 10 14 10
3 10 11 16 6 1 8 12 7 10 12 5
sub 22 32 21 24 20 15

11 11 1 1 18 20 18 12 6 10 8 4
14 11 20 21 17 20 19 12 6 10 7 2
4 10 1 1 6 1 7 12 3 10 7 2
sub 21 2 21 24 20 4

2 10 10 16 12 20 6 12 18 11 2 1
10 10 16 16 12 20 19 12 7 10 4 2
9 10 17 16 16 20 16 12 6 10 6 2
sub 10 16 20 12 10 2

Rd 1 85 103 103 76 82 30

[QuikSand]

HEAT TWO

ROUND TWO


Airhog albionmoonlight Desnudo

1 10 1 1 1 1
2 10 2 1 2 1
3 10 3 1 3 1
4 10 4 1 4 1
5 10 5 1 5 1
6 10 6 1 6 1
7 10 7 1 7 1
8 10 8 1 8 1
9 10 9 16 9 1
10 10 10 16 10 1
11 11 11 16 11 20
12 11 12 16 12 20
13 11 13 16 13 20
14 11 14 16 14 20
15 11 15 16 15 20
16 11 16 16 16 20
17 11 17 16 17 20
18 11 18 16 18 20
19 11 19 21 19 20
20 11 20 21 20 20

Rd 2 2 10 5 1 8 1
5 10 18 16 20 20
1 10 9 16 17 20
sub 10 16 20

8 10 10 16 1 1
10 10 16 16 2 1
7 10 17 16 7 1
12 11 12 16 16 20
1 10 19 21 19 20
sub 10 16 7.333333333

3 10 2 1 20 20
17 11 7 1 8 1
8 10 16 16 15 20
6 10 5 1 16 20
9 10 15 16 11 20
5 10 10 16 10 1
15 11 5 1 2 1
sub 10.2 7 12.4

13 11 9 16 13 20
7 10 2 1 14 20
17 11 16 16 2 1
1 10 18 16 11 20
12 11 6 1 1 1
sub 11 16 20

17 11 11 16 20 20
1 10 20 21 9 1
17 11 19 21 10 1
15 11 5 1 5 1
11 11 16 16 2 1
sub 10 1 1

Rd 2 51.2 56 60.73333333

[QuikSand]

HEAT TWO

ROUND THREE


albionmoonlight Desnudo

1 1 1 1
2 1 2 1
3 1 3 1
4 1 4 1
5 1 5 1
6 1 6 1
7 1 7 1
8 1 8 1
9 16 9 1
10 16 10 1
11 16 11 20
12 16 12 20
13 16 13 20
14 16 14 20
15 16 15 20
16 16 16 20
17 16 17 20
18 16 18 20
19 21 19 20
20 21 20 20

Rd 3 10 16 11 20
0 1.1

17 16 15 20
0 1.1

17 16 10 1
2.6 0

20 21 14 20
1.1 0

18 16 1 1
2.6 0

10 16 16 20
0 1.1

2 1 18 20
0 3

14 16 10 1
2.6 0

8 1 12 20
0 3

15 16 13 20
0 1.1

Rd 3 8.9 10.4

[albionmoonlight]

[contemplating refinements]

[QuikSand]

Well, 3 out of the original 12 entries were identical -- half 10s, half 11s. Luck of the draw split those two between 3rd and 4th in this event, and the identical entry managed to miss out on 3rd place by only 2 points in the first round.

Seems like they had a little bad luck here, but that strategy might well be fairly good. Decent chance to eke out of the first round, fairly unlikely to be in last place out of three in round two, and then it's basically a coin flip in round three -- perhaps the 10/11 design would actually be a long-term winner, given this specific setup. (It also cannot get crushed in any given roll in round three -- totally immune to a big opposing score, but could still get a big score if it rolls a 10.11 against a 1 or 2)

[QuikSand]

Okay, Desnudo's winning die -- another fairly obvious setup. This one, obviously, is feast or famine. In the first round, it could rather easily end up in last place... or first. Its biggest weaknesses are probably that (1) a spot of bad luck in round two could easily put it in last place, where a "safer" die would be far more likely to survive, and (2) its likelihood of rolling 1s make it prone to giving up the big bonus score in round three, meaning that it may need to win more than its fair share of rolls -- Desnudo got lucky this time and scores a 3.0 twice... but that was in large part a function of the opposing die also having lots of 1s.

[QuikSand]

Okay, I am probably at the end of my rope with regard to Excel. I have this et up to manually recalculate the rolls, and then I paste the resulting data off to the side as a static value. Works fine for a single iteration... not so well if I'd eventualy like to open this up to, say, running and recording 1,000 results of the same six contestants.

Is there anyone more wizardlike than I with Excel who might be interested in helping with the spreadsheet? (Or using some other program, if you prefer?) I honestly think this game would get more interesting if the luck element went down, and the design element went up -- and I think more trials is probably the best way to do that.

[albionmoonlight]

Putting 21s on my die was an idea based on the fact that a lot of people would, without really thinking about it, stop at 20. Now that people realize that 20 is not a limit, I wonder if that would be worth it. Also, I got kind of lucky that one of them hit in the first place because I only had two on the die.

[TredWel]

Running n interations of these trials wouldn't be at all hard to program. I can tackle it later today.

[Eaglesfan27]

My die did horribly.

[albionmoonlight]

I agree that this starts to get interesting once it gets up to 1000+ trials and luck is reduced.

Here's a question, since part of this is, I presume, to provoke discussion.

Is there a "best" or "perfect" die for this game (once luck is minimized), or can any die be beaten by someone who sees it first and designs a die to beat it.

Put another way. Is there some die "X" that is the best possible, or is it a situation where die X would (tend to) beat die Y would (tend to) beat die Z would (tend to) beat die X, etc.?

[QuikSand]

albion... what got me thinking about this puzzle was an old classic probability puzzle about the reciprocal nature of die-rolling. The challenge there is to design three six-sided dice such that, on balance, A beats B, B beats C, but C beats A. It's not that tough to do... if you know the properties of the die you will face head to head, it's not tough to design a die that would be likely to beat it.

That got me thinking about this whole design-a-die concept... and here we are.

But as a f'rinstance... if you knew you were going to face my "Mean 313" die (with five 3s and fifteen 13s) in the final round, you could pretty easily design a die that would have a very good shot to beat it, I bet. (Lots of 14s would seem like a good start)

Thus the three-round process that requires a different set of skills to win.


As for what's optimum -- perhaps you have to have a method to determine how various results are weighed. Do you shoot for just the best overall chance to win the whole contest? Or do 2nd and 3rd place finishes count to some degree as well?

And, I suspect, back to the original thinking, the "perfect" die for this contest might well be a decided underdog if the other comptitors knew what they were going up against.

[TredWel]

Here are the results of Heat 1, run 10000 times.


Bone #0 (QuikSand's Mean 313)
First Round Losses: 3969
First Round Average Score: 86.05
Second Round Losses: 1975
Second Round Average Score: 52.33029
Final Round Losses: 1204
Final Round Wins: 2852

Bone #1 (KWhit)
First Round Losses: 5051
First Round Average Score: 84.3041
Second Round Losses: 1941
Second Round Average Score: 51.534046
Final Round Losses: 1848
Final Round Wins: 1160

Bone #2 (EaglesFan27)
First Round Losses: 4584
First Round Average Score: 84.9504
Second Round Losses: 1907
Second Round Average Score: 51.736706
Final Round Losses: 1645
Final Round Wins: 1864

Bone #3 (Vince)
First Round Losses: 4536
First Round Average Score: 85.6154
Second Round Losses: 1910
Second Round Average Score: 52.250916
Final Round Losses: 1922
Final Round Wins: 1632

Bone #4 (TredWel)
First Round Losses: 5308
First Round Average Score: 83.9778
Second Round Losses: 1341
Second Round Average Score: 51.764492
Final Round Losses: 1832
Final Round Wins: 1519

Bone #5 (Celeval)
First Round Losses: 6552
First Round Average Score: 83.7206
Second Round Losses: 926
Second Round Average Score: 51.963165
Final Round Losses: 1549
Final Round Wins: 973

[KWhit]

My bone sucks.

[TredWel]

Heat 2, run 10000 times.


Bone #0 (Airhog)
First Round Losses: 4845
First Round Average Score: 84.0188
Second Round Losses: 1108
Second Round Average Score: 51.75461
Final Round Losses: 2136
Final Round Wins: 1911

Bone #1 (albionmoonlight)
First Round Losses: 6098
First Round Average Score: 81.275
Second Round Losses: 1583
Second Round Average Score: 49.78242
Final Round Losses: 1382
Final Round Wins: 937

Bone #2 (Desnudo)
First Round Losses: 4837
First Round Average Score: 84.5434
Second Round Losses: 2095
Second Round Average Score: 51.80961
Final Round Losses: 2217
Final Round Wins: 851

Bone #3 (Buzzbee)
First Round Losses: 3017
First Round Average Score: 85.976
Second Round Losses: 2481
Second Round Average Score: 52.27624
Final Round Losses: 558
Final Round Wins: 3944

Bone #4 (Daimyo)
First Round Losses: 5413
First Round Average Score: 83.9781
Second Round Losses: 1083
Second Round Average Score: 51.74602
Final Round Losses: 1768
Final Round Wins: 1736

Bone #5 (paperstreetsoap)
First Round Losses: 5790
First Round Average Score: 78.7316
Second Round Losses: 1650
Second Round Average Score: 51.477196
Final Round Losses: 1939
Final Round Wins: 621

[QuikSand]

Impressibe results from Buzzbee's entry... wonder how much of that is a function of having two opponents in his group that yield 10s and 11s (which his frequent 12s beat)?

I'm guessing that my Mean 313 might beat his... just because the 13 v. 12 matchup favors mine.

I'm starting to think that there's just no such thing as "best" here...

[QuikSand]

Thanks for the effort, TredWel.

[albionmoonlight]

Maybe the next step is to design some dice to go up against Buzz' die. Then, if/when one of those wins, design dice to go up against the winner. Rinse, repeat until we get a perfect die or get back to Buzz' die.

Of course, if everyone feels that there is no such thing as best (that's how I feel), then it may not be worth running the experiment.

[Marmel]

How would a regular 1-20 die score in the 10,000 trials?

[QuikSand]

I agree, another trial with TredWel's system woudl be interesting.

I nominate:

Buzzbee's 12-heavy die
QuikSand's Mean 313
A standard 1-20 die
A die with 10x10 and 10x11 (three people submitted this idea)
Desnudo's die from Heat Two
EaglesFan27's die fron Heat One

[albionmoonlight]

I agree, another trial with TredWel's system woudl be interesting.

I nominate:

Buzzbee's 12-heavy die
QuikSand's Mean 313
A standard 1-20 die
A die with 10x10 and 10x11 (three people submitted this idea)
Desnudo's die from Heat Two
EaglesFan27's die fron Heat One
Maybe throw a 14 heavy die into the mix, though, instead of Desnudo's die?

4
4
4
4
4
4
4
14
14
14
14
14
14
14
14
14
14
14
14
14

[Buzzbee]

Interesting results.

Some notes regarding my die:

I took the 210 and divided it by 20 and of course came up with 10.5, as I'm guessing many people did. I started to go with 11's, but figured others might as well. Since bumping it up a number would only reduce the 'low' numbers by one face, it seemed to be a good strategy. That was really about all the thought I put into it.

I'm wondering if 15 might produce good long term results because it splits the difference between 10 and 20. However, like many, I believe deep down that there is no single 'best' die, just better when compared to specific others. It seems that since all are operating with a total of 210 points, the distribution will become irrelevant over time. Of course the .1 addition that QS made to the rules might skew this a bit.

[QuikSand]

Now you're just picking on me.

(sounds fine to me)

[TredWel]

I agree, another trial with TredWel's system woudl be interesting.

I nominate:

Buzzbee's 12-heavy die
QuikSand's Mean 313
A standard 1-20 die
A die with 10x10 and 10x11 (three people submitted this idea)
Desnudo's die from Heat Two
EaglesFan27's die fron Heat One
Here's 10000 trials of this case:


Bone #0 (Buzzbee's 12-heavy die)
First Round Losses: 3572
First Round Average Score: 85.964
Second Round Losses: 2104
Second Round Average Score: 52.416306
Final Round Losses: 2028
Final Round Wins: 2296

Bone #1 (QuikSand's Mean 313)
First Round Losses: 4688
First Round Average Score: 85.872
Second Round Losses: 1642
Second Round Average Score: 52.118412
Final Round Losses: 805
Final Round Wins: 2865

Bone #2 (Standard 1-20 die)
First Round Losses: 5276
First Round Average Score: 83.8609
Second Round Losses: 1714
Second Round Average Score: 51.685646
Final Round Losses: 1474
Final Round Wins: 1536

Bone #3 (10x10 & 10x11)
First Round Losses: 5982
First Round Average Score: 83.9898
Second Round Losses: 1037
Second Round Average Score: 51.74166
Final Round Losses: 2092
Final Round Wins: 889

Bone #4 (Desnudo's Heat 2 die)
First Round Losses: 5415
First Round Average Score: 84.2318
Second Round Losses: 1852
Second Round Average Score: 52.12999
Final Round Losses: 2108
Final Round Wins: 625

Bone #5 (EaglesFan27's Heat 1 die)
First Round Losses: 5067
First Round Average Score: 85.006
Second Round Losses: 1651
Second Round Average Score: 51.692276
Final Round Losses: 1493
Final Round Wins: 1789

[TredWel]

Here's the data with the 14-heavy die instead of Desnudo's die


Bone #0 (12-heavy)
First Round Losses: 3654
First Round Average Score: 85.98
Second Round Losses: 2246
Second Round Average Score: 52.39521
Final Round Losses: 2455
Final Round Wins: 1645

Bone #1 (Mean 3-13)
First Round Losses: 4530
First Round Average Score: 86.057
Second Round Losses: 1759
Second Round Average Score: 52.200367
Final Round Losses: 1400
Final Round Wins: 2311

Bone #2 (Standard)
First Round Losses: 5357
First Round Average Score: 83.7256
Second Round Losses: 1682
Second Round Average Score: 51.926987
Final Round Losses: 1627
Final Round Wins: 1334

Bone #3 (10-11)
First Round Losses: 6103
First Round Average Score: 83.9752
Second Round Losses: 1003
Second Round Average Score: 51.75545
Final Round Losses: 2228
Final Round Wins: 666

Bone #4 (4-14)
First Round Losses: 5160
First Round Average Score: 85.351
Second Round Losses: 1606
Second Round Average Score: 52.147728
Final Round Losses: 782
Final Round Wins: 2452

Bone #5 (EF27)
First Round Losses: 5196
First Round Average Score: 84.8112
Second Round Losses: 1704
Second Round Average Score: 51.79517
Final Round Losses: 1508
Final Round Wins: 1592

[QuikSand]

Yes, I figured that the 14-heavy die would probbaly beat out mine.

TredWel... how much of this are you willing to go through? I'd think another worthy trial might be to create a 15-heavy die, and add that in place of EagleFan's...

sides 1 through 9 = 5
sides 10 through 20 = 15

I guess the logic here is that the 5s beat the lower score of the other top dice, while the 15s beat their higher scores as well. Of course, you sacrifice the benefit of having quite as many higher numbers here... I wonder if this one wins, or if we have reached the "tipping point."

[TredWel]

I've got no problem running as many as we need to figure this out. The program takes 10 seconds to run, and I just copy and paste the output.

Here's the results with the 5-15 die added:


Bone #0 (2-12)
First Round Losses: 3915
First Round Average Score: 85.604
Second Round Losses: 2244
Second Round Average Score: 52.237633
Final Round Losses: 2364
Final Round Wins: 1477

Bone #1 (3-13)
First Round Losses: 4574
First Round Average Score: 85.9
Second Round Losses: 1700
Second Round Average Score: 52.365646
Final Round Losses: 1639
Final Round Wins: 2087

Bone #2 (Standard)
First Round Losses: 5295
First Round Average Score: 83.8256
Second Round Losses: 1745
Second Round Average Score: 51.795963
Final Round Losses: 1641
Final Round Wins: 1319

Bone #3 (10-11)
First Round Losses: 5857
First Round Average Score: 84.0031
Second Round Losses: 1048
Second Round Average Score: 51.74777
Final Round Losses: 2326
Final Round Wins: 769

Bone #4 (4-14)
First Round Losses: 5073
First Round Average Score: 85.507
Second Round Losses: 1681
Second Round Average Score: 52.166225
Final Round Losses: 1168
Final Round Wins: 2078

Bone #5 (5-15)
First Round Losses: 5286
First Round Average Score: 84.667
Second Round Losses: 1582
Second Round Average Score: 52.305473
Final Round Losses: 862
Final Round Wins: 2270

[Huckleberry]

I'm enjoying this. Just following along, but I have a question.

What do y'all think caused the 313 to overtake the 414 in the latest run?

[TredWel]

I attribute it to the fact that 313 had about 500 more cracks at the final round than 414. 313 is the best of these six dice in the first two rounds, so it makes sense that it would have more chances to record a victory. 313 has a 56% victory rate in the final round, while 414 has a 64% victory rate.

[albionmoonlight]

Time for the Super 16?

[QuikSand]

So, do you keep going with the same concept -- increasing both sides by one? A 16/6 die is going to have to be pretty messy... only nine 16s to eleven 6s. At that point, it's actually more likely to get the lower of the two rolls, which (I'd think) defeats the original purpose.

I do find it interesting that the "Mean 313" die does seem to have the highest first round score (long term) of anything we've tried. (Or is that just my imagination?)

[albionmoonlight]

Maybe eleven 16s and seven 4s and two 3s? (I think that adds up to 210).

[Eaglesfan27]

Here are the results of Heat 1, run 10000 times.


Bone #0 (QuikSand's Mean 313)
First Round Losses: 3969
First Round Average Score: 86.05
Second Round Losses: 1975
Second Round Average Score: 52.33029
Final Round Losses: 1204
Final Round Wins: 2852

Bone #1 (KWhit)
First Round Losses: 5051
First Round Average Score: 84.3041
Second Round Losses: 1941
Second Round Average Score: 51.534046
Final Round Losses: 1848
Final Round Wins: 1160

Bone #2 (EaglesFan27)
First Round Losses: 4584
First Round Average Score: 84.9504
Second Round Losses: 1907
Second Round Average Score: 51.736706
Final Round Losses: 1645
Final Round Wins: 1864

Bone #3 (Vince)
First Round Losses: 4536
First Round Average Score: 85.6154
Second Round Losses: 1910
Second Round Average Score: 52.250916
Final Round Losses: 1922
Final Round Wins: 1632

Bone #4 (TredWel)
First Round Losses: 5308
First Round Average Score: 83.9778
Second Round Losses: 1341
Second Round Average Score: 51.764492
Final Round Losses: 1832
Final Round Wins: 1519

Bone #5 (Celeval)
First Round Losses: 6552
First Round Average Score: 83.7206
Second Round Losses: 926
Second Round Average Score: 51.963165
Final Round Losses: 1549
Final Round Wins: 973

Interesting results, my bone didn't do as bad as I thought it would with multiple repetitions.

[Vince]

Heh, my bone did better than I thought as well. I didn't even think about the 'penalty' of rolling a one -- interesting results all around.

[Daimyo]

I've been playing with this this afternoon in Excel and the best round 1 die I can get in terms of average score is 1x5,6x5,15x5,20x5 with 90.7. Second best I've found is 1x3,6x7,15x7,20x3 with an average of 89.1. Third is 1x4,6x6,15x6,20x4. I've been trying several different patterns and that one seems the best so far.

Those three are also 2, 3, 4 in terms of odds of getting out of the first round, but the best has been 11x19,1x1... that one has a low average, but it's so consistant and almost always does just enough to finish in the top half.

Haven't even looked at round 2 scoring yet.