Strange dice.

[Guv]

There have been some posts asking mathematical questions, and others have reminded me of some counterintuitive probability problems. Start with the most difficult one first.

It is assumed that the dice described below are "true" dice. Id est: Each face has the same probability of being face up (1/6 or odds of 5 to one against).

Imagine three custom made dice which do not have 1-6 on the faces like standard dice. One die is red, one blue, and one green. If you were to examine these dice, you would notice that no two dice use the same numbers. You would also notice that each die has less that 6 different numbers on it.

Imagine playing and betting on a game in which one player owns the blue, the other owns the green die. The winner of each roll is the one whose die has the larger number. Neither player is bright enough to examine the two dice and figure out the probabilities. However, the owner of the green die notices that he loses about 5 out of nine rolls, with no ties (In fact, the odds favor the blue die over the green die by 5 to 4). The losing player demands to use the blue die. The other player agrees, but says he will use the red die. Once again, the same player loses about 5 out of nine rolls, with no ties (In fact, the odds favor red over blue by 5 to 4). Now, the losing player demands to use the red die and insists that the other player use the green die. Can you guess or calculate the odds in a game using the red & green die?

I can describe a set of three dice which conform to the above conditions, but I am not sure that it is the only possible "unique" set. The set that I have in mind can be used to figure out how to make trivially different sets. It will be interesting if anybody comes up with more than one set of dice conforming to the above conditions.

On to easier problems using a set of identical dice. Each die has "T" on three faces, and "H" on three faces. I could have used 3 zeros & 3 ones to make it easier for the real binary-thinking computer nerds to understand. I also could have suggested 3 different lewd pictures of a nude female and three of a nude male to make it easier for others of you to visualize the dice.

If I roll one such die 4 times, what is the probability (or odds) of getting "H" four times? How about getting "THHH?" Are odds the same? What about rolling 4 dice at one? Do the odds on rolling "H" four times change? Is it the same or different from "H" 3 times & "T" once?

Now consider a slightly different situation. Suppose we roll one die millions of times, recording the sequence of rolls (theoretical mathematicians do not worry about how long such procedures take). We expect the sequence to include "H" approximately as often as "T." We do not expect "H" & "T" to alternate (Id est: We do not expect "HTHTHTHTHTHTHTHT..."). Suppose we examine the recorded sequence, and count how many times we see "HHHH" and how many times we see "THHH." Do we expect to see the same number of each in an extremely long sequence? Do we expect to see one sequence occur before the other?

If anyone shows an interest, I will post answers to the above questions.

[kb244]

Are you taking Discrete math or something?

[Sam Finch]

hmm, I'm going to try and up the number of replies on this post by seeing if anybody can guess the number of different sets of dice that will work for this question, by the way, we define a set of dice by labeling the sides of each die, eg the blue die has sides B1,B2,B3,B4,B5.B6 B1 being the lowest value on the die, B6 being the highest (some values will equal others on the same die eg B4 might = B5) A set of Dice is defined by Putting the sides B1...B6 R1...R6 and G1...G6 in order, with the lowest value first. (Just to keep things simple no 2 sides on different die can be the same)

So, How Many sets are there?

[kb244]

well that depends, since no sides are the same, and each dice is said to have 6 sides, my wild guess would be 46656

also as far as the winning set, could I assume that each dice different numbers are in order, in such a way that one dice would always win since it's lowest number is higher than the other two's highest number for example

Green = 1 2 3 4 5 6
Red = 7 8 9 10 11 12
Blue = 13 14 15 16 17 18

[kb244]

I may have read you wrong, but are we saying of each color we have a single die, or are we saying that of each color we have a pair of die, but when rolled neither number are ever the same.

[Guv]

kb244, There are exactly three dice, one red, one blue, one green.

With the numbers you picked, the blue die would always win over the green die. It would not average five wins and four losses.

The game is played by rolling two dice and comparing the numbers which are face up. The die with the higher number face up wins.
If you were to examine these dice, you would notice that no two dice use the same numbers. You would also notice that each die has less that 6 different numbers on it.

Note that the above indicates that the same number can appear more than once on a given die, but cannot appear on two different dice. The blue die could have 2 ones and 4 twos, in which case, neither the red nor the green die could have a one or a two. The blue die could also have 2 ones, 2 twos, & 2 threes. Neither of these blue dice could be used in a set which meets the probability conditions, but would conform to the above quoted sentences.

[HarbringerOfVole]

Actually, the statement

Quote: If you were to examine these dice, you would notice that no two dice use the same numbers. You would also notice that each die has less that 6 different numbers on it.

Says that no two dice uses the same numbers, it dosn't state that no two dice share a number, a completly different statment. The original statement mearly means that no two dice are identical.

Taking it as you ment it one solution is: green 1,1,1,4,7,7
blue 3,3,3,3,6,6
red 2,5,5,5,5,5
This obeys the rules of each die not sharing any numbers, each die having lessthan 6 different numbers, and the odds shown in the game. The odds for red vs green is that green loses 13 out of 36 games, so red wins 23 out of 36 games, or that the odds favour the red die 23 to 13.
Another possible solution is green: 3,3,3,3,7,7
blue: 2,6,6,6,6,6
red: 1,4,4,8,8,8

this time with the odds favouring the red die 13 to 5, (or for eaiser comparison, it wins 26 out of 36 games.) This set is v. similer to first, so as said, no significant differce.
Or, green 1,2,4,8,14,16
blue 3,5,7,9,11,13
red 6,6,10,10,10,10 (as with above, sorry I forgot to mention it, this last set can eaisly be changed, for example 6,6,6,10,10,15 or 6,6,6,6,15,15, as the critical numbers are the losing numbers - which, in this set are (due to restriction rules) 6,10 and 12. To obtain odds of 5 to 4, the die must lose 16 out of 36 games, and the number of games lost in 16 by the numbers are 6=4, 10=2, 12=1, and as long as they add up to 16, then the set is permissible. Another example is 6,6,6,10,12,12.) Iam not sure if this is significantly different, Guv, you'd better tell me!

.

[HarbringerOfVole]

Sorry, forgot to post odds on the last red v green game:
with the first set of numbers (th others produce different odds, for obvious reason, the critical losing numbers are differnt, and have differnt values)
the odds favour the red die to the tune of 11 to 7. (or it wins 22 times out of 36).

A note of interest, can any one come up with a set that has the green odds on to beat the red?

[HarbringerOfVole]

Completly forgot about other questions :)


The odds of THHH (in that order) is 1 in 16. the odds of getting 3H and 1T is 1 in 4. The probability of 4H is 1 in 16.
If 4 dice are rolled at once, then the odds do not change, except that order is irrelevant, but now which die gets which number is relevant.
I am guessing that in a very long sequence of events as said, you woulg get no more THHH than HHHH, as they have an equal probability, as order is important. However, they can overlap, eg THHHH would show both of the events. I would not expect to see a large degree of swapping ie THHH HHHH THHH HHHH (even if separated by other groups or over lapping, but agin, just a guess.)

[Sam Finch]

Seeing as nobody except kb224 tried to guess how many solutions there were I shall pronounce him the winner.

There are in fact 38293 different solutions to the problem, (I can list them if you like) and they are all essentially different and can all be used to generate an infinite number of trivially different solutions.

I'd agree with HarbringerOfVole on the THHH problem, and I'll have a looksie if there are any solutions where green beats red.

but for now a puzzle for you all.

If I take 3 dice,1 red, 1 greed and one blue and paint numbers on them all at random what is the probability that it will be a solution to Guv's puzzle?

[HarryW]

How's this?

P(solution) = (1/6)^36 * 38293 = 3.7125676659743535988763677376978e-24



I'd like to take you up on that offer of listing them all too :)

[HarryW]

Oh but it depends on the limit of the random numbers you paint on. I assumed 6 but I just remembered HarbingerOfVole used more. If there is no limit then I guess the probability is 0.

[kb244]

I assume is the propability of all the combinations subtracting the impossible combinations.

[Sam Finch]

Harry's right it is 0, but I'm not satisfied with his reasoning why.

[Guv]

Sam, I would like to see a few of your 38293 solutions. I would really like to know how you arrived at 38293, which is the product of two primes (257, 149).

I guess it is time to post some answers.

Easy ones first. Harbringer is partially correct on the H/T dice. The odds are 15 to 1 against (probability 1/16) for rolling HHHH with four dice and for rolling it by rolling one die four times. Chances for THHH by rolling one die four times is the same: Probability is 1/16 or 15 to one against. He is also correct about T once and H three times with four dice at once. Probability 1/4 or 3 to one against. Order matters here. When rolling one die four times, you could roll THHH, HTHH, HHTH, or HHHT. All four of these rolls consist of one T & 3 H's. Together these four events are equivalent to rolling one T and 3 H's when rolling 4 dice at once.

I mentioned these dice primarily due to situations relating to a long string of recorded results.

First, HHHH occurs more often that THHH. Note that if HHHHH occurs (H five times in a row), HHHH (four times) occurs twice. Similarly: If H occurs 6, 7 or more times in a row (which can happen occasionally) there are clustered occurrences of HHHH. Such clusteres cannot occur for THHH. Hence HHHH occurs more often that THHH.

Second, If HHHH does not occur at the very start of the recorded string, it cannot occur prior to THHH! Due to clustering, the less frequent event (THHH) is more likely to occur first. In fact, it is likely to occur first if you start examining the string at a randomly selected position. Unless HHHH occurs at the spot you pick, it will not occur first.

Interesting: The less likely event is more likely to occur first. There is a real life situation involving (admittedly strange) train schedules. Suppose a train leaves for Town X 10 times per hour, but all leave between 15 and 30 minutes after the hour. A train for town Y leaves at ten minutes after each hour. If you arrive at the station at a random time, it is more likely that the next train will go to town Y, even though there are more trains to town X.

Now the red, blue, and green dice. A solution is related to the magic square of order 3, which also can be used as a delightful example of an isomorphic relationship. Perhaps isomorphisms should be the subject of another post (By the way, isomorphic is recognized by my spell checker, but isomorphism is not?).

The dice are as follows.
Red: 8, 1, 6 (Opposite faces have the same number).
Blu: 3, 5, 7 (Opposite faces have the same number).
Grn: 4, 9, 2 (Opposite faces have the same number).

The above are rows of a magic square. The Columns can also be used to construct dice with the same properties. Note that red averages 5 wins out of nine rolls versus blue; Blue wins 5 out 9 versus green; and Green wins 5 out of 9 versus red.

The above indicates that "Is more likely than" is not transitive like "equals" or "greater than" relationships! I do not know about the rest of you, but I was astonished the first time I encountered this anomaly about probability. Some might not call it an anomaly, but I consider it to be strange, peculiar, counterintuitive, et cetera.

I imagine that an infinite number of sets of dice can be constructed to conform to the conditions. It seems that adding a constant to each value on a face would maintain conformance to the conditions. For example, add ten to each value.

R: 18, 11, 16
B: 13, 15, 17
G: 14, 19, 12

I have never tried to determine if sets of dice conforming to the above can be constructing without using numbers from the magic square or numbers derived from them.

[Sam Finch]

Sam F***ed up.

Sorry, 38293 was wrong, that was the number of solutions for Blue beats Green 5:4 and Blue beats Red 5:4.

I might redo the algorithm for doing it the right way, but I might not (The algorithm for calculating it the right way is probably a little harder).

Just at a guess I think it'll be around 10 000 for this way.

but here are a few

Red {2,2,2,2,5,5}: Green {1,1,1,4,7,7}: Blue: {3,3,3,3,6,6}
Red {2,2,2,2,2,7}: Green {1,1,1,4,6,8}: Blue: {3,3,3,3,5,5}
Red {2,4,4,7,7,7}: Green {1,1,3,6,9,9}: Blue: {5,5,5,5,8,8}
Red {2,4,4,4,7,10}: Green {1,1,3,6,9,11}: Blue: {5,5,5,5,8,8}
Red {2,4,4,4,4,9}: Green {1,1,3,6,8,8}: Blue: {5,5,5,5,7,7}
Red {2,2,6,6,6,6}: Green {1,1,3,5,8,8}: Blue: {4,4,4,4,7,7}
Red {2,2,4,7,7,10}: Green {1,1,3,6,9,11}: Blue: {5,5,5,5,8,8}
Red {2,2,4,4,9,9}: Green {1,1,3,6,8,10}: Blue: {5,5,5,5,7,7}
Red {2,2,4,4,7,10}: Green {1,1,3,6,9,9}: Blue: {5,5,5,5,8,8}
Red {2,2,2,6,9,9}: Green {1,1,3,5,8,10}: Blue: {4,4,4,4,7,7}
Red {2,2,2,6,6,9}: Green {1,1,3,5,8,8}: Blue: {4,4,4,4,7,7}
Red {2,2,2,4,9,11}: Green {1,1,3,6,8,10}: Blue: {5,5,5,5,7,7}
Red {2,2,2,2,8,8}: Green {1,1,3,5,7,7}: Blue: {4,4,4,4,6,6}
Red {2,4,7,7,7,7}: Green {1,3,3,6,9,9}: Blue: {5,5,5,5,8,8}
Red {2,4,4,7,7,10}: Green {1,3,3,6,9,11}: Blue: {5,5,5,5,8,8}
Red {2,4,4,4,9,9}: Green {1,3,3,6,8,10}: Blue: {5,5,5,5,7,7}
Red {2,4,4,4,7,10}: Green {1,3,3,6,9,9}: Blue: {5,5,5,5,8,8}
Red {2,4,6,9,9,12}: Green {1,3,5,8,11,11}: Blue: {7,7,7,7,10,10}
Red {2,4,4,6,11,11}: Green {1,3,5,8,10,10}: Blue: {7,7,7,7,9,9}
Red {2,2,6,6,9,9}: Green {1,3,3,5,8,10}: Blue: {4,4,4,4,7,7}
Red {2,2,6,6,6,9}: Green {1,3,3,5,8,8}: Blue: {4,4,4,4,7,7}
Red {2,2,4,9,9,9}: Green {1,3,3,6,8,10}: Blue: {5,5,5,5,7,7}
Red {2,2,4,7,10,12}: Green {1,3,3,6,9,11}: Blue: {5,5,5,5,8,8}
Red {2,2,4,4,9,9}: Green {1,3,3,6,8,8}: Blue: {5,5,5,5,7,7}
Red {2,2,4,8,11,11}: Green {1,3,5,7,10,10}: Blue: {6,6,6,6,9,9}
Red {2,2,2,8,10,10}: Green {1,3,3,5,7,9}: Blue: {4,4,4,4,6,6}
Red {1,5,5,5,5,5}: Green {2,2,2,4,7,7}: Blue: {3,3,3,3,6,6}
Red {1,3,6,6,6,9}: Green {2,2,2,5,8,10}: Blue: {4,4,4,4,7,7}
Red {1,3,3,6,9,9}: Green {2,2,2,5,8,10}: Blue: {4,4,4,4,7,7}
Red {1,3,3,6,6,9}: Green {2,2,2,5,8,8}: Blue: {4,4,4,4,7,7}
Red {1,3,3,3,8,10}: Green {2,2,2,5,7,9}: Blue: {4,4,4,4,6,6}
Red {1,3,7,7,10,10}: Green {2,2,4,6,9,11}: Blue: {5,5,5,5,8,8}
Red {1,3,7,7,7,10}: Green {2,2,4,6,9,9}: Blue: {5,5,5,5,8,8}
Red {1,3,5,10,10,10}: Green {2,2,4,7,9,11}: Blue: {6,6,6,6,8,8}
Red {1,3,5,5,10,10}: Green {2,2,4,7,9,9}: Blue: {6,6,6,6,8,8}
Red {1,3,3,9,9,11}: Green {2,2,4,6,8,10}: Blue: {5,5,5,5,7,7}
Red {1,3,3,7,10,10}: Green {2,2,4,6,9,9}: Blue: {5,5,5,5,8,8}
Red {1,3,7,10,10,10}: Green {2,4,4,6,9,11}: Blue: {5,5,5,5,8,8}
Red {1,3,7,7,10,12}: Green {2,4,4,6,9,11}: Blue: {5,5,5,5,8,8}
Red {1,3,5,10,10,12}: Green {2,4,4,7,9,11}: Blue: {6,6,6,6,8,8}
Red {1,3,5,8,11,11}: Green {2,4,4,7,10,10}: Blue: {6,6,6,6,9,9}
Red {1,3,3,9,9,9}: Green {2,4,4,6,8,8}: Blue: {5,5,5,5,7,7}
Red {1,1,7,7,7,7}: Green {2,2,2,4,6,8}: Blue: {3,3,3,3,5,5}
Red {1,1,5,8,8,10}: Green {2,2,2,4,7,9}: Blue: {3,3,3,3,6,6}
Red {1,1,5,5,8,8}: Green {2,2,2,4,7,7}: Blue: {3,3,3,3,6,6}
Red {1,1,3,8,10,10}: Green {2,2,2,5,7,9}: Blue: {4,4,4,4,6,6}
Red {1,1,3,9,9,9}: Green {2,2,4,6,8,8}: Blue: {5,5,5,5,7,7}
Red {2,2,2,2,5,5}: Green {1,1,1,4,6,8}: Blue: {3,3,3,3,3,7}
Red {2,2,2,2,2,5}: Green {1,1,1,4,4,7}: Blue: {3,3,3,3,3,6}
Red {2,4,4,7,7,7}: Green {1,1,3,6,8,10}: Blue: {5,5,5,5,5,9}
Red {2,4,4,4,7,9}: Green {1,1,3,6,8,11}: Blue: {5,5,5,5,5,10}
Red {2,4,4,4,4,9}: Green {1,1,3,6,6,8}: Blue: {5,5,5,5,5,7}
Red {2,2,6,6,6,6}: Green {1,1,3,5,7,9}: Blue: {4,4,4,4,4,8}
Red {2,2,4,7,7,9}: Green {1,1,3,6,8,11}: Blue: {5,5,5,5,5,10}
Red {2,2,4,4,7,7}: Green {1,1,3,6,6,9}: Blue: {5,5,5,5,5,8}
Red {2,2,4,4,7,11}: Green {1,1,3,6,8,10}: Blue: {5,5,5,5,5,9}
Red {2,2,2,6,8,8}: Green {1,1,3,5,7,10}: Blue: {4,4,4,4,4,9}
Red {2,2,2,6,6,10}: Green {1,1,3,5,7,9}: Blue: {4,4,4,4,4,8}
Red {2,2,2,4,7,10}: Green {1,1,3,6,6,9}: Blue: {5,5,5,5,5,8}
Red {2,2,2,2,8,8}: Green {1,1,3,5,5,7}: Blue: {4,4,4,4,4,6}
Red {2,4,7,7,7,7}: Green {1,3,3,6,8,10}: Blue: {5,5,5,5,5,9}
Red {2,4,4,7,7,9}: Green {1,3,3,6,8,11}: Blue: {5,5,5,5,5,10}
Red {2,4,4,4,7,7}: Green {1,3,3,6,6,9}: Blue: {5,5,5,5,5,8}
Red {2,4,4,4,7,11}: Green {1,3,3,6,8,10}: Blue: {5,5,5,5,5,9}
Red {2,4,6,6,9,12}: Green {1,3,5,8,8,11}: Blue: {7,7,7,7,7,10}
Red {2,4,4,8,8,8}: Green {1,3,5,7,7,10}: Blue: {6,6,6,6,6,9}
Red {2,4,4,6,11,11}: Green {1,3,5,8,8,10}: Blue: {7,7,7,7,7,9}
Red {2,2,6,6,8,8}: Green {1,3,3,5,7,10}: Blue: {4,4,4,4,4,9}
Red {2,2,6,6,6,10}: Green {1,3,3,5,7,9}: Blue: {4,4,4,4,4,8}
Red {2,2,4,7,7,7}: Green {1,3,3,6,6,9}: Blue: {5,5,5,5,5,8}
Red {2,2,4,7,9,12}: Green {1,3,3,6,8,11}: Blue: {5,5,5,5,5,10}
Red {2,2,4,4,9,9}: Green {1,3,3,6,6,8}: Blue: {5,5,5,5,5,7}
Red {2,2,4,8,8,11}: Green {1,3,5,7,7,10}: Blue: {6,6,6,6,6,9}
Red {2,2,2,6,9,9}: Green {1,3,3,5,5,8}: Blue: {4,4,4,4,4,7}
Red {1,5,5,5,5,5}: Green {2,2,2,4,6,8}: Blue: {3,3,3,3,3,7}
Red {1,3,6,6,6,8}: Green {2,2,2,5,7,10}: Blue: {4,4,4,4,4,9}
Red {1,3,3,6,8,8}: Green {2,2,2,5,7,10}: Blue: {4,4,4,4,4,9}
Red {1,3,3,6,6,10}: Green {2,2,2,5,7,9}: Blue: {4,4,4,4,4,8}
Red {1,3,3,3,6,9}: Green {2,2,2,5,5,8}: Blue: {4,4,4,4,4,7}
Red {1,3,7,7,9,9}: Green {2,2,4,6,8,11}: Blue: {5,5,5,5,5,10}
Red {1,3,7,7,7,11}: Green {2,2,4,6,8,10}: Blue: {5,5,5,5,5,9}
Red {1,3,5,8,8,8}: Green {2,2,4,7,7,10}: Blue: {6,6,6,6,6,9}
Red {1,3,5,5,10,10}: Green {2,2,4,7,7,9}: Blue: {6,6,6,6,6,8}
Red {1,3,3,7,7,10}: Green {2,2,4,6,6,9}: Blue: {5,5,5,5,5,8}
Red {1,3,3,7,11,11}: Green {2,2,4,6,8,10}: Blue: {5,5,5,5,5,9}
Red {1,3,7,9,9,9}: Green {2,4,4,6,8,11}: Blue: {5,5,5,5,5,10}
Red {1,3,7,7,9,12}: Green {2,4,4,6,8,11}: Blue: {5,5,5,5,5,10}
Red {1,3,5,8,8,11}: Green {2,4,4,7,7,10}: Blue: {6,6,6,6,6,9}
Red {1,3,5,8,12,12}: Green {2,4,4,7,9,11}: Blue: {6,6,6,6,6,10}
Red {1,3,5,9,12,12}: Green {2,4,6,8,8,11}: Blue: {7,7,7,7,7,10}
Red {1,3,3,9,9,9}: Green {2,4,4,6,6,8}: Blue: {5,5,5,5,5,7}
Red {1,1,5,5,5,5}: Green {2,2,2,4,4,7}: Blue: {3,3,3,3,3,6}
Red {1,1,5,7,7,10}: Green {2,2,2,4,6,9}: Blue: {3,3,3,3,3,8}
Red {1,1,5,5,9,9}: Green {2,2,2,4,6,8}: Blue: {3,3,3,3,3,7}
Red {1,1,3,6,9,9}: Green {2,2,2,5,5,8}: Blue: {4,4,4,4,4,7}
Red {1,1,3,9,9,9}: Green {2,2,4,6,6,8}: Blue: {5,5,5,5,5,7}
Red {4,4,4,4,7,7}: Green {1,1,3,6,9,9}: Blue: {2,5,5,8,8,8}
Red {4,4,4,4,4,9}: Green {1,1,3,6,8,10}: Blue: {2,5,5,7,7,7}
Red {2,5,5,8,8,8}: Green {1,1,4,7,10,10}: Blue: {3,6,6,9,9,9}
Red {2,5,5,5,8,11}: Green {1,1,4,7,10,12}: Blue: {3,6,6,9,9,9}
Red {2,5,5,5,5,10}: Green {1,1,4,7,9,9}: Blue: {3,6,6,8,8,8}
Red {2,2,7,7,7,7}: Green {1,1,4,6,9,9}: Blue: {3,5,5,8,8,8}
Red {2,2,5,8,8,11}: Green {1,1,4,7,10,12}: Blue: {3,6,6,9,9,9}
Red {2,2,5,5,10,10}: Green {1,1,4,7,9,11}: Blue: {3,6,6,8,8,8}
Red {2,2,5,5,8,11}: Green {1,1,4,7,10,10}: Blue: {3,6,6,9,9,9}
Red {2,2,2,7,10,10}: Green {1,1,4,6,9,11}: Blue: {3,5,5,8,8,8}
Red {2,2,2,7,7,10}: Green {1,1,4,6,9,9}: Blue: {3,5,5,8,8,8}
Red {2,2,2,5,10,12}: Green {1,1,4,7,9,11}: Blue: {3,6,6,8,8,8}
Red {2,2,2,2,9,9}: Green {1,1,4,6,8,8}: Blue: {3,5,5,7,7,7}
Red {2,6,9,9,9,9}: Green {1,3,5,8,11,11}: Blue: {4,7,7,10,10,10}
Red {2,6,6,6,9,12}: Green {1,3,5,8,11,11}: Blue: {4,7,7,10,10,10}
Red {2,4,7,10,10,13}: Green {1,3,6,9,12,12}: Blue: {5,8,8,11,11,11}
Red {2,4,4,7,12,12}: Green {1,3,6,9,11,11}: Blue: {5,8,8,10,10,10}
Red {2,2,8,8,8,11}: Green {1,3,5,7,10,10}: Blue: {4,6,6,9,9,9}
Red {2,2,6,6,11,11}: Green {1,3,5,8,10,10}: Blue: {4,7,7,9,9,9}
Red {2,2,4,9,12,12}: Green {1,3,6,8,11,11}: Blue: {5,7,7,10,10,10}
Red {1,7,7,7,7,7}: Green {2,2,4,6,9,9}: Blue: {3,5,5,8,8,8}
Red {1,5,8,8,8,11}: Green {2,2,4,7,10,12}: Blue: {3,6,6,9,9,9}
Red {1,5,5,8,11,11}: Green {2,2,4,7,10,12}: Blue: {3,6,6,9,9,9}
Red {1,5,5,8,8,11}: Green {2,2,4,7,10,10}: Blue: {3,6,6,9,9,9}
Red {1,5,5,5,10,12}: Green {2,2,4,7,9,11}: Blue: {3,6,6,8,8,8}
Red {1,3,8,8,11,11}: Green {2,2,5,7,10,12}: Blue: {4,6,6,9,9,9}
Red {1,3,8,8,8,11}: Green {2,2,5,7,10,10}: Blue: {4,6,6,9,9,9}
Red {1,3,6,11,11,11}: Green {2,2,5,8,10,12}: Blue: {4,7,7,9,9,9}
Red {1,3,6,6,11,11}: Green {2,2,5,8,10,10}: Blue: {4,7,7,9,9,9}
Red {1,3,3,10,10,12}: Green {2,2,5,7,9,11}: Blue: {4,6,6,8,8,8}
Red {1,3,3,8,11,11}: Green {2,2,5,7,10,10}: Blue: {4,6,6,9,9,9}
Red {1,3,7,10,13,13}: Green {2,4,6,9,12,12}: Blue: {5,8,8,11,11,11}
Red {1,3,3,11,11,11}: Green {2,4,6,8,10,10}: Blue: {5,7,7,9,9,9}
Red {1,1,9,9,9,9}: Green {2,2,4,6,8,10}: Blue: {3,5,5,7,7,7}
Red {1,1,7,10,10,12}: Green {2,2,4,6,9,11}: Blue: {3,5,5,8,8,8}
Red {1,1,7,7,10,10}: Green {2,2,4,6,9,9}: Blue: {3,5,5,8,8,8}
Red {1,1,5,10,12,12}: Green {2,2,4,7,9,11}: Blue: {3,6,6,8,8,8}
Red {1,1,3,10,10,10}: Green {2,2,5,7,9,9}: Blue: {4,6,6,8,8,8}
Red {4,4,4,4,7,7}: Green {1,1,3,6,9,11}: Blue: {2,5,5,5,8,10}
Red {4,4,4,4,4,9}: Green {1,1,3,6,8,11}: Blue: {2,5,5,5,7,10}
Red {2,5,5,8,8,8}: Green {1,1,4,7,10,12}: Blue: {3,6,6,6,9,11}
Red {2,5,5,5,8,11}: Green {1,1,4,7,10,13}: Blue: {3,6,6,6,9,12}
Red {2,5,5,5,5,12}: Green {1,1,4,7,9,11}: Blue: {3,6,6,6,8,10}
Red {2,2,7,7,7,7}: Green {1,1,4,6,9,11}: Blue: {3,5,5,5,8,10}
Red {2,2,5,8,8,11}: Green {1,1,4,7,10,13}: Blue: {3,6,6,6,9,12}
Red {2,2,5,5,10,10}: Green {1,1,4,7,9,12}: Blue: {3,6,6,6,8,11}
Red {2,2,5,5,8,13}: Green {1,1,4,7,10,12}: Blue: {3,6,6,6,9,11}
Red {2,2,2,7,10,10}: Green {1,1,4,6,9,12}: Blue: {3,5,5,5,8,11}
Red {2,2,2,7,7,12}: Green {1,1,4,6,9,11}: Blue: {3,5,5,5,8,10}
Red {2,2,2,5,10,13}: Green {1,1,4,7,9,12}: Blue: {3,6,6,6,8,11}
Red {2,2,2,2,11,11}: Green {1,1,4,6,8,10}: Blue: {3,5,5,5,7,9}
Red {1,7,7,7,7,7}: Green {2,2,4,6,9,11}: Blue: {3,5,5,5,8,10}
Red {1,5,8,8,8,11}: Green {2,2,4,7,10,13}: Blue: {3,6,6,6,9,12}
Red {1,5,5,8,11,11}: Green {2,2,4,7,10,13}: Blue: {3,6,6,6,9,12}
Red {1,5,5,8,8,13}: Green {2,2,4,7,10,12}: Blue: {3,6,6,6,9,11}
Red {1,5,5,5,10,13}: Green {2,2,4,7,9,12}: Blue: {3,6,6,6,8,11}
Red {1,3,8,8,11,11}: Green {2,2,5,7,10,13}: Blue: {4,6,6,6,9,12}
Red {1,3,8,8,8,13}: Green {2,2,5,7,10,12}: Blue: {4,6,6,6,9,11}
Red {1,3,6,11,11,11}: Green {2,2,5,8,10,13}: Blue: {4,7,7,7,9,12}
Red {1,3,6,6,13,13}: Green {2,2,5,8,10,12}: Blue: {4,7,7,7,9,11}
Red {1,3,3,10,10,13}: Green {2,2,5,7,9,12}: Blue: {4,6,6,6,8,11}
Red {1,3,3,8,13,13}: Green {2,2,5,7,10,12}: Blue: {4,6,6,6,9,11}
Red {1,1,9,9,9,9}: Green {2,2,4,6,8,11}: Blue: {3,5,5,5,7,10}
Red {1,1,7,10,10,13}: Green {2,2,4,6,9,12}: Blue: {3,5,5,5,8,11}
Red {1,1,7,7,12,12}: Green {2,2,4,6,9,11}: Blue: {3,5,5,5,8,10}
Red {1,1,5,10,13,13}: Green {2,2,4,7,9,12}: Blue: {3,6,6,6,8,11}
Red {1,1,3,12,12,12}: Green {2,2,5,7,9,11}: Blue: {4,6,6,6,8,10}
Red {4,4,4,4,7,7}: Green {1,1,3,6,8,8}: Blue: {2,5,5,5,5,9}
Red {4,4,4,4,4,7}: Green {1,1,3,6,6,8}: Blue: {2,5,5,5,5,9}
Red {2,5,5,8,8,8}: Green {1,1,4,7,9,9}: Blue: {3,6,6,6,6,10}
Red {2,5,5,5,8,10}: Green {1,1,4,7,9,11}: Blue: {3,6,6,6,6,12}
Red {2,5,5,5,5,8}: Green {1,1,4,7,7,7}: Blue: {3,6,6,6,6,9}
Red {2,2,7,7,7,7}: Green {1,1,4,6,8,8}: Blue: {3,5,5,5,5,9}
Red {2,2,5,8,8,10}: Green {1,1,4,7,9,11}: Blue: {3,6,6,6,6,12}
Red {2,2,5,5,8,8}: Green {1,1,4,7,7,9}: Blue: {3,6,6,6,6,10}
Red {2,2,5,5,8,10}: Green {1,1,4,7,9,9}: Blue: {3,6,6,6,6,11}
Red {2,2,2,7,9,9}: Green {1,1,4,6,8,10}: Blue: {3,5,5,5,5,11}
Red {2,2,2,7,7,9}: Green {1,1,4,6,8,8}: Blue: {3,5,5,5,5,10}
Red {2,2,2,5,8,10}: Green {1,1,4,7,7,9}: Blue: {3,6,6,6,6,11}
Red {2,2,2,2,7,7}: Green {1,1,4,6,6,6}: Blue: {3,5,5,5,5,8}
Red {2,6,9,9,9,9}: Green {1,3,5,8,10,10}: Blue: {4,7,7,7,7,11}
Red {2,6,6,6,9,9}: Green {1,3,5,8,8,10}: Blue: {4,7,7,7,7,11}
Red {2,6,6,6,9,11}: Green {1,3,5,8,10,10}: Blue: {4,7,7,7,7,12}
Red {2,4,7,10,10,12}: Green {1,3,6,9,11,11}: Blue: {5,8,8,8,8,13}
Red {2,4,7,7,10,12}: Green {1,3,6,9,9,11}: Blue: {5,8,8,8,8,13}
Red {2,4,4,9,9,9}: Green {1,3,6,8,8,10}: Blue: {5,7,7,7,7,11}
Red {2,4,4,7,10,10}: Green {1,3,6,9,9,9}: Blue: {5,8,8,8,8,11}
Red {2,2,8,8,8,10}: Green {1,3,5,7,9,9}: Blue: {4,6,6,6,6,11}
Red {2,2,6,9,9,9}: Green {1,3,5,8,8,10}: Blue: {4,7,7,7,7,11}
Red {2,2,6,6,9,9}: Green {1,3,5,8,8,8}: Blue: {4,7,7,7,7,10}
Red {2,2,4,9,9,11}: Green {1,3,6,8,8,10}: Blue: {5,7,7,7,7,12}
Red {2,2,4,9,11,11}: Green {1,3,6,8,10,10}: Blue: {5,7,7,7,7,12}
Red {2,2,2,8,10,10}: Green {1,3,5,7,7,9}: Blue: {4,6,6,6,6,11}
Red {1,7,7,7,7,7}: Green {2,2,4,6,8,8}: Blue: {3,5,5,5,5,9}
Red {1,5,8,8,8,10}: Green {2,2,4,7,9,11}: Blue: {3,6,6,6,6,12}
Red {1,5,5,8,10,10}: Green {2,2,4,7,9,11}: Blue: {3,6,6,6,6,12}
Red {1,5,5,8,8,10}: Green {2,2,4,7,9,9}: Blue: {3,6,6,6,6,11}
Red {1,5,5,5,8,10}: Green {2,2,4,7,7,9}: Blue: {3,6,6,6,6,11}
Red {1,3,8,8,10,10}: Green {2,2,5,7,9,11}: Blue: {4,6,6,6,6,12}
Red {1,3,8,8,8,10}: Green {2,2,5,7,9,9}: Blue: {4,6,6,6,6,11}
Red {1,3,6,9,9,9}: Green {2,2,5,8,8,10}: Blue: {4,7,7,7,7,11}
Red {1,3,6,6,9,9}: Green {2,2,5,8,8,8}: Blue: {4,7,7,7,7,10}
Red {1,3,3,8,8,10}: Green {2,2,5,7,7,9}: Blue: {4,6,6,6,6,11}
Red {1,3,3,8,10,10}: Green {2,2,5,7,9,9}: Blue: {4,6,6,6,6,11}
Red {1,3,7,10,10,12}: Green {2,4,6,9,9,11}: Blue: {5,8,8,8,8,13}
Red {1,3,7,10,12,12}: Green {2,4,6,9,11,11}: Blue: {5,8,8,8,8,13}
Red {1,3,5,10,12,12}: Green {2,4,7,9,9,11}: Blue: {6,8,8,8,8,13}
Red {1,3,3,9,9,9}: Green {2,4,6,8,8,8}: Blue: {5,7,7,7,7,10}
Red {1,1,7,7,7,7}: Green {2,2,4,6,6,8}: Blue: {3,5,5,5,5,9}
Red {1,1,7,9,9,11}: Green {2,2,4,6,8,10}: Blue: {3,5,5,5,5,12}
Red {1,1,7,7,9,9}: Green {2,2,4,6,8,8}: Blue: {3,5,5,5,5,10}
Red {1,1,5,8,10,10}: Green {2,2,4,7,7,9}: Blue: {3,6,6,6,6,11}
Red {1,1,3,8,8,8}: Green {2,2,5,7,7,7}: Blue: {4,6,6,6,6,9}
Red {4,4,4,4,6,6}: Green {1,1,3,5,8,8}: Blue: {2,2,7,7,7,7}
Red {4,4,4,4,4,8}: Green {1,1,3,5,7,9}: Blue: {2,2,6,6,6,6}
Red {2,5,5,7,7,7}: Green {1,1,4,6,9,9}: Blue: {3,3,8,8,8,8}
Red {2,5,5,5,7,10}: Green {1,1,4,6,9,11}: Blue: {3,3,8,8,8,8}
Red {2,5,5,5,5,9}: Green {1,1,4,6,8,8}: Blue: {3,3,7,7,7,7}
Red {2,2,5,5,5,5}: Green {1,1,4,4,7,7}: Blue: {3,3,6,6,6,6}
Red {2,2,5,7,7,10}: Green {1,1,4,6,9,11}: Blue: {3,3,8,8,8,8}
Red {2,2,5,5,9,9}: Green {1,1,4,6,8,10}: Blue: {3,3,7,7,7,7}
Red {2,2,5,5,7,10}: Green {1,1,4,6,9,9}: Blue: {3,3,8,8,8,8}
Red {2,2,2,5,8,8}: Green {1,1,4,4,7,9}: Blue: {3,3,6,6,6,6}
Red {2,2,2,5,5,8}: Green {1,1,4,4,7,7}: Blue: {3,3,6,6,6,6}
Red {2,2,2,5,9,11}: Green {1,1,4,6,8,10}: Blue: {3,3,7,7,7,7}
Red {2,2,2,2,7,7}: Green {1,1,4,4,6,6}: Blue: {3,3,5,5,5,5}
Red {2,6,8,8,8,8}: Green {1,3,5,7,10,10}: Blue: {4,4,9,9,9,9}
Red {2,6,6,6,8,11}: Green {1,3,5,7,10,10}: Blue: {4,4,9,9,9,9}
Red {2,4,7,7,7,10}: Green {1,3,6,6,9,11}: Blue: {5,5,8,8,8,8}
Red {2,4,7,9,9,12}: Green {1,3,6,8,11,11}: Blue: {5,5,10,10,10,10}
Red {2,4,4,9,9,9}: Green {1,3,6,6,8,10}: Blue: {5,5,7,7,7,7}
Red {2,4,4,7,10,12}: Green {1,3,6,6,9,11}: Blue: {5,5,8,8,8,8}
Red {2,4,4,7,11,11}: Green {1,3,6,8,10,10}: Blue: {5,5,9,9,9,9}
Red {2,2,6,6,9,9}: Green {1,3,5,5,8,10}: Blue: {4,4,7,7,7,7}
Red {2,2,6,6,6,9}: Green {1,3,5,5,8,8}: Blue: {4,4,7,7,7,7}
Red {2,2,6,6,10,10}: Green {1,3,5,7,9,9}: Blue: {4,4,8,8,8,8}
Red {2,2,4,9,9,11}: Green {1,3,6,6,8,10}: Blue: {5,5,7,7,7,7}
Red {2,2,4,7,10,10}: Green {1,3,6,6,9,9}: Blue: {5,5,8,8,8,8}
Red {2,2,2,8,10,10}: Green {1,3,5,5,7,9}: Blue: {4,4,6,6,6,6}
Red {1,5,5,5,5,5}: Green {2,2,4,4,7,7}: Blue: {3,3,6,6,6,6}
Red {1,5,7,7,7,10}: Green {2,2,4,6,9,11}: Blue: {3,3,8,8,8,8}
Red {1,5,5,7,10,10}: Green {2,2,4,6,9,11}: Blue: {3,3,8,8,8,8}
Red {1,5,5,7,7,10}: Green {2,2,4,6,9,9}: Blue: {3,3,8,8,8,8}
Red {1,5,5,5,9,11}: Green {2,2,4,6,8,10}: Blue: {3,3,7,7,7,7}
Red {1,3,6,6,9,9}: Green {2,2,5,5,8,10}: Blue: {4,4,7,7,7,7}
Red {1,3,6,6,6,9}: Green {2,2,5,5,8,8}: Blue: {4,4,7,7,7,7}
Red {1,3,6,10,10,10}: Green {2,2,5,7,9,11}: Blue: {4,4,8,8,8,8}
Red {1,3,6,6,10,10}: Green {2,2,5,7,9,9}: Blue: {4,4,8,8,8,8}
Red {1,3,3,8,8,10}: Green {2,2,5,5,7,9}: Blue: {4,4,6,6,6,6}
Red {1,3,3,6,9,9}: Green {2,2,5,5,8,8}: Blue: {4,4,7,7,7,7}
Red {1,3,7,10,10,10}: Green {2,4,6,6,9,11}: Blue: {5,5,8,8,8,8}
Red {1,3,7,7,10,12}: Green {2,4,6,6,9,11}: Blue: {5,5,8,8,8,8}
Red {1,3,7,9,12,12}: Green {2,4,6,8,11,11}: Blue: {5,5,10,10,10,10}
Red {1,3,5,10,12,12}: Green {2,4,7,7,9,11}: Blue: {6,6,8,8,8,8}
Red {1,3,3,9,9,9}: Green {2,4,6,6,8,8}: Blue: {5,5,7,7,7,7}
Red {1,1,7,7,7,7}: Green {2,2,4,4,6,8}: Blue: {3,3,5,5,5,5}
Red {1,1,5,8,8,10}: Green {2,2,4,4,7,9}: Blue: {3,3,6,6,6,6}
Red {1,1,5,5,8,8}: Green {2,2,4,4,7,7}: Blue: {3,3,6,6,6,6}
Red {1,1,5,9,11,11}: Green {2,2,4,6,8,10}: Blue: {3,3,7,7,7,7}
Red {1,1,3,8,8,8}: Green {2,2,5,5,7,7}: Blue: {4,4,6,6,6,6}
Red {4,4,4,4,7,7}: Green {1,1,3,6,9,11}: Blue: {2,2,5,8,8,10}
Red {4,4,4,4,4,9}: Green {1,1,3,6,8,11}: Blue: {2,2,5,7,7,10}
Red {2,5,5,8,8,8}: Green {1,1,4,7,10,12}: Blue: {3,3,6,9,9,11}
Red {2,5,5,5,8,11}: Green {1,1,4,7,10,13}: Blue: {3,3,6,9,9,12}
Red {2,5,5,5,5,12}: Green {1,1,4,7,9,11}: Blue: {3,3,6,8,8,10}
Red {2,2,7,7,7,7}: Green {1,1,4,6,9,11}: Blue: {3,3,5,8,8,10}
Red {2,2,5,8,8,11}: Green {1,1,4,7,10,13}: Blue: {3,3,6,9,9,12}
Red {2,2,5,5,10,10}: Green {1,1,4,7,9,12}: Blue: {3,3,6,8,8,11}
Red {2,2,5,5,8,13}: Green {1,1,4,7,10,12}: Blue: {3,3,6,9,9,11}
Red {2,2,2,7,10,10}: Green {1,1,4,6,9,12}: Blue: {3,3,5,8,8,11}
Red {2,2,2,7,7,12}: Green {1,1,4,6,9,11}: Blue: {3,3,5,8,8,10}
Red {2,2,2,5,10,13}: Green {1,1,4,7,9,12}: Blue: {3,3,6,8,8,11}
Red {2,2,2,2,11,11}: Green {1,1,4,6,8,10}: Blue: {3,3,5,7,7,9}
Red {1,7,7,7,7,7}: Green {2,2,4,6,9,11}: Blue: {3,3,5,8,8,10}
Red {1,5,8,8,8,11}: Green {2,2,4,7,10,13}: Blue: {3,3,6,9,9,12}
Red {1,5,5,8,11,11}: Green {2,2,4,7,10,13}: Blue: {3,3,6,9,9,12}
Red {1,5,5,8,8,13}: Green {2,2,4,7,10,12}: Blue: {3,3,6,9,9,11}
Red {1,5,5,5,10,13}: Green {2,2,4,7,9,12}: Blue: {3,3,6,8,8,11}
Red {1,3,8,8,11,11}: Green {2,2,5,7,10,13}: Blue: {4,4,6,9,9,12}
Red {1,3,8,8,8,13}: Green {2,2,5,7,10,12}: Blue: {4,4,6,9,9,11}
Red {1,3,6,11,11,11}: Green {2,2,5,8,10,13}: Blue: {4,4,7,9,9,12}
Red {1,3,6,6,13,13}: Green {2,2,5,8,10,12}: Blue: {4,4,7,9,9,11}
Red {1,3,3,10,10,13}: Green {2,2,5,7,9,12}: Blue: {4,4,6,8,8,11}
Red {1,3,3,8,13,13}: Green {2,2,5,7,10,12}: Blue: {4,4,6,9,9,11}
Red {1,1,9,9,9,9}: Green {2,2,4,6,8,11}: Blue: {3,3,5,7,7,10}
Red {1,1,7,10,10,13}: Green {2,2,4,6,9,12}: Blue: {3,3,5,8,8,11}
Red {1,1,7,7,12,12}: Green {2,2,4,6,9,11}: Blue: {3,3,5,8,8,10}
Red {1,1,5,10,13,13}: Green {2,2,4,7,9,12}: Blue: {3,3,6,8,8,11}
Red {1,1,3,12,12,12}: Green {2,2,5,7,9,11}: Blue: {4,4,6,8,8,10}
Red {4,4,4,4,7,7}: Green {1,1,3,6,8,10}: Blue: {2,2,5,5,9,9}
Red {4,4,4,4,4,7}: Green {1,1,3,6,6,9}: Blue: {2,2,5,5,8,8}
Red {2,5,5,8,8,8}: Green {1,1,4,7,9,11}: Blue: {3,3,6,6,10,10}
Red {2,5,5,5,8,10}: Green {1,1,4,7,9,12}: Blue: {3,3,6,6,11,11}
Red {2,5,5,5,5,10}: Green {1,1,4,7,7,9}: Blue: {3,3,6,6,8,8}
Red {2,2,7,7,7,7}: Green {1,1,4,6,8,10}: Blue: {3,3,5,5,9,9}
Red {2,2,5,8,8,10}: Green {1,1,4,7,9,12}: Blue: {3,3,6,6,11,11}
Red {2,2,5,5,8,8}: Green {1,1,4,7,7,10}: Blue: {3,3,6,6,9,9}
Red {2,2,5,5,8,12}: Green {1,1,4,7,9,11}: Blue: {3,3,6,6,10,10}
Red {2,2,2,7,9,9}: Green {1,1,4,6,8,11}: Blue: {3,3,5,5,10,10}
Red {2,2,2,7,7,11}: Green {1,1,4,6,8,10}: Blue: {3,3,5,5,9,9}
Red {2,2,2,5,8,11}: Green {1,1,4,7,7,10}: Blue: {3,3,6,6,9,9}
Red {2,2,2,2,9,9}: Green {1,1,4,6,6,8}: Blue: {3,3,5,5,7,7}
Red {2,6,6,6,9,9}: Green {1,3,5,8,8,11}: Blue: {4,4,7,7,10,10}
Red {2,4,7,7,10,13}: Green {1,3,6,9,9,12}: Blue: {5,5,8,8,11,11}
Red {2,4,4,9,9,9}: Green {1,3,6,8,8,11}: Blue: {5,5,7,7,10,10}
Red {2,4,4,7,12,12}: Green {1,3,6,9,9,11}: Blue: {5,5,8,8,10,10}
Red {2,2,6,9,9,9}: Green {1,3,5,8,8,11}: Blue: {4,4,7,7,10,10}
Red {2,2,6,6,11,11}: Green {1,3,5,8,8,10}: Blue: {4,4,7,7,9,9}
Red {2,2,4,9,9,12}: Green {1,3,6,8,8,11}: Blue: {5,5,7,7,10,10}
Red {2,2,2,8,11,11}: Green {1,3,5,7,7,10}: Blue: {4,4,6,6,9,9}
Red {1,7,7,7,7,7}: Green {2,2,4,6,8,10}: Blue: {3,3,5,5,9,9}
Red {1,5,8,8,8,10}: Green {2,2,4,7,9,12}: Blue: {3,3,6,6,11,11}
Red {1,5,5,8,10,10}: Green {2,2,4,7,9,12}: Blue: {3,3,6,6,11,11}
Red {1,5,5,8,8,12}: Green {2,2,4,7,9,11}: Blue: {3,3,6,6,10,10}
Red {1,5,5,5,8,11}: Green {2,2,4,7,7,10}: Blue: {3,3,6,6,9,9}
Red {1,3,8,8,10,10}: Green {2,2,5,7,9,12}: Blue: {4,4,6,6,11,11}
Red {1,3,8,8,8,12}: Green {2,2,5,7,9,11}: Blue: {4,4,6,6,10,10}
Red {1,3,6,9,9,9}: Green {2,2,5,8,8,11}: Blue: {4,4,7,7,10,10}
Red {1,3,6,6,11,11}: Green {2,2,5,8,8,10}: Blue: {4,4,7,7,9,9}
Red {1,3,3,8,8,11}: Green {2,2,5,7,7,10}: Blue: {4,4,6,6,9,9}
Red {1,3,3,8,12,12}: Green {2,2,5,7,9,11}: Blue: {4,4,6,6,10,10}
Red {1,3,7,10,10,13}: Green {2,4,6,9,9,12}: Blue: {5,5,8,8,11,11}
Red {1,3,5,10,13,13}: Green {2,4,7,9,9,12}: Blue: {6,6,8,8,11,11}
Red {1,3,3,11,11,11}: Green {2,4,6,8,8,10}: Blue: {5,5,7,7,9,9}
Red {1,1,7,7,7,7}: Green {2,2,4,6,6,9}: Blue: {3,3,5,5,8,8}
Red {1,1,7,9,9,12}: Green {2,2,4,6,8,11}: Blue: {3,3,5,5,10,10}
Red {1,1,7,7,11,11}: Green {2,2,4,6,8,10}: Blue: {3,3,5,5,9,9}
Red {1,1,5,8,11,11}: Green {2,2,4,7,7,10}: Blue: {3,3,6,6,9,9}
Red {1,1,3,10,10,10}: Green {2,2,5,7,7,9}: Blue: {4,4,6,6,8,8}
Red {4,4,4,4,7,7}: Green {1,1,3,6,9,9}: Blue: {2,2,5,5,8,10}
Red {4,4,4,4,4,9}: Green {1,1,3,6,8,10}: Blue: {2,2,5,5,7,11}
Red {2,5,5,8,8,8}: Green {1,1,4,7,10,10}: Blue: {3,3,6,6,9,11}
Red {2,5,5,5,8,11}: Green {1,1,4,7,10,12}: Blue: {3,3,6,6,9,13}
Red {2,5,5,5,5,10}: Green {1,1,4,7,9,9}: Blue: {3,3,6,6,8,11}
Red {2,2,7,7,7,7}: Green {1,1,4,6,9,9}: Blue: {3,3,5,5,8,10}
Red {2,2,5,8,8,11}: Green {1,1,4,7,10,12}: Blue: {3,3,6,6,9,13}
Red {2,2,5,5,10,10}: Green {1,1,4,7,9,11}: Blue: {3,3,6,6,8,12}
Red {2,2,5,5,8,11}: Green {1,1,4,7,10,10}: Blue: {3,3,6,6,9,12}
Red {2,2,2,7,10,10}: Green {1,1,4,6,9,11}: Blue: {3,3,5,5,8,12}
Red {2,2,2,7,7,10}: Green {1,1,4,6,9,9}: Blue: {3,3,5,5,8,11}
Red {2,2,2,5,10,12}: Green {1,1,4,7,9,11}: Blue: {3,3,6,6,8,13}
Red {2,2,2,2,9,9}: Green {1,1,4,6,8,8}: Blue: {3,3,5,5,7,10}
Red {2,6,9,9,9,9}: Green {1,3,5,8,11,11}: Blue: {4,4,7,7,10,12}
Red {2,6,6,6,9,12}: Green {1,3,5,8,11,11}: Blue: {4,4,7,7,10,13}
Red {2,4,7,10,10,13}: Green {1,3,6,9,12,12}: Blue: {5,5,8,8,11,14}
Red {2,4,4,7,12,12}: Green {1,3,6,9,11,11}: Blue: {5,5,8,8,10,13}
Red {2,2,8,8,8,11}: Green {1,3,5,7,10,10}: Blue: {4,4,6,6,9,12}
Red {2,2,6,6,11,11}: Green {1,3,5,8,10,10}: Blue: {4,4,7,7,9,12}
Red {2,2,4,9,12,12}: Green {1,3,6,8,11,11}: Blue: {5,5,7,7,10,13}
Red {1,7,7,7,7,7}: Green {2,2,4,6,9,9}: Blue: {3,3,5,5,8,10}
Red {1,5,8,8,8,11}: Green {2,2,4,7,10,12}: Blue: {3,3,6,6,9,13}
Red {1,5,5,8,11,11}: Green {2,2,4,7,10,12}: Blue: {3,3,6,6,9,13}
Red {1,5,5,8,8,11}: Green {2,2,4,7,10,10}: Blue: {3,3,6,6,9,12}
Red {1,5,5,5,10,12}: Green {2,2,4,7,9,11}: Blue: {3,3,6,6,8,13}
Red {1,3,8,8,11,11}: Green {2,2,5,7,10,12}: Blue: {4,4,6,6,9,13}
Red {1,3,8,8,8,11}: Green {2,2,5,7,10,10}: Blue: {4,4,6,6,9,12}
Red {1,3,6,11,11,11}: Green {2,2,5,8,10,12}: Blue: {4,4,7,7,9,13}
Red {1,3,6,6,11,11}: Green {2,2,5,8,10,10}: Blue: {4,4,7,7,9,12}
Red {1,3,3,10,10,12}: Green {2,2,5,7,9,11}: Blue: {4,4,6,6,8,13}
Red {1,3,3,8,11,11}: Green {2,2,5,7,10,10}: Blue: {4,4,6,6,9,12}
Red {1,3,7,10,13,13}: Green {2,4,6,9,12,12}: Blue: {5,5,8,8,11,14}
Red {1,3,3,11,11,11}: Green {2,4,6,8,10,10}: Blue: {5,5,7,7,9,12}
Red {1,1,9,9,9,9}: Green {2,2,4,6,8,10}: Blue: {3,3,5,5,7,11}
Red {1,1,7,10,10,12}: Green {2,2,4,6,9,11}: Blue: {3,3,5,5,8,13}
Red {1,1,7,7,10,10}: Green {2,2,4,6,9,9}: Blue: {3,3,5,5,8,11}
Red {1,1,5,10,12,12}: Green {2,2,4,7,9,11}: Blue: {3,3,6,6,8,13}
Red {1,1,3,10,10,10}: Green {2,2,5,7,9,9}: Blue: {4,4,6,6,8,11}
Red {4,4,4,4,6,6}: Green {1,1,3,5,8,10}: Blue: {2,2,2,7,9,9}
Red {4,4,4,4,4,8}: Green {1,1,3,5,7,10}: Blue: {2,2,2,6,9,9}
Red {2,5,5,7,7,7}: Green {1,1,4,6,9,11}: Blue: {3,3,3,8,10,10}
Red {2,5,5,5,7,10}: Green {1,1,4,6,9,12}: Blue: {3,3,3,8,11,11}
Red {2,5,5,5,5,11}: Green {1,1,4,6,8,10}: Blue: {3,3,3,7,9,9}
Red {2,2,5,5,5,5}: Green {1,1,4,4,7,9}: Blue: {3,3,3,6,8,8}
Red {2,2,5,7,7,10}: Green {1,1,4,6,9,12}: Blue: {3,3,3,8,11,11}
Red {2,2,5,5,9,9}: Green {1,1,4,6,8,11}: Blue: {3,3,3,7,10,10}
Red {2,2,5,5,7,12}: Green {1,1,4,6,9,11}: Blue: {3,3,3,8,10,10}
Red {2,2,2,5,8,8}: Green {1,1,4,4,7,10}: Blue: {3,3,3,6,9,9}
Red {2,2,2,5,5,10}: Green {1,1,4,4,7,9}: Blue: {3,3,3,6,8,8}
Red {2,2,2,5,9,12}: Green {1,1,4,6,8,11}: Blue: {3,3,3,7,10,10}
Red {2,2,2,2,9,9}: Green {1,1,4,4,6,8}: Blue: {3,3,3,5,7,7}
Red {2,4,7,7,7,10}: Green {1,3,6,6,9,12}: Blue: {5,5,5,8,11,11}
Red {2,4,4,9,9,9}: Green {1,3,6,6,8,11}: Blue: {5,5,5,7,10,10}
Red {2,4,4,7,10,13}: Green {1,3,6,6,9,12}: Blue: {5,5,5,8,11,11}
Red {2,2,6,6,9,9}: Green {1,3,5,5,8,11}: Blue: {4,4,4,7,10,10}
Red {2,2,6,6,6,11}: Green {1,3,5,5,8,10}: Blue: {4,4,4,7,9,9}
Red {2,2,4,9,9,12}: Green {1,3,6,6,8,11}: Blue: {5,5,5,7,10,10}
Red {2,2,4,7,12,12}: Green {1,3,6,6,9,11}: Blue: {5,5,5,8,10,10}
Red {2,2,2,8,11,11}: Green {1,3,5,5,7,10}: Blue: {4,4,4,6,9,9}
Red {1,5,5,5,5,5}: Green {2,2,4,4,7,9}: Blue: {3,3,3,6,8,8}
Red {1,5,7,7,7,10}: Green {2,2,4,6,9,12}: Blue: {3,3,3,8,11,11}
Red {1,5,5,7,10,10}: Green {2,2,4,6,9,12}: Blue: {3,3,3,8,11,11}
Red {1,5,5,7,7,12}: Green {2,2,4,6,9,11}: Blue: {3,3,3,8,10,10}
Red {1,5,5,5,9,12}: Green {2,2,4,6,8,11}: Blue: {3,3,3,7,10,10}
Red {1,3,6,6,9,9}: Green {2,2,5,5,8,11}: Blue: {4,4,4,7,10,10}
Red {1,3,6,6,6,11}: Green {2,2,5,5,8,10}: Blue: {4,4,4,7,9,9}
Red {1,3,6,10,10,10}: Green {2,2,5,7,9,12}: Blue: {4,4,4,8,11,11}
Red {1,3,6,6,12,12}: Green {2,2,5,7,9,11}: Blue: {4,4,4,8,10,10}
Red {1,3,3,8,8,11}: Green {2,2,5,5,7,10}: Blue: {4,4,4,6,9,9}
Red {1,3,3,6,11,11}: Green {2,2,5,5,8,10}: Blue: {4,4,4,7,9,9}
Red {1,3,7,10,10,10}: Green {2,4,6,6,9,12}: Blue: {5,5,5,8,11,11}
Red {1,3,7,7,10,13}: Green {2,4,6,6,9,12}: Blue: {5,5,5,8,11,11}
Red {1,3,5,10,13,13}: Green {2,4,7,7,9,12}: Blue: {6,6,6,8,11,11}
Red {1,3,3,11,11,11}: Green {2,4,6,6,8,10}: Blue: {5,5,5,7,9,9}
Red {1,1,7,7,7,7}: Green {2,2,4,4,6,9}: Blue: {3,3,3,5,8,8}
Red {1,1,5,8,8,11}: Green {2,2,4,4,7,10}: Blue: {3,3,3,6,9,9}
Red {1,1,5,5,10,10}: Green {2,2,4,4,7,9}: Blue: {3,3,3,6,8,8}
Red {1,1,5,9,12,12}: Green {2,2,4,6,8,11}: Blue: {3,3,3,7,10,10}
Red {1,1,3,10,10,10}: Green {2,2,5,5,7,9}: Blue: {4,4,4,6,8,8}
Red {4,4,4,4,6,6}: Green {1,1,3,5,8,8}: Blue: {2,2,2,7,7,9}
Red {4,4,4,4,4,8}: Green {1,1,3,5,7,9}: Blue: {2,2,2,6,6,10}
Red {2,5,5,7,7,7}: Green {1,1,4,6,9,9}: Blue: {3,3,3,8,8,10}
Red {2,5,5,5,7,10}: Green {1,1,4,6,9,11}: Blue: {3,3,3,8,8,12}
Red {2,5,5,5,5,9}: Green {1,1,4,6,8,8}: Blue: {3,3,3,7,7,10}
Red {2,2,5,5,5,5}: Green {1,1,4,4,7,7}: Blue: {3,3,3,6,6,8}
Red {2,2,5,7,7,10}: Green {1,1,4,6,9,11}: Blue: {3,3,3,8,8,12}
Red {2,2,5,5,9,9}: Green {1,1,4,6,8,10}: Blue: {3,3,3,7,7,11}
Red {2,2,5,5,7,10}: Green {1,1,4,6,9,9}: Blue: {3,3,3,8,8,11}
Red {2,2,2,5,8,8}: Green {1,1,4,4,7,9}: Blue: {3,3,3,6,6,10}
Red {2,2,2,5,5,8}: Green {1,1,4,4,7,7}: Blue: {3,3,3,6,6,9}
Red {2,2,2,5,9,11}: Green {1,1,4,6,8,10}: Blue: {3,3,3,7,7,12}
Red {2,2,2,2,7,7}: Green {1,1,4,4,6,6}: Blue: {3,3,3,5,5,8}
Red {2,6,8,8,8,8}: Green {1,3,5,7,10,10}: Blue: {4,4,4,9,9,11}
Red {2,6,6,6,8,11}: Green {1,3,5,7,10,10}: Blue: {4,4,4,9,9,12}
Red {2,4,7,7,7,10}: Green {1,3,6,6,9,11}: Blue: {5,5,5,8,8,12}
Red {2,4,7,9,9,12}: Green {1,3,6,8,11,11}: Blue: {5,5,5,10,10,13}
Red {2,4,4,9,9,9}: Green {1,3,6,6,8,10}: Blue: {5,5,5,7,7,11}
Red {2,4,4,7,10,12}: Green {1,3,6,6,9,11}: Blue: {5,5,5,8,8,13}
Red {2,4,4,7,11,11}: Green {1,3,6,8,10,10}: Blue: {5,5,5,9,9,12}
Red {2,2,6,6,9,9}: Green {1,3,5,5,8,10}: Blue: {4,4,4,7,7,11}
Red {2,2,6,6,6,9}: Green {1,3,5,5,8,8}: Blue: {4,4,4,7,7,10}
Red {2,2,6,6,10,10}: Green {1,3,5,7,9,9}: Blue: {4,4,4,8,8,11}
Red {2,2,4,9,9,11}: Green {1,3,6,6,8,10}: Blue: {5,5,5,7,7,12}
Red {2,2,4,7,10,10}: Green {1,3,6,6,9,9}: Blue: {5,5,5,8,8,11}
Red {2,2,2,8,10,10}: Green {1,3,5,5,7,9}: Blue: {4,4,4,6,6,11}
Red {1,5,5,5,5,5}: Green {2,2,4,4,7,7}: Blue: {3,3,3,6,6,8}
Red {1,5,7,7,7,10}: Green {2,2,4,6,9,11}: Blue: {3,3,3,8,8,12}
Red {1,5,5,7,10,10}: Green {2,2,4,6,9,11}: Blue: {3,3,3,8,8,12}
Red {1,5,5,7,7,10}: Green {2,2,4,6,9,9}: Blue: {3,3,3,8,8,11}
Red {1,5,5,5,9,11}: Green {2,2,4,6,8,10}: Blue: {3,3,3,7,7,12}
Red {1,3,6,6,9,9}: Green {2,2,5,5,8,10}: Blue: {4,4,4,7,7,11}
Red {1,3,6,6,6,9}: Green {2,2,5,5,8,8}: Blue: {4,4,4,7,7,10}
Red {1,3,6,10,10,10}: Green {2,2,5,7,9,11}: Blue: {4,4,4,8,8,12}
Red {1,3,6,6,10,10}: Green {2,2,5,7,9,9}: Blue: {4,4,4,8,8,11}
Red {1,3,3,8,8,10}: Green {2,2,5,5,7,9}: Blue: {4,4,4,6,6,11}
Red {1,3,3,6,9,9}: Green {2,2,5,5,8,8}: Blue: {4,4,4,7,7,10}
Red {1,3,7,10,10,10}: Green {2,4,6,6,9,11}: Blue: {5,5,5,8,8,12}
Red {1,3,7,7,10,12}: Green {2,4,6,6,9,11}: Blue: {5,5,5,8,8,13}
Red {1,3,7,9,12,12}: Green {2,4,6,8,11,11}: Blue: {5,5,5,10,10,13}
Red {1,3,5,10,12,12}: Green {2,4,7,7,9,11}: Blue: {6,6,6,8,8,13}
Red {1,3,3,9,9,9}: Green {2,4,6,6,8,8}: Blue: {5,5,5,7,7,10}
Red {1,1,7,7,7,7}: Green {2,2,4,4,6,8}: Blue: {3,3,3,5,5,9}
Red {1,1,5,8,8,10}: Green {2,2,4,4,7,9}: Blue: {3,3,3,6,6,11}
Red {1,1,5,5,8,8}: Green {2,2,4,4,7,7}: Blue: {3,3,3,6,6,9}
Red {1,1,5,9,11,11}: Green {2,2,4,6,8,10}: Blue: {3,3,3,7,7,12}
Red {1,1,3,8,8,8}: Green {2,2,5,5,7,7}: Blue: {4,4,4,6,6,9}
Red {4,4,4,4,7,7}: Green {1,1,3,6,8,10}: Blue: {2,2,2,5,9,11}
Red {4,4,4,4,4,7}: Green {1,1,3,6,6,9}: Blue: {2,2,2,5,8,10}
Red {2,5,5,8,8,8}: Green {1,1,4,7,9,11}: Blue: {3,3,3,6,10,12}
Red {2,5,5,5,8,10}: Green {1,1,4,7,9,12}: Blue: {3,3,3,6,11,13}
Red {2,5,5,5,5,10}: Green {1,1,4,7,7,9}: Blue: {3,3,3,6,8,11}
Red {2,2,7,7,7,7}: Green {1,1,4,6,8,10}: Blue: {3,3,3,5,9,11}
Red {2,2,5,8,8,10}: Green {1,1,4,7,9,12}: Blue: {3,3,3,6,11,13}
Red {2,2,5,5,8,8}: Green {1,1,4,7,7,10}: Blue: {3,3,3,6,9,11}
Red {2,2,5,5,8,12}: Green {1,1,4,7,9,11}: Blue: {3,3,3,6,10,13}
Red {2,2,2,7,9,9}: Green {1,1,4,6,8,11}: Blue: {3,3,3,5,10,12}
Red {2,2,2,7,7,11}: Green {1,1,4,6,8,10}: Blue: {3,3,3,5,9,12}
Red {2,2,2,5,8,11}: Green {1,1,4,7,7,10}: Blue: {3,3,3,6,9,12}
Red {2,2,2,2,9,9}: Green {1,1,4,6,6,8}: Blue: {3,3,3,5,7,10}
Red {2,6,6,6,9,9}: Green {1,3,5,8,8,11}: Blue: {4,4,4,7,10,12}
Red {2,4,7,7,10,13}: Green {1,3,6,9,9,12}: Blue: {5,5,5,8,11,14}
Red {2,4,4,9,9,9}: Green {1,3,6,8,8,11}: Blue: {5,5,5,7,10,12}
Red {2,4,4,7,12,12}: Green {1,3,6,9,9,11}: Blue: {5,5,5,8,10,13}
Red {2,2,6,9,9,9}: Green {1,3,5,8,8,11}: Blue: {4,4,4,7,10,12}
Red {2,2,6,6,11,11}: Green {1,3,5,8,8,10}: Blue: {4,4,4,7,9,12}
Red {2,2,4,9,9,12}: Green {1,3,6,8,8,11}: Blue: {5,5,5,7,10,13}
Red {2,2,2,8,11,11}: Green {1,3,5,7,7,10}: Blue: {4,4,4,6,9,12}
Red {1,7,7,7,7,7}: Green {2,2,4,6,8,10}: Blue: {3,3,3,5,9,11}
Red {1,5,8,8,8,10}: Green {2,2,4,7,9,12}: Blue: {3,3,3,6,11,13}
Red {1,5,5,8,10,10}: Green {2,2,4,7,9,12}: Blue: {3,3,3,6,11,13}
Red {1,5,5,8,8,12}: Green {2,2,4,7,9,11}: Blue: {3,3,3,6,10,13}
Red {1,5,5,5,8,11}: Green {2,2,4,7,7,10}: Blue: {3,3,3,6,9,12}
Red {1,3,8,8,10,10}: Green {2,2,5,7,9,12}: Blue: {4,4,4,6,11,13}
Red {1,3,8,8,8,12}: Green {2,2,5,7,9,11}: Blue: {4,4,4,6,10,13}
Red {1,3,6,9,9,9}: Green {2,2,5,8,8,11}: Blue: {4,4,4,7,10,12}
Red {1,3,6,6,11,11}: Green {2,2,5,8,8,10}: Blue: {4,4,4,7,9,12}
Red {1,3,3,8,8,11}: Green {2,2,5,7,7,10}: Blue: {4,4,4,6,9,12}
Red {1,3,3,8,12,12}: Green {2,2,5,7,9,11}: Blue: {4,4,4,6,10,13}
Red {1,3,7,10,10,13}: Green {2,4,6,9,9,12}: Blue: {5,5,5,8,11,14}
Red {1,3,5,10,13,13}: Green {2,4,7,9,9,12}: Blue: {6,6,6,8,11,14}
Red {1,3,3,11,11,11}: Green {2,4,6,8,8,10}: Blue: {5,5,5,7,9,12}
Red {1,1,7,7,7,7}: Green {2,2,4,6,6,9}: Blue: {3,3,3,5,8,10}
Red {1,1,7,9,9,12}: Green {2,2,4,6,8,11}: Blue: {3,3,3,5,10,13}
Red {1,1,7,7,11,11}: Green {2,2,4,6,8,10}: Blue: {3,3,3,5,9,12}
Red {1,1,5,8,11,11}: Green {2,2,4,7,7,10}: Blue: {3,3,3,6,9,12}
Red {1,1,3,10,10,10}: Green {2,2,5,7,7,9}: Blue: {4,4,4,6,8,11}
Red {4,4,4,4,6,6}: Green {1,1,3,5,7,7}: Blue: {2,2,2,2,8,8}
Red {4,4,4,4,4,6}: Green {1,1,3,5,5,7}: Blue: {2,2,2,2,8,8}
Red {2,5,5,7,7,7}: Green {1,1,4,6,8,8}: Blue: {3,3,3,3,9,9}
Red {2,5,5,5,7,9}: Green {1,1,4,6,8,10}: Blue: {3,3,3,3,11,11}
Red {2,5,5,5,5,7}: Green {1,1,4,6,6,6}: Blue: {3,3,3,3,8,8}
Red {2,2,5,5,5,5}: Green {1,1,4,4,6,6}: Blue: {3,3,3,3,7,7}
Red {2,2,5,7,7,9}: Green {1,1,4,6,8,10}: Blue: {3,3,3,3,11,11}
Red {2,2,5,5,7,7}: Green {1,1,4,6,6,8}: Blue: {3,3,3,3,9,9}
Red {2,2,5,5,7,9}: Green {1,1,4,6,8,8}: Blue: {3,3,3,3,10,10}
Red {2,2,2,5,7,7}: Green {1,1,4,4,6,8}: Blue: {3,3,3,3,9,9}
Red {2,2,2,5,5,7}: Green {1,1,4,4,6,6}: Blue: {3,3,3,3,8,8}
Red {2,2,2,5,7,9}: Green {1,1,4,6,6,8}: Blue: {3,3,3,3,10,10}
Red {2,2,2,2,5,5}: Green {1,1,4,4,4,4}: Blue: {3,3,3,3,6,6}
Red {2,6,8,8,8,8}: Green {1,3,5,7,9,9}: Blue: {4,4,4,4,10,10}
Red {2,6,6,6,8,8}: Green {1,3,5,7,7,9}: Blue: {4,4,4,4,10,10}
Red {2,6,6,6,8,10}: Green {1,3,5,7,9,9}: Blue: {4,4,4,4,11,11}


I've just cut the list down a bit because it was probably going to get annoying.

[Edited by Sam Finch on 10-04-2000 at 04:57 AM]

[HarbringerOfVole]

Here is an interesting solution:
Green 2,2,2,5,8,8
Blue 3,3,3,3,7,7
Red 1,4,4,4,4,6

You may notice thatthe odds fof green against red are 5 to 4 in favour of green, with all the other restrctions being met.
On another note, which colour (using the above set) would you chose if 2 dice of each colour were rolled? Will the odds be the same, or different? If differnet, is it possible to make dice that conform the rules of the problem, and have the 5:4 green:red odds, that do keep the same odds? If so what about 3 dice? Or N dice? Might keep you occupied....

[Guv]

Sam & Harbringer, your answers seem to conform to the original rules. I got a little confused while checking odds on the three possible die combinations for several sets of dice. I thought that maybe some colors might have to be reassigned to be exactly in conformance, but at least two out of 3 games showed 5 to 4 odds as required (the basic idea in the original description of the dice).

The interesting characteristic of my 3 dice is the lack of transitivity. Illustrating this property of probability was the reason I posted. Nobody seems to think this is strange?

I only checked a few of Sam's Dice. The ones I checked did not display this characteristic.

Harbringer, unless I got the colors confused, it looks as though your red die beats both of the others, and the blue die loses to both of the others.

It looks as though the original conditions did not provide enough information to determine the odds for the third pair of dice. I suspected this to be the case, but never checked it out. You fellows answered that question for me.

[KWell]

Here is all the solution if the dices have numbers between 1 and 6

Blue:114444 Green:222225 Red:333366
Blue:114444 Green:222225=20,16
Red:333366 Blue:114444=20,16
Red:333366 Green:222225=32,4

Blue:114444 Green:222226 Red:333355
Blue:114444 Green:222226=20,16
Red:333355 Blue:114444=20,16
Red:333355 Green:222226=30,6

Blue:114444 Green:333335 Red:222266
Blue:114444 Green:333335=20,16
Red:222266 Blue:114444=20,16
Red:222266 Green:333335=12,24

Blue:114444 Green:333336 Red:222255
Blue:114444 Green:333336=20,16
Red:222255 Blue:114444=20,16
Red:222255 Green:333336=10,26

Blue:333366 Green:114444 Red:255555
Blue:333366 Green:114444=20,16
Red:255555 Blue:333366=20,16
Red:255555 Green:114444=32,4

Blue:333366 Green:115555 Red:244444
Blue:333366 Green:115555=20,16
Red:244444 Blue:333366=20,16
Red:244444 Green:115555=12,24

Blue:333366 Green:224444 Red:155555
Blue:333366 Green:224444=20,16
Red:155555 Blue:333366=20,16
Red:155555 Green:224444=30,6

Blue:333366 Green:225555 Red:144444
Blue:333366 Green:225555=20,16
Red:144444 Blue:333366=20,16
Red:144444 Green:225555=10,26

[HarbringerOfVole]

To simply show odds:
In order for the odds to be 5:4, then out of 36 gams, the winning dice must win 20, the losing dice must win 16,

So,
Green 2,2,2,5,8,8
Blue 3,3,3,3,7,7
Red 1,4,4,4,4,6

In orderf for the odds to be 5:4, then out of 36 games a die must win 20, so the other die must win 16 (no ties.)

So, blue against green
If blue rolls a 3, it beats the 2, not the 5, or the 8, so it wins 3 games for each 3(there are 4 3s), so 12 wins.
If it rolls a 7, it beats the 2's and the 5, 4 games for each 7,(there are 2 7s) so 8 wins.
This gives a total of 20 wins out of 36(it is easier to see with a probability diagrem, go draw one if your interested!)
From the other point of view, to test no maths errors,
Green against blue
If green rolls a two, dosn't beat anything, no wins.
If green rolls a 5, it beats the 3's, so 4 wins.
If green rolls an 8, it beats all the blues numbers, so 6*2 wins, 12.
This gives 16 wins.
There are no ties. 20:16 wins, in the favour of blue, divide by 4, what do you get??????????????
This clearly gives 5:4 odds!!!!!!!!!

Okay, lets take green and red, from reds side first.
Red rolls a 1, 0 wins.
Red rolls a 4, beats the 2s, three wins per 4, 12 wins.
Red rolls a 6, beats the 2s and the 5, 4 wins per 6, 4 wins.
4+12=16.
Want me to do it form th e other perpesective?????? I will.
Green rolls a 2, beats the 1, 1 win per 2, 3 wins.
Green rolls a 5, beats the 4s and the 1, 5 wins per 5, 5 wins.
Green rolls an 8, beats the 1's the 4's and the 6, 6 wins per 8, 12 wins.
12+5+3=20.
No ties.
20:16 to green, divide by 4, 5:4.
How hard is that???????????????

To really labour the point, I will do blue and red.
Blue rolls a 3, beats the 1, 1 win per 3, 4 wins.
Blue rolls a 7, beta the 1,4s and the 6, 6 wins per 7, 12 wins.
12+4=16.
Red rolls a 1, beats nada, 0 wins.
Red rolls a 4, beats the 3s, 4 wins per 4, 16 wins.
Red rolls a 6, beats the 3s, 4 wins per 6, 4 wins.
16+4=20
No ties.
20:16 odds in favour of red, divide by 4, 5:4 odds.

So, blue beats green, green beats red, and red beats blue, as stated, all in a 5:4 ratio, so though you may know a lot about complex math, you don't seem to be very good at simple arithmitic!(No offence, my friend studying math at Cambridge
has problems adding two small integers together!! Dosn't seem to affect his math ability to adversly though!)

[Guv]

Harbringer, you are correct. I should never have tried to count the wins & losses in my head.

You did not have to go to all that trouble to convince me. Either a vicious accusation of error or a friendly comment, would have convinced me to recheck.

[HarbringerOfVole]

Sorry about that, got a little carried away! :)
Normally I am quite polite!

[Guv]

Harbringer, I did not mean to imply that you were impolite. Your post was not interpreted as a putdown. I was merely trying to inject a little humor into a suggestion that you could have convinced me that I had made a mistake without showing all the details.

HarbringerOfVole jogs something in my memory, but I can not conjure up any data. Is this the name of some historical, fictional, or mythical person?

I keep checking your posts for the spelling, because my memory wants me to type "Harbinger" (no "r" in 5th character), and suggests that this part is an archaic profession or trade.

[HarbringerOfVole]

Guv,
As my spelling is rather bad, I would imagine that the correct spelling is 'Harbinger', and that would fit better with the pronuciation.
The word Harbinger(or Harbringer??) means, if I remember correctly (no dictionary), a fortelling, esp. of doom or destruction, so it is like an omen, I don't think there was actually a Harbinger of Vole, (there could be, but I independantly came up with it, or subconciosly copied it!).
Id imagine that 'Guv' is spelt correctly though!

[Guv]

Harbringer, of course Guv is spelled correctly. My real name is Gouverneur. Isn't "Guv" the way all Americans would Anglicize this name? Some Britishers, too. Don't they say "guvn'r" or some such word?

For those of us who do not catch on quickly, why not say "Famous last words: They could not hit an elephant at this dist ..."?