Calculating the pass-line bet in
craps and crapless craps
Marty Wollner
SpikerSystems
Discflicker.com
4/22/2009 6:14:36 AM
1) Calculating the pass-line bet in crapless craps
The ways table for two dice is:
Point = 2 Ways = 1 1-1
Point = 3 Ways = 2 1-2, 2-1
Point = 4 Ways = 3 1-3, 2-2, 3-1
Point = 5 Ways = 4 1-4, 2-3, 3-2, 4-1
Point = 6 Ways = 5 1-5, 2-4, 3-3, 4-2, 5-1
Point = 7 Ways = 6 1-6, 2-5, 3-4, 4-3, 5-2, 6-1
Point = 8 Ways = 5 2-6, 3-5, 4-4, 5-3, 6-2
Point = 9 Ways = 4 3-6, 4-5, 5-4, 6-3
Point = 10 Ways = 3 4-6, 5-5, 6-4
Point = 11 Ways = 2 5-6, 6-5
Point = 12 Ways = 1 6-6
Note that the odds of hitting a seven are 6 out of 36 (6/36 or 1/6), because there are 36 total rolls, 6 of which are sevens; 1-6, 2-5, 3-4, 4-3, 5-2, 6-1
A pass line winner occurs when the seven is hit on the first roll (called the Comeout Roll), or if not, the number that comes is repeated before a seven.
Unlike traditional craps, in crapless craps, the 2,3,11,and 12 are simply treated as points, just like the 4,5,6,8,9,10.
Players never loose on the comeout roll; the only way to loose is comeout on a point, then throw a seven before hitting it. This is called a Seven-Out.
Calculate the pass line bet odds by summing up the frequency (f1) of all the winners, starting with the 2 - 6 points. These occur if the number is rolled,
and then repeated before a seven-out...
For the 2 point, f1 = 1 / 36 (the frequency of a 2 occuring) TIMES 1 / 7 (the odds of hitting the point ... 1 ways vs 6 ways to make a seven)
same as 1 / 252 ... = 3.96825396825397E-03
For the 3 point, f1 = 2 / 36 (the frequency of a 3 occuring) TIMES 2 / 8 (the odds of hitting the point ... 2 ways vs 6 ways to make a seven)
same as 4 / 288 ... = 1.38888888888889E-02
For the 4 point, f1 = 3 / 36 (the frequency of a 4 occuring) TIMES 3 / 9 (the odds of hitting the point ... 3 ways vs 6 ways to make a seven)
same as 9 / 324 ... = 2.77777777777778E-02
For the 5 point, f1 = 4 / 36 (the frequency of a 5 occuring) TIMES 4 / 10 (the odds of hitting the point ... 4 ways vs 6 ways to make a seven)
same as 16 / 360 ... = 4.44444444444444E-02
For the 6 point, f1 = 5 / 36 (the frequency of a 6 occuring) TIMES 5 / 11 (the odds of hitting the point ... 5 ways vs 6 ways to make a seven)
same as 25 / 396 ... = 6.31313131313131E-02
f1 sum total = .153210678210678
Multiply by 2 to add the upper numbers (8,9,10,11,12) = .306421356421356
The front line winners include the sevens (6/36 or 1/6), = .166666666666667
Adding the front line winners, f1 = .473088023088023
NEXT: calculate this by counting loosers..
2) Pass-line bet in crapless craps by counting loosing rolls
These occur if the number is rolled, and then a seven comes before it is repeated...
Calculate this by summing up the frequency (f1) of all the loosers, starting with the 2 - 6 points.
For the 2 point, f1 = 1 / 36 (the frequency of a 2 occuring) TIMES 6 / 7 ... the odds of hitting a seven (6 ways) vs the point ( 1 ways)
same as 6 / 252 ... = 2.38095238095238E-02
For the 3 point, f1 = 2 / 36 (the frequency of a 3 occuring) TIMES 6 / 8 ... the odds of hitting a seven (6 ways) vs the point ( 2 ways)
same as 12 / 288 ... = 4.16666666666667E-02
For the 4 point, f1 = 3 / 36 (the frequency of a 4 occuring) TIMES 6 / 9 ... the odds of hitting a seven (6 ways) vs the point ( 3 ways)
same as 18 / 324 ... = 5.55555555555556E-02
For the 5 point, f1 = 4 / 36 (the frequency of a 5 occuring) TIMES 6 / 10 ... the odds of hitting a seven (6 ways) vs the point ( 4 ways)
same as 24 / 360 ... = 6.66666666666667E-02
For the 6 point, f1 = 5 / 36 (the frequency of a 6 occuring) TIMES 6 / 11 ... the odds of hitting a seven (6 ways) vs the point ( 5 ways)
same as 30 / 396 ... = 7.57575757575758E-02
f1 sum total = .263455988455988
Multiply by 2 to add the upper numbers (8,9,10,11,12) = .526911976911977
Thats it.. don't bother with the front line winners, we're only counting loosers
So, we have .473088023088023 ways to win and .526911976911977 ways to loose.
They should total up, right? ... .473088023088023 + .526911976911977 = 1
NEXT: calculate this for traditional craps...
3) Pass-line bet in traditional craps by counting winning rolls
These occur if a 7 or 11 is thrown on the comeout roll, or, if not a 2, 3, or 12, by repeating the point number rolled before a seven.
Calculate this by summing up the frequency (f1) of all the winners, starting with the 4 - 6 points.
For the 4 point, f1 = 3 / 36 (the frequency of a 4 occuring) TIMES 3 / 9 (the odds of hitting the point ... 3 ways vs 6 ways to make a seven)
same as 9 / 324 ... = 2.77777777777778E-02
For the 5 point, f1 = 4 / 36 (the frequency of a 5 occuring) TIMES 4 / 10 (the odds of hitting the point ... 4 ways vs 6 ways to make a seven)
same as 16 / 360 ... = 4.44444444444444E-02
For the 6 point, f1 = 5 / 36 (the frequency of a 6 occuring) TIMES 5 / 11 (the odds of hitting the point ... 5 ways vs 6 ways to make a seven)
same as 25 / 396 ... = 6.31313131313131E-02
f1 sum total = .135353535353535
Multiply by 2 to add the upper numbers (8,9,10) = .270707070707071
The front line winners include the sevens (6/36 or 1/6) plus the 11 (2/36 or 1/18), = .222222222222222
Adding the front line winners, f1 = .492929292929293
Thats it.. don't bother with the 2, 3, or 12 because they are not winners.
NEXT: calculate this by counting loosers..
4) Pass-line bet in traditional craps by counting loosing rolls
These occur if a 2, 3, or 12 is thrown on the comeout roll, or the number is rolled, and then a seven comes before it is repeated...
Calculate this by summing up the frequency (f1) of all the loosers, starting with the 4 - 6 points.
For the 4 point, f1 = 3 / 36 (the frequency of a 4 occuring) TIMES 6 / 9 ... the odds of hitting a seven (6 ways) vs the point ( 3 ways)
same as 18 / 324 ... = 5.55555555555556E-02
For the 5 point, f1 = 4 / 36 (the frequency of a 5 occuring) TIMES 6 / 10 ... the odds of hitting a seven (6 ways) vs the point ( 4 ways)
same as 24 / 360 ... = 6.66666666666667E-02
For the 6 point, f1 = 5 / 36 (the frequency of a 6 occuring) TIMES 6 / 11 ... the odds of hitting a seven (6 ways) vs the point ( 5 ways)
same as 30 / 396 ... = 7.57575757575758E-02
f1 sum total = .197979797979798
Multiply by 2 to add the upper numbers (8,9,10) = .395959595959596
The comeout roll loosers include the two (1/36) plus the three (2/36 or 1/18) plus the 12 (1/36), = .111111111111111
Adding the comeout roll loosers, f1 = .507070707070707
Thats it.. don't bother with the comeout roll winners, we're only counting loosers
So, we have .492929292929293 ways to win and .507070707070707 ways to loose.
They should total up, right? ... .492929292929293 + .507070707070707 = 1.
Should these really add up? Traditional Craps has a Don't Pass bet.
If the pass line looses 50.7070707070707 percent of the time, why not just play the Don't Pass bet?
Casinos are not built to loose, so they always have the advantage with every wager they make.
NEXT: learn calculating the Don't Pass bet in traditional craps by counting winning rolls
5) Don't Pass bet in traditional craps by counting winning rolls
The Don't Pass bet in traditional craps is not the exact opposide of the Pass bet.
On the comeout roll, a 12 is not a winner or a looser, its just a push.
This means that the 12 is not part of the 36 numbers in play during the comeout roll, leaving only 35 numbers to consider.
Calculate this by summing up the frequency (f1) of the winning rolls ...
For the 4 point, f1 = 3 / 35 (the frequency of a 4 occuring on the comeout roll) TIMES 6 / 9 ... the odds of hitting a seven (6 ways) vs the point (3 ways)
same as 18 / 315 ... = 5.71428571428571E-02
For the 5 point, f1 = 4 / 35 (the frequency of a 5 occuring on the comeout roll) TIMES 6 / 10 ... the odds of hitting a seven (6 ways) vs the point (4 ways)
same as 24 / 350 ... = 6.85714285714286E-02
For the 6 point, f1 = 5 / 35 (the frequency of a 6 occuring on the comeout roll) TIMES 6 / 11 ... the odds of hitting a seven (6 ways) vs the point (5 ways)
same as 30 / 385 ... = 7.79220779220779E-02
f1 sum total = .203636363636364
Multiply by 2 to add the upper numbers (8,9,10) = .407272727272727
The comeout roll winners include the two (1/35) plus the three (2/35), = 8.57142857142857E-02
Adding the comeout roll winners, f1 = .492987012987013
Thats it.. don't bother with the comeout roll loosers, we're only counting winners
So, now we have .492929292929293 ways to win by playing the pass line and .492987012987013 ways to win by playing the don't pass line.
Should they total up? ... .492929292929293 + .492987012987013 = .985916305916306 ... now that's better.. the house always needs an edge.
We calculated the odds of loosing the Pass bet in crapless craps, but there are no Don't Pass bets in that game.
This is because there are no numbers to use to offset the house odds on the don't bets.
The truth is, Vegas wasn 't built on winners. As a player you don't have any possible chance to win, on average.
Either you believe in magic, or you think mathematical averages don't apply to you, especially over time.
If you accept the fact that you can't win, but still want to play for entertainment value, then your
goal as a player is to find the bets that will allow you to play as long as possible. A slight difference in the
payback of a bet makes a huge difference in how long you can play.
NEXT: learn how to make the bets with the best paybacks, but first, let's try something different...
6) Pass-line bet in 3-dice Crapless Craps (3DCC)
This is one of my new games available from SpikerSystems via DiscFlicker.com.
It is played just like 2 dice crapless craps (2DCC) but with 3 dice. The Ways table for 3 dice is as follows:
Point = 3 Ways = 1
Point = 4 Ways = 3
Point = 5 Ways = 6
Point = 6 Ways = 10
Point = 7 Ways = 15
Point = 8 Ways = 21
Point = 9 Ways = 25
Point = 10 Ways = 27
Point = 11 Ways = 27
Point = 12 Ways = 25
Point = 13 Ways = 21
Point = 14 Ways = 15
Point = 15 Ways = 10
Point = 16 Ways = 6
Point = 17 Ways = 3
Point = 18 Ways = 1
Let's try using the 2 center rolls both for a table-out just like the 7 table out in 2DC is in the middle.
We will use both the numbers 10 AND 11 for the 2DCC 'seven', what we will call the 'Table-Outs'.
With 27 Table-outs for each of the 10 and 11, there are 54 total Table-Outs.
Now the point numbers include 3,4,5,6,7,8,9 and then 12,13,14,15,16,17,18
Instead of 36 roll combinations there are 216 (6 * 6 * 6).
Instead of the table-out occuring 6 out of 36 times, it occurs 54 out of 216 times.
Calculate this by summing up the frequency (f1) of all the winners, starting with the 3 - 9 points.
These occur if the number is rolled, and then repeated before a 10 or 11 'Table-Outs'...
For the 3 point, f1 = 1 / 216 (the frequency of a 3 occuring) TIMES 1 / 55 (the odds of hitting the point ... 1 ways vs 54 ways to make a 10 or 11)
same as 1 / 11880 ... = 8.41750841750842E-05
For the 4 point, f1 = 3 / 216 (the frequency of a 4 occuring) TIMES 3 / 57 (the odds of hitting the point ... 3 ways vs 54 ways to make a 10 or 11)
same as 9 / 12312 ... = 7.30994152046784E-04
For the 5 point, f1 = 6 / 216 (the frequency of a 5 occuring) TIMES 6 / 60 (the odds of hitting the point ... 6 ways vs 54 ways to make a 10 or 11)
same as 36 / 12960 ... = 2.77777777777778E-03
For the 6 point, f1 = 10 / 216 (the frequency of a 6 occuring) TIMES 10 / 64 (the odds of hitting the point ... 10 ways vs 54 ways to make a 10 or 11)
same as 100 / 13824 ... = 7.2337962962963E-03
For the 7 point, f1 = 15 / 216 (the frequency of a 7 occuring) TIMES 15 / 69 (the odds of hitting the point ... 15 ways vs 54 ways to make a 10 or 11)
same as 225 / 14904 ... = 1.50966183574879E-02
For the 8 point, f1 = 21 / 216 (the frequency of a 8 occuring) TIMES 21 / 75 (the odds of hitting the point ... 21 ways vs 54 ways to make a 10 or 11)
same as 441 / 16200 ... = 2.72222222222222E-02
For the 9 point, f1 = 25 / 216 (the frequency of a 9 occuring) TIMES 25 / 79 (the odds of hitting the point ... 25 ways vs 54 ways to make a 10 or 11)
same as 625 / 17064 ... = 3.66268166901078E-02
f1 sum total = 8.97724005801139E-02
Multiply by 2 to add the upper numbers (12,13,14,15,16,17,18) = .179544801160228
The front line winners include the 10s and 11s (54/216 or 1/4), = .25
Adding the front line winners, f1 = .429544801160228
NEXT: calculate this by counting lossers..
7) Pass-line bet in 3DCC 10-11 Outs counting loosers
Calculate this by summing up the frequency (f1) of all the loosers, starting with the 3 - 9 points.
These occur if the number is rolled, and then a 10 or 11 'Table-Outs' occurs before the number is repeated...
For the 3 point, f1 = 1 / 216 (the frequency of a 3 occuring) TIMES 54 / 55 (54 ways to make a 10 or 11 vs the odds of hitting the point ... 1 ways)
same as 1 / 11880 ... = 4.54545454545455E-03
For the 4 point, f1 = 3 / 216 (the frequency of a 4 occuring) TIMES 54 / 57 (54 ways to make a 10 or 11 vs the odds of hitting the point ... 3 ways)
same as 9 / 12312 ... = 1.31578947368421E-02
For the 5 point, f1 = 6 / 216 (the frequency of a 5 occuring) TIMES 54 / 60 (54 ways to make a 10 or 11 vs the odds of hitting the point ... 6 ways)
same as 36 / 12960 ... = .025
For the 6 point, f1 = 10 / 216 (the frequency of a 6 occuring) TIMES 54 / 64 (54 ways to make a 10 or 11 vs the odds of hitting the point ... 10 ways)
same as 100 / 13824 ... = .0390625
For the 7 point, f1 = 15 / 216 (the frequency of a 7 occuring) TIMES 54 / 69 (54 ways to make a 10 or 11 vs the odds of hitting the point ... 15 ways)
same as 225 / 14904 ... = 5.43478260869565E-02
For the 8 point, f1 = 21 / 216 (the frequency of a 8 occuring) TIMES 54 / 75 (54 ways to make a 10 or 11 vs the odds of hitting the point ... 21 ways)
same as 441 / 16200 ... = .07
For the 9 point, f1 = 25 / 216 (the frequency of a 9 occuring) TIMES 54 / 79 (54 ways to make a 10 or 11 vs the odds of hitting the point ... 25 ways)
same as 625 / 17064 ... = 7.91139240506329E-02
f1 sum total = .285227599419886
Multiply by 2 to add the upper numbers (12,13,14,15,16,17,18) = .570455198839772
They should total up, right? ... .429544801160228 + .570455198839772 = 1
NEXT: calculate this using the 9 and 12 as Table Outs..
8) Pass-line bet in 3DCC using the 9 and 12 as Table Outs
Instead of the table-out occuring 6 out of 36 times, it occurs 50 out of 216 times.
Calculate this by summing up the frequency (f1) of all the winners, starting with the 3 - 10 points, but skip the 9.
These occur if the number is rolled, and then repeated before a 9 or 12 'Table-Outs'...
For the 3 point, f1 = 1 / 216 (the frequency of a 3 occuring) TIMES 1 / 51 (the odds of hitting the point ... 1 ways vs 50 ways to make a 9 or 12)
same as 1 / 11016 ... = 9.07770515613653E-05
For the 4 point, f1 = 3 / 216 (the frequency of a 4 occuring) TIMES 3 / 53 (the odds of hitting the point ... 3 ways vs 50 ways to make a 9 or 12)
same as 9 / 11448 ... = 7.86163522012579E-04
For the 5 point, f1 = 6 / 216 (the frequency of a 5 occuring) TIMES 6 / 56 (the odds of hitting the point ... 6 ways vs 50 ways to make a 9 or 12)
same as 36 / 12096 ... = 2.97619047619048E-03
For the 6 point, f1 = 10 / 216 (the frequency of a 6 occuring) TIMES 10 / 60 (the odds of hitting the point ... 10 ways vs 50 ways to make a 9 or 12)
same as 100 / 12960 ... = 7.71604938271605E-03
For the 7 point, f1 = 15 / 216 (the frequency of a 7 occuring) TIMES 15 / 65 (the odds of hitting the point ... 15 ways vs 50 ways to make a 9 or 12)
same as 225 / 14040 ... = .016025641025641
For the 8 point, f1 = 21 / 216 (the frequency of a 8 occuring) TIMES 21 / 71 (the odds of hitting the point ... 21 ways vs 50 ways to make a 9 or 12)
same as 441 / 15336 ... = 2.87558685446009E-02
For the 9 point, f1 = 25 / 216 (the frequency of a 9 occuring) TIMES 25 / 75 (the odds of hitting the point ... 25 ways vs 50 ways to make a 9 or 12)
same as 625 / 16200 ... = 3.85802469135802E-02
Skipping summing the 9 point
For the 10 point, f1 = 27 / 216 (the frequency of a 10 occuring) TIMES 27 / 77 (the odds of hitting the point ... 27 ways vs 50 ways to make a 9 or 12)
same as 729 / 16632 ... = 4.38311688311688E-02
f1 sum total = .100181858833891
Multiply by 2 to add the upper numbers (11,13,14,15,16,17,18) = .200363717667783
The front line winners include the 9s and 12s, each of which have 25 ways (50/216), = .231481481481481
Adding the front line winners, f1 = .431845199149264
NEXT: calculate this by counting lossers..
9) Pass-line bet in 3DCC using the 9 and 12 as Table Outs counting loosers
Instead of the table-out occuring 6 out of 36 times, it occurs 50 out of 216 times.
Calculate this by summing up the frequency (f1) of all the loosers, starting with the 3 - 10 points, but skip the 9.
These occur if the number is rolled, and then a 9 or 12 'Table-Outs' occurs before the number is repeated...
For the 3 point, f1 = 1 / 216 (the frequency of a 3 occuring) TIMES 1 / 51 (50 ways to make a 9 or 12 vs the odds of hitting the point ... 1 ways)
same as 1 / 11016 ... = 4.53885257806826E-03
For the 4 point, f1 = 3 / 216 (the frequency of a 4 occuring) TIMES 3 / 53 (50 ways to make a 9 or 12 vs the odds of hitting the point ... 3 ways)
same as 9 / 11448 ... = 1.31027253668763E-02
For the 5 point, f1 = 6 / 216 (the frequency of a 5 occuring) TIMES 6 / 56 (50 ways to make a 9 or 12 vs the odds of hitting the point ... 6 ways)
same as 36 / 12096 ... = 2.48015873015873E-02
For the 6 point, f1 = 10 / 216 (the frequency of a 6 occuring) TIMES 10 / 60 (50 ways to make a 9 or 12 vs the odds of hitting the point ... 10 ways)
same as 100 / 12960 ... = 3.85802469135802E-02
For the 7 point, f1 = 15 / 216 (the frequency of a 7 occuring) TIMES 15 / 65 (50 ways to make a 9 or 12 vs the odds of hitting the point ... 15 ways)
same as 225 / 14040 ... = 5.34188034188034E-02
For the 8 point, f1 = 21 / 216 (the frequency of a 8 occuring) TIMES 21 / 71 (50 ways to make a 9 or 12 vs the odds of hitting the point ... 21 ways)
same as 441 / 15336 ... = 6.84663536776213E-02
For the 9 point, f1 = 25 / 216 (the frequency of a 9 occuring) TIMES 25 / 75 (50 ways to make a 9 or 12 vs the odds of hitting the point ... 25 ways)
same as 625 / 16200 ... = 7.71604938271605E-02
Skipping summing the 9 point
For the 10 point, f1 = 27 / 216 (the frequency of a 10 occuring) TIMES 27 / 77 (50 ways to make a 9 or 12 vs the odds of hitting the point ... 27 ways)
same as 729 / 16632 ... = 8.11688311688312E-02
f1 sum total = .284077400425368
Multiply by 2 to add the upper numbers (11,13,14,15,16,17,18) = .568154800850736
They should total up, right? ... .431845199149264 + .568154800850736 = 1
NEXT: calculate this using the 8 and 13 as Table Outs..
10) Pass-line bet in 3DCC using the 8 and 13 as Table Outs
Instead of the table-out occuring 6 out of 36 times, it occurs 42 out of 216 times.
Calculate this by summing up the frequency (f1) of all the winners, starting with the 3 - 10 points, but skip the 8.
These occur if the number is rolled, and then repeated before an 8 or 13 'Table-Outs'...
For the 3 point, f1 = 1 / 216 (the frequency of a 3 occuring) TIMES 1 / 43 (the odds of hitting the point ... 1 ways vs 42 ways to make a 8 or 13)
same as 1 / 9288 ... = 1.07665805340224E-04
For the 4 point, f1 = 3 / 216 (the frequency of a 4 occuring) TIMES 3 / 45 (the odds of hitting the point ... 3 ways vs 42 ways to make a 8 or 13)
same as 9 / 9720 ... = 9.25925925925926E-04
For the 5 point, f1 = 6 / 216 (the frequency of a 5 occuring) TIMES 6 / 48 (the odds of hitting the point ... 6 ways vs 42 ways to make a 8 or 13)
same as 36 / 10368 ... = 3.47222222222222E-03
For the 6 point, f1 = 10 / 216 (the frequency of a 6 occuring) TIMES 10 / 52 (the odds of hitting the point ... 10 ways vs 42 ways to make a 8 or 13)
same as 100 / 11232 ... = 8.9031339031339E-03
For the 7 point, f1 = 15 / 216 (the frequency of a 7 occuring) TIMES 15 / 57 (the odds of hitting the point ... 15 ways vs 42 ways to make a 8 or 13)
same as 225 / 12312 ... = 1.82748538011696E-02
For the 8 point, f1 = 21 / 216 (the frequency of a 8 occuring) TIMES 21 / 63 (the odds of hitting the point ... 21 ways vs 42 ways to make a 8 or 13)
same as 441 / 13608 ... = 3.24074074074074E-02
Skipping summing the 8 point
For the 9 point, f1 = 25 / 216 (the frequency of a 9 occuring) TIMES 25 / 67 (the odds of hitting the point ... 25 ways vs 42 ways to make a 8 or 13)
same as 625 / 14472 ... = 4.31868435599779E-02
For the 10 point, f1 = 27 / 216 (the frequency of a 10 occuring) TIMES 27 / 69 (the odds of hitting the point ... 27 ways vs 42 ways to make a 8 or 13)
same as 729 / 14904 ... = 4.89130434782609E-02
f1 sum total = .123783688696031
Multiply by 2 to add the upper numbers (11,12,14,15,16,17,18) = .247567377392061
The front line winners include the 8s and 13s, each of which have 21 ways (42/216), = .194444444444444
Adding the front line winners, f1 = .442011821836506
NEXT: calculate this by counting lossers..
11) Pass-line bet in 3DCC using the 8 and 13 as Table Outs counting loosers
Instead of the table-out occuring 6 out of 36 times, it occurs 42 out of 216 times.
Calculate this by summing up the frequency (f1) of all the loosers, starting with the 3 - 10 points, but skip the 8.
These occur if the number is rolled, and then a 8 or 13 'Table-Outs' occurs before the number is repeated...
For the 3 point, f1 = 1 / 216 (the frequency of a 3 occuring) TIMES 1 / 43 (42 ways to make a 8 or 13 vs the odds of hitting the point ... 1 ways)
same as 1 / 9288 ... = 4.52196382428941E-03
For the 4 point, f1 = 3 / 216 (the frequency of a 4 occuring) TIMES 3 / 45 (42 ways to make a 8 or 13 vs the odds of hitting the point ... 3 ways)
same as 9 / 9720 ... = .012962962962963
For the 5 point, f1 = 6 / 216 (the frequency of a 5 occuring) TIMES 6 / 48 (42 ways to make a 8 or 13 vs the odds of hitting the point ... 6 ways)
same as 36 / 10368 ... = 2.43055555555556E-02
For the 6 point, f1 = 10 / 216 (the frequency of a 6 occuring) TIMES 10 / 52 (42 ways to make a 8 or 13 vs the odds of hitting the point ... 10 ways)
same as 100 / 11232 ... = 3.73931623931624E-02
For the 7 point, f1 = 15 / 216 (the frequency of a 7 occuring) TIMES 15 / 57 (42 ways to make a 8 or 13 vs the odds of hitting the point ... 15 ways)
same as 225 / 12312 ... = 5.11695906432749E-02
For the 8 point, f1 = 21 / 216 (the frequency of a 8 occuring) TIMES 21 / 63 (42 ways to make a 8 or 13 vs the odds of hitting the point ... 21 ways)
same as 441 / 13608 ... = 6.48148148148148E-02
Skipping summing the 8 point
For the 9 point, f1 = 25 / 216 (the frequency of a 9 occuring) TIMES 25 / 67 (42 ways to make a 8 or 13 vs the odds of hitting the point ... 25 ways)
same as 625 / 14472 ... = 7.25538971807629E-02
For the 10 point, f1 = 27 / 216 (the frequency of a 10 occuring) TIMES 27 / 69 (42 ways to make a 8 or 13 vs the odds of hitting the point ... 27 ways)
same as 729 / 14904 ... = 7.60869565217391E-02
f1 sum total = .278994089081747
Multiply by 2 to add the upper numbers (11,12,14,15,16,17,18) = .557988178163494
They should total up, right? ... .442011821836506 + .557988178163494 = 1
NEXT: calculate this using the 7 and 14 as Table Outs..
12) Pass-line bet in 3DCC using the 7 and 14 as Table Outs
Instead of the table-out occuring 6 out of 36 times, it occurs 30 out of 216 times.
Calculate this by summing up the frequency (f1) of all the winners, starting with the 3 - 10 points, but skip the 7.
These occur if the number is rolled, and then repeated before a 7 or 14 'Table-Outs'...
For the 3 point, f1 = 1 / 216 (the frequency of a 3 occuring) TIMES 1 / 31 (the odds of hitting the point ... 1 ways vs 30 ways to make a 7 or 14)
same as 1 / 6696 ... = 1.49342891278375E-04
For the 4 point, f1 = 3 / 216 (the frequency of a 4 occuring) TIMES 3 / 33 (the odds of hitting the point ... 3 ways vs 30 ways to make a 7 or 14)
same as 9 / 7128 ... = 1.26262626262626E-03
For the 5 point, f1 = 6 / 216 (the frequency of a 5 occuring) TIMES 6 / 36 (the odds of hitting the point ... 6 ways vs 30 ways to make a 7 or 14)
same as 36 / 7776 ... = 4.62962962962963E-03
For the 6 point, f1 = 10 / 216 (the frequency of a 6 occuring) TIMES 10 / 40 (the odds of hitting the point ... 10 ways vs 30 ways to make a 7 or 14)
same as 100 / 8640 ... = 1.15740740740741E-02
For the 7 point, f1 = 15 / 216 (the frequency of a 7 occuring) TIMES 15 / 45 (the odds of hitting the point ... 15 ways vs 30 ways to make a 7 or 14)
same as 225 / 9720 ... = 2.31481481481481E-02
Skipping summing the 7 point
For the 8 point, f1 = 21 / 216 (the frequency of a 8 occuring) TIMES 21 / 51 (the odds of hitting the point ... 21 ways vs 30 ways to make a 7 or 14)
same as 441 / 11016 ... = 4.00326797385621E-02
For the 9 point, f1 = 25 / 216 (the frequency of a 9 occuring) TIMES 25 / 55 (the odds of hitting the point ... 25 ways vs 30 ways to make a 7 or 14)
same as 625 / 11880 ... = 5.26094276094276E-02
For the 10 point, f1 = 27 / 216 (the frequency of a 10 occuring) TIMES 27 / 57 (the odds of hitting the point ... 27 ways vs 30 ways to make a 7 or 14)
same as 729 / 12312 ... = 5.92105263157895E-02
f1 sum total = .169468306521388
Multiply by 2 to add the upper numbers (11,12,13,15,16,17,18) = .338936613042775
The front line winners include the 7s and 14s, each of which have 15 ways (30/216), = .138888888888889
Adding the front line winners, f1 = .477825501931664
NEXT: calculate this by counting lossers..
13) Pass-line bet in 3DCC using the 7 and 14 as Table Outs counting loosers
Instead of the table-out occuring 6 out of 36 times, it occurs 30 out of 216 times.
Calculate this by summing up the frequency (f1) of all the loosers, starting with the 3 - 10 points, but skip the 7.
These occur if the number is rolled, and then a 7 or 14 'Table-Outs' occurs before the number is repeated...
For the 3 point, f1 = 1 / 216 (the frequency of a 3 occuring) TIMES 1 / 31 (30 ways to make a 7 or 14 vs the odds of hitting the point ... 1 ways)
same as 1 / 6696 ... = 4.48028673835125E-03
For the 4 point, f1 = 3 / 216 (the frequency of a 4 occuring) TIMES 3 / 33 (30 ways to make a 7 or 14 vs the odds of hitting the point ... 3 ways)
same as 9 / 7128 ... = 1.26262626262626E-02
For the 5 point, f1 = 6 / 216 (the frequency of a 5 occuring) TIMES 6 / 36 (30 ways to make a 7 or 14 vs the odds of hitting the point ... 6 ways)
same as 36 / 7776 ... = 2.31481481481481E-02
For the 6 point, f1 = 10 / 216 (the frequency of a 6 occuring) TIMES 10 / 40 (30 ways to make a 7 or 14 vs the odds of hitting the point ... 10 ways)
same as 100 / 8640 ... = 3.47222222222222E-02
For the 7 point, f1 = 15 / 216 (the frequency of a 7 occuring) TIMES 15 / 45 (30 ways to make a 7 or 14 vs the odds of hitting the point ... 15 ways)
same as 225 / 9720 ... = 4.62962962962963E-02
Skipping summing the 7 point
For the 8 point, f1 = 21 / 216 (the frequency of a 8 occuring) TIMES 21 / 51 (30 ways to make a 7 or 14 vs the odds of hitting the point ... 21 ways)
same as 441 / 11016 ... = 5.71895424836601E-02
For the 9 point, f1 = 25 / 216 (the frequency of a 9 occuring) TIMES 25 / 55 (30 ways to make a 7 or 14 vs the odds of hitting the point ... 25 ways)
same as 625 / 11880 ... = 6.31313131313131E-02
For the 10 point, f1 = 27 / 216 (the frequency of a 10 occuring) TIMES 27 / 57 (30 ways to make a 7 or 14 vs the odds of hitting the point ... 27 ways)
same as 729 / 12312 ... = 6.57894736842105E-02
f1 sum total = .261087249034168
Multiply by 2 to add the upper numbers (11,12,13,15,16,17,18) = .522174498068336
They should total up, right? ... .477825501931664 + .522174498068336 = 1
NEXT: calculate this using the 7 and 14 and all the tripples as Table Outs..
14) Pass-line bet in 3DCC using the 7 and 14 and all tripples as Table Outs
Instead of the table-out occuring 6 out of 36 times, it occurs 36 out of 216 times.
Calculate this by summing up the frequency (f1) of all the winners, starting with the 3 - 10 points, but skipping the 3d 7 point rolls and all the tripple rolls.
These occur if the number is rolled, and then repeated before a 7 or 14 or tripple 'Table-Outs'...
f1 = (iWays / 216) * (iWays / (iWays + 36)) = 0
For the 3 point, f1 = 0 / 216 (the frequency of a a non-tripple3 occuring) TIMES 0 / 36 (the odds of hitting the point ...0 ways vs 36 ways to make a 7 or 14 or tripple)
same as 0 / 7776 ... = 0
f1 = (iWays / 216) * (iWays / (iWays + 36)) = 1.06837606837607E-03
For the 4 point, f1 = 3 / 216 (the frequency of a 4 occuring) TIMES 3 / 39 (the odds of hitting the point ...3 ways vs 36 ways to make a 7 or 14 or tripple)
same as 9 / 8424 ... = 1.06837606837607E-03
f1 = (iWays / 216) * (iWays / (iWays + 36)) = 3.96825396825397E-03
For the 5 point, f1 = 6 / 216 (the frequency of a 5 occuring) TIMES 6 / 42 (the odds of hitting the point ...6 ways vs 36 ways to make a 7 or 14 or tripple)
same as 36 / 9072 ... = 3.96825396825397E-03
f1 = (iWays / 216) * (iWays / (iWays + 36)) = 8.33333333333333E-03
For the 6 point, f1 = 9 / 216 (the frequency of a a non-tripple6 occuring) TIMES 9 / 45 (the odds of hitting the point ...9 ways vs 36 ways to make a 7 or 14 or tripple)
same as 81 / 9720 ... = 8.33333333333333E-03
f1 = (iWays / 216) * (iWays / (iWays + 36)) = 2.04248366013072E-02
For the 7 point, f1 = 15 / 216 (the frequency of a 7 occuring) TIMES 15 / 51 (the odds of hitting the point ...15 ways vs 36 ways to make a 7 or 14 or tripple)
same as 225 / 11016 ... = 2.04248366013072E-02
Skipping summing the 7 point
f1 = (iWays / 216) * (iWays / (iWays + 36)) = 3.58187134502924E-02
For the 8 point, f1 = 21 / 216 (the frequency of a 8 occuring) TIMES 21 / 57 (the odds of hitting the point ...21 ways vs 36 ways to make a 7 or 14 or tripple)
same as 441 / 12312 ... = 3.58187134502924E-02
f1 = (iWays / 216) * (iWays / (iWays + 36)) = 4.44444444444444E-02
For the 9 point, f1 = 24 / 216 (the frequency of a a non-tripple9 occuring) TIMES 24 / 60 (the odds of hitting the point ...24 ways vs 36 ways to make a 7 or 14 or tripple)
same as 576 / 12960 ... = 4.44444444444444E-02
f1 = (iWays / 216) * (iWays / (iWays + 36)) = 5.35714285714286E-02
For the 10 point, f1 = 27 / 216 (the frequency of a 10 occuring) TIMES 27 / 63 (the odds of hitting the point ...27 ways vs 36 ways to make a 7 or 14 or tripple)
same as 729 / 13608 ... = 5.35714285714286E-02
f1 sum total = .147204549836129
Multiply by 2 to add the upper numbers (11,12,13,15,16,17,18) = .294409099672258
The front line winners include the 6 tripples, each of which have 1 way (6/216), plus the 7s and 14s , each of which have 15 ways (30/216), = 36/215 .166666666666667
Adding the front line winners, f1 = .461075766338924
NEXT: calculate this by counting lossers..
15) Pass-line bet in 3DCC using 7, 14, trips Table-Outs counting loosers
Instead of the table-out occuring 6 out of 36 times, it occurs 36 out of 216 times.
Calculate this by summing up the frequency (f1) of all the loosers, starting with the 3 - 10 points, but skipping the 3d 7 point rolls and all the tripple rolls.
These occur if the number is rolled, and then a 7 or 14 or a tripple 'Table-Outs' occurs before the number is repeated...
For the 3 point, f1 = 0 / 216 (the frequency of a a non-tripple 3 occuring) TIMES 0 / 36 (36 ways to make a 7 or 14 or tripple vs the odds of hitting the point ... 0 ways)
same as 0 / 7776 ... = 0
For the 4 point, f1 = 3 / 216 (the frequency of a 4 occuring) TIMES 3 / 39 (36 ways to make a 7 or 14 or tripple vs the odds of hitting the point ... 3 ways)
same as 9 / 8424 ... = 1.28205128205128E-02
For the 5 point, f1 = 6 / 216 (the frequency of a 5 occuring) TIMES 6 / 42 (36 ways to make a 7 or 14 or tripple vs the odds of hitting the point ... 6 ways)
same as 36 / 9072 ... = 2.38095238095238E-02
For the 6 point, f1 = 9 / 216 (the frequency of a a non-tripple 6 occuring) TIMES 9 / 45 (36 ways to make a 7 or 14 or tripple vs the odds of hitting the point ... 9 ways)
same as 81 / 9720 ... = 3.33333333333333E-02
For the 7 point, f1 = 15 / 216 (the frequency of a 7 occuring) TIMES 15 / 51 (36 ways to make a 7 or 14 or tripple vs the odds of hitting the point ... 15 ways)
same as 225 / 11016 ... = 4.90196078431373E-02
Skipping summing the 7 point
For the 8 point, f1 = 21 / 216 (the frequency of a 8 occuring) TIMES 21 / 57 (36 ways to make a 7 or 14 or tripple vs the odds of hitting the point ... 21 ways)
same as 441 / 12312 ... = 6.14035087719298E-02
For the 9 point, f1 = 24 / 216 (the frequency of a a non-tripple 9 occuring) TIMES 24 / 60 (36 ways to make a 7 or 14 or tripple vs the odds of hitting the point ... 24 ways)
same as 576 / 12960 ... = 6.66666666666667E-02
For the 10 point, f1 = 27 / 216 (the frequency of a 10 occuring) TIMES 27 / 63 (36 ways to make a 7 or 14 or tripple vs the odds of hitting the point ... 27 ways)
same as 729 / 13608 ... = 7.14285714285714E-02
f1 sum total = .269462116830538
Multiply by 2 to add the upper numbers (11,12,13,15,16,17,18) = .538924233661076
They should total up, right? ... .461075766338924 + .538924233661076 = 1
NEXT: a discussion on mapping a three dice roll into a two dice outcome
16) Mapping a 3 dice roll into a 2 dice or 1 die outcome
If 3 dice are thrown, the roll can also be used for 2 dice and for 1 dice games.
Of the 36 rolls that permutate from 2 dice, there are really only 21 distinct dice combinations.
1 1 1
2 1 2
3 1 3
4 1 4
5 1 5
6 1 6
8 2 2
9 2 3
10 2 4
11 2 5
12 2 6
15 3 3
16 3 4
17 3 5
18 3 6
22 4 4
23 4 5
24 4 6
29 5 5
30 5 6
36 6 6
Similarly, of the 216 rolls that permutate from 3 dice, there are really only 56 distinct dice combinations.
We need to map those 56 distinct 3 dice roll combinations into all 36 2 dice rolls.
We also need to map the 21 distinct 2 dice roll combinations into all 6 single dice rolls.
3 | 1-1-1 :1|
4 | 1-1-2 :3|
5 | 1-1-3 :3 | 1-2-2 :3|
6 | 1-1-4 :3 | 1-2-3 :6 | 2-2-2 :1|
7 | 1-1-5 :3 | 1-2-4 :6 | 1-3-3 :3 | 2-2-3 :3|
8 | 1-1-6 :3 | 1-2-5 :6 | 1-3-4 :6 | 2-2-4 :3 | 2-3-3 :3|
9 | 1-2-6 :6 | 1-3-5 :6 | 1-4-4 :3 | 2-2-5 :3 | 2-3-4 :6 | 3-3-3 :1|
10 | 1-3-6 :6 | 1-4-5 :6 | 2-2-6 :3 | 2-3-5 :6 | 2-4-4 :3 | 3-3-4 :3|
11 | 1-4-6 :6 | 1-5-5 :3 | 2-3-6 :6 | 2-4-5 :6 | 3-3-5 :3 | 3-4-4 :3|
12 | 1-5-6 :6 | 2-4-6 :6 | 2-5-5 :3 | 3-3-6 :3 | 3-4-5 :6 | 4-4-4 :1|
13 | 1-6-6 :3 | 2-5-6 :6 | 3-4-6 :6 | 3-5-5 :3 | 4-4-5 :3|
14 | 2-6-6 :3 | 3-5-6 :6 | 4-4-6 :3 | 4-5-5 :3|
15 | 3-6-6 :3 | 4-5-6 :6 | 5-5-5 :1|
16 | 4-6-6 :3 | 5-5-6 :3|
17 | 5-6-6 :3|
18 | 6-6-6 :1|
NEXT: mapping table...
17) Mapping of 3 dice roll into a 2 dice or 1 die outcomes table
index 1: 1-1-1 point = 3 ways = 1 Maps to 2 Dice index 6 = 1-6 point = 7
index 2: 1-1-2 point = 4 ways = 3 Maps to 2 Dice index 1 = 1-1 point = 2
index 3: 1-1-3 point = 5 ways = 6 Maps to 2 Dice index 1 = 1-1 point = 2
index 4: 1-1-4 point = 6 ways = 10 Maps to 2 Dice index 2 = 1-2 point = 3
index 5: 1-1-5 point = 7 ways = 15 Maps to 2 Dice index 6 = 1-6 point = 7
index 6: 1-1-6 point = 8 ways = 21 Maps to 2 Dice index 3 = 1-3 point = 4
index 7: 1-2-1 point = 4 ways = 3 Maps to 2 Dice index 1 = 1-1 point = 2
index 8: 1-2-2 point = 5 ways = 6 Maps to 2 Dice index 2 = 1-2 point = 3
index 9: 1-2-3 point = 6 ways = 10 Maps to 2 Dice index 7 = 2-1 point = 3
index 10: 1-2-4 point = 7 ways = 15 Maps to 2 Dice index 11 = 2-5 point = 7
index 11: 1-2-5 point = 8 ways = 21 Maps to 2 Dice index 8 = 2-2 point = 4
index 12: 1-2-6 point = 9 ways = 25 Maps to 2 Dice index 9 = 2-3 point = 5
index 13: 1-3-1 point = 5 ways = 6 Maps to 2 Dice index 1 = 1-1 point = 2
index 14: 1-3-2 point = 6 ways = 10 Maps to 2 Dice index 7 = 2-1 point = 3
index 15: 1-3-3 point = 7 ways = 15 Maps to 2 Dice index 16 = 3-4 point = 7
index 16: 1-3-4 point = 8 ways = 21 Maps to 2 Dice index 13 = 3-1 point = 4
index 17: 1-3-5 point = 9 ways = 25 Maps to 2 Dice index 14 = 3-2 point = 5
index 18: 1-3-6 point = 10 ways = 27 Maps to 2 Dice index 5 = 1-5 point = 6
index 19: 1-4-1 point = 6 ways = 10 Maps to 2 Dice index 2 = 1-2 point = 3
index 20: 1-4-2 point = 7 ways = 15 Maps to 2 Dice index 11 = 2-5 point = 7
index 21: 1-4-3 point = 8 ways = 21 Maps to 2 Dice index 13 = 3-1 point = 4
index 22: 1-4-4 point = 9 ways = 25 Maps to 2 Dice index 4 = 1-4 point = 5
index 23: 1-4-5 point = 10 ways = 27 Maps to 2 Dice index 10 = 2-4 point = 6
index 24: 1-4-6 point = 11 ways = 27 Maps to 2 Dice index 32 = 6-2 point = 8
index 25: 1-5-1 point = 7 ways = 15 Maps to 2 Dice index 6 = 1-6 point = 7
index 26: 1-5-2 point = 8 ways = 21 Maps to 2 Dice index 8 = 2-2 point = 4
index 27: 1-5-3 point = 9 ways = 25 Maps to 2 Dice index 14 = 3-2 point = 5
index 28: 1-5-4 point = 10 ways = 27 Maps to 2 Dice index 10 = 2-4 point = 6
index 29: 1-5-5 point = 11 ways = 27 Maps to 2 Dice index 18 = 3-6 point = 9
index 30: 1-5-6 point = 12 ways = 25 Maps to 2 Dice index 28 = 5-4 point = 9
index 31: 1-6-1 point = 8 ways = 21 Maps to 2 Dice index 3 = 1-3 point = 4
index 32: 1-6-2 point = 9 ways = 25 Maps to 2 Dice index 9 = 2-3 point = 5
index 33: 1-6-3 point = 10 ways = 27 Maps to 2 Dice index 5 = 1-5 point = 6
index 34: 1-6-4 point = 11 ways = 27 Maps to 2 Dice index 32 = 6-2 point = 8
index 35: 1-6-5 point = 12 ways = 25 Maps to 2 Dice index 28 = 5-4 point = 9
index 36: 1-6-6 point = 13 ways = 21 Maps to 2 Dice index 34 = 6-4 point = 10
index 37: 2-1-1 point = 4 ways = 3 Maps to 2 Dice index 1 = 1-1 point = 2
index 38: 2-1-2 point = 5 ways = 6 Maps to 2 Dice index 2 = 1-2 point = 3
index 39: 2-1-3 point = 6 ways = 10 Maps to 2 Dice index 7 = 2-1 point = 3
index 40: 2-1-4 point = 7 ways = 15 Maps to 2 Dice index 11 = 2-5 point = 7
index 41: 2-1-5 point = 8 ways = 21 Maps to 2 Dice index 8 = 2-2 point = 4
index 42: 2-1-6 point = 9 ways = 25 Maps to 2 Dice index 9 = 2-3 point = 5
index 43: 2-2-1 point = 5 ways = 6 Maps to 2 Dice index 2 = 1-2 point = 3
index 44: 2-2-2 point = 6 ways = 10 Maps to 2 Dice index 6 = 1-6 point = 7
index 45: 2-2-3 point = 7 ways = 15 Maps to 2 Dice index 16 = 3-4 point = 7
index 46: 2-2-4 point = 8 ways = 21 Maps to 2 Dice index 3 = 1-3 point = 4
index 47: 2-2-5 point = 9 ways = 25 Maps to 2 Dice index 19 = 4-1 point = 5
index 48: 2-2-6 point = 10 ways = 27 Maps to 2 Dice index 19 = 4-1 point = 5
index 49: 2-3-1 point = 6 ways = 10 Maps to 2 Dice index 7 = 2-1 point = 3
index 50: 2-3-2 point = 7 ways = 15 Maps to 2 Dice index 16 = 3-4 point = 7
index 51: 2-3-3 point = 8 ways = 21 Maps to 2 Dice index 4 = 1-4 point = 5
index 52: 2-3-4 point = 9 ways = 25 Maps to 2 Dice index 25 = 5-1 point = 6
index 53: 2-3-5 point = 10 ways = 27 Maps to 2 Dice index 15 = 3-3 point = 6
index 54: 2-3-6 point = 11 ways = 27 Maps to 2 Dice index 27 = 5-3 point = 8
index 55: 2-4-1 point = 7 ways = 15 Maps to 2 Dice index 11 = 2-5 point = 7
index 56: 2-4-2 point = 8 ways = 21 Maps to 2 Dice index 3 = 1-3 point = 4
index 57: 2-4-3 point = 9 ways = 25 Maps to 2 Dice index 25 = 5-1 point = 6
index 58: 2-4-4 point = 10 ways = 27 Maps to 2 Dice index 20 = 4-2 point = 6
index 59: 2-4-5 point = 11 ways = 27 Maps to 2 Dice index 22 = 4-4 point = 8
index 60: 2-4-6 point = 12 ways = 25 Maps to 2 Dice index 23 = 4-5 point = 9
index 61: 2-5-1 point = 8 ways = 21 Maps to 2 Dice index 8 = 2-2 point = 4
index 62: 2-5-2 point = 9 ways = 25 Maps to 2 Dice index 19 = 4-1 point = 5
index 63: 2-5-3 point = 10 ways = 27 Maps to 2 Dice index 15 = 3-3 point = 6
index 64: 2-5-4 point = 11 ways = 27 Maps to 2 Dice index 22 = 4-4 point = 8
index 65: 2-5-5 point = 12 ways = 25 Maps to 2 Dice index 33 = 6-3 point = 9
index 66: 2-5-6 point = 13 ways = 21 Maps to 2 Dice index 29 = 5-5 point = 10
index 67: 2-6-1 point = 9 ways = 25 Maps to 2 Dice index 9 = 2-3 point = 5
index 68: 2-6-2 point = 10 ways = 27 Maps to 2 Dice index 19 = 4-1 point = 5
index 69: 2-6-3 point = 11 ways = 27 Maps to 2 Dice index 27 = 5-3 point = 8
index 70: 2-6-4 point = 12 ways = 25 Maps to 2 Dice index 23 = 4-5 point = 9
index 71: 2-6-5 point = 13 ways = 21 Maps to 2 Dice index 29 = 5-5 point = 10
index 72: 2-6-6 point = 14 ways = 15 Maps to 2 Dice index 31 = 6-1 point = 7
index 73: 3-1-1 point = 5 ways = 6 Maps to 2 Dice index 1 = 1-1 point = 2
index 74: 3-1-2 point = 6 ways = 10 Maps to 2 Dice index 7 = 2-1 point = 3
index 75: 3-1-3 point = 7 ways = 15 Maps to 2 Dice index 16 = 3-4 point = 7
index 76: 3-1-4 point = 8 ways = 21 Maps to 2 Dice index 13 = 3-1 point = 4
index 77: 3-1-5 point = 9 ways = 25 Maps to 2 Dice index 14 = 3-2 point = 5
index 78: 3-1-6 point = 10 ways = 27 Maps to 2 Dice index 5 = 1-5 point = 6
index 79: 3-2-1 point = 6 ways = 10 Maps to 2 Dice index 7 = 2-1 point = 3
index 80: 3-2-2 point = 7 ways = 15 Maps to 2 Dice index 16 = 3-4 point = 7
index 81: 3-2-3 point = 8 ways = 21 Maps to 2 Dice index 4 = 1-4 point = 5
index 82: 3-2-4 point = 9 ways = 25 Maps to 2 Dice index 25 = 5-1 point = 6
index 83: 3-2-5 point = 10 ways = 27 Maps to 2 Dice index 15 = 3-3 point = 6
index 84: 3-2-6 point = 11 ways = 27 Maps to 2 Dice index 27 = 5-3 point = 8
index 85: 3-3-1 point = 7 ways = 15 Maps to 2 Dice index 16 = 3-4 point = 7
index 86: 3-3-2 point = 8 ways = 21 Maps to 2 Dice index 4 = 1-4 point = 5
index 87: 3-3-3 point = 9 ways = 25 Maps to 2 Dice index 6 = 1-6 point = 7
index 88: 3-3-4 point = 10 ways = 27 Maps to 2 Dice index 20 = 4-2 point = 6
index 89: 3-3-5 point = 11 ways = 27 Maps to 2 Dice index 17 = 3-5 point = 8
index 90: 3-3-6 point = 12 ways = 25 Maps to 2 Dice index 18 = 3-6 point = 9
index 91: 3-4-1 point = 8 ways = 21 Maps to 2 Dice index 13 = 3-1 point = 4
index 92: 3-4-2 point = 9 ways = 25 Maps to 2 Dice index 25 = 5-1 point = 6
index 93: 3-4-3 point = 10 ways = 27 Maps to 2 Dice index 20 = 4-2 point = 6
index 94: 3-4-4 point = 11 ways = 27 Maps to 2 Dice index 17 = 3-5 point = 8
index 95: 3-4-5 point = 12 ways = 25 Maps to 2 Dice index 12 = 2-6 point = 8
index 96: 3-4-6 point = 13 ways = 21 Maps to 2 Dice index 24 = 4-6 point = 10
index 97: 3-5-1 point = 9 ways = 25 Maps to 2 Dice index 14 = 3-2 point = 5
index 98: 3-5-2 point = 10 ways = 27 Maps to 2 Dice index 15 = 3-3 point = 6
index 99: 3-5-3 point = 11 ways = 27 Maps to 2 Dice index 17 = 3-5 point = 8
index 100: 3-5-4 point = 12 ways = 25 Maps to 2 Dice index 12 = 2-6 point = 8
index 101: 3-5-5 point = 13 ways = 21 Maps to 2 Dice index 34 = 6-4 point = 10
index 102: 3-5-6 point = 14 ways = 15 Maps to 2 Dice index 21 = 4-3 point = 7
index 103: 3-6-1 point = 10 ways = 27 Maps to 2 Dice index 5 = 1-5 point = 6
index 104: 3-6-2 point = 11 ways = 27 Maps to 2 Dice index 27 = 5-3 point = 8
index 105: 3-6-3 point = 12 ways = 25 Maps to 2 Dice index 18 = 3-6 point = 9
index 106: 3-6-4 point = 13 ways = 21 Maps to 2 Dice index 24 = 4-6 point = 10
index 107: 3-6-5 point = 14 ways = 15 Maps to 2 Dice index 21 = 4-3 point = 7
index 108: 3-6-6 point = 15 ways = 10 Maps to 2 Dice index 35 = 6-5 point = 11
index 109: 4-1-1 point = 6 ways = 10 Maps to 2 Dice index 2 = 1-2 point = 3
index 110: 4-1-2 point = 7 ways = 15 Maps to 2 Dice index 11 = 2-5 point = 7
index 111: 4-1-3 point = 8 ways = 21 Maps to 2 Dice index 13 = 3-1 point = 4
index 112: 4-1-4 point = 9 ways = 25 Maps to 2 Dice index 4 = 1-4 point = 5
index 113: 4-1-5 point = 10 ways = 27 Maps to 2 Dice index 10 = 2-4 point = 6
index 114: 4-1-6 point = 11 ways = 27 Maps to 2 Dice index 32 = 6-2 point = 8
index 115: 4-2-1 point = 7 ways = 15 Maps to 2 Dice index 11 = 2-5 point = 7
index 116: 4-2-2 point = 8 ways = 21 Maps to 2 Dice index 3 = 1-3 point = 4
index 117: 4-2-3 point = 9 ways = 25 Maps to 2 Dice index 25 = 5-1 point = 6
index 118: 4-2-4 point = 10 ways = 27 Maps to 2 Dice index 20 = 4-2 point = 6
index 119: 4-2-5 point = 11 ways = 27 Maps to 2 Dice index 22 = 4-4 point = 8
index 120: 4-2-6 point = 12 ways = 25 Maps to 2 Dice index 23 = 4-5 point = 9
index 121: 4-3-1 point = 8 ways = 21 Maps to 2 Dice index 13 = 3-1 point = 4
index 122: 4-3-2 point = 9 ways = 25 Maps to 2 Dice index 25 = 5-1 point = 6
index 123: 4-3-3 point = 10 ways = 27 Maps to 2 Dice index 20 = 4-2 point = 6
index 124: 4-3-4 point = 11 ways = 27 Maps to 2 Dice index 17 = 3-5 point = 8
index 125: 4-3-5 point = 12 ways = 25 Maps to 2 Dice index 12 = 2-6 point = 8
index 126: 4-3-6 point = 13 ways = 21 Maps to 2 Dice index 24 = 4-6 point = 10
index 127: 4-4-1 point = 9 ways = 25 Maps to 2 Dice index 4 = 1-4 point = 5
index 128: 4-4-2 point = 10 ways = 27 Maps to 2 Dice index 20 = 4-2 point = 6
index 129: 4-4-3 point = 11 ways = 27 Maps to 2 Dice index 17 = 3-5 point = 8
index 130: 4-4-4 point = 12 ways = 25 Maps to 2 Dice index 31 = 6-1 point = 7
index 131: 4-4-5 point = 13 ways = 21 Maps to 2 Dice index 33 = 6-3 point = 9
index 132: 4-4-6 point = 14 ways = 15 Maps to 2 Dice index 26 = 5-2 point = 7
index 133: 4-5-1 point = 10 ways = 27 Maps to 2 Dice index 10 = 2-4 point = 6
index 134: 4-5-2 point = 11 ways = 27 Maps to 2 Dice index 22 = 4-4 point = 8
index 135: 4-5-3 point = 12 ways = 25 Maps to 2 Dice index 12 = 2-6 point = 8
index 136: 4-5-4 point = 13 ways = 21 Maps to 2 Dice index 33 = 6-3 point = 9
index 137: 4-5-5 point = 14 ways = 15 Maps to 2 Dice index 26 = 5-2 point = 7
index 138: 4-5-6 point = 15 ways = 10 Maps to 2 Dice index 30 = 5-6 point = 11
index 139: 4-6-1 point = 11 ways = 27 Maps to 2 Dice index 32 = 6-2 point = 8
index 140: 4-6-2 point = 12 ways = 25 Maps to 2 Dice index 23 = 4-5 point = 9
index 141: 4-6-3 point = 13 ways = 21 Maps to 2 Dice index 24 = 4-6 point = 10
index 142: 4-6-4 point = 14 ways = 15 Maps to 2 Dice index 26 = 5-2 point = 7
index 143: 4-6-5 point = 15 ways = 10 Maps to 2 Dice index 30 = 5-6 point = 11
index 144: 4-6-6 point = 16 ways = 6 Maps to 2 Dice index 36 = 6-6 point = 12
index 145: 5-1-1 point = 7 ways = 15 Maps to 2 Dice index 6 = 1-6 point = 7
index 146: 5-1-2 point = 8 ways = 21 Maps to 2 Dice index 8 = 2-2 point = 4
index 147: 5-1-3 point = 9 ways = 25 Maps to 2 Dice index 14 = 3-2 point = 5
index 148: 5-1-4 point = 10 ways = 27 Maps to 2 Dice index 10 = 2-4 point = 6
index 149: 5-1-5 point = 11 ways = 27 Maps to 2 Dice index 18 = 3-6 point = 9
index 150: 5-1-6 point = 12 ways = 25 Maps to 2 Dice index 28 = 5-4 point = 9
index 151: 5-2-1 point = 8 ways = 21 Maps to 2 Dice index 8 = 2-2 point = 4
index 152: 5-2-2 point = 9 ways = 25 Maps to 2 Dice index 19 = 4-1 point = 5
index 153: 5-2-3 point = 10 ways = 27 Maps to 2 Dice index 15 = 3-3 point = 6
index 154: 5-2-4 point = 11 ways = 27 Maps to 2 Dice index 22 = 4-4 point = 8
index 155: 5-2-5 point = 12 ways = 25 Maps to 2 Dice index 33 = 6-3 point = 9
index 156: 5-2-6 point = 13 ways = 21 Maps to 2 Dice index 29 = 5-5 point = 10
index 157: 5-3-1 point = 9 ways = 25 Maps to 2 Dice index 14 = 3-2 point = 5
index 158: 5-3-2 point = 10 ways = 27 Maps to 2 Dice index 15 = 3-3 point = 6
index 159: 5-3-3 point = 11 ways = 27 Maps to 2 Dice index 17 = 3-5 point = 8
index 160: 5-3-4 point = 12 ways = 25 Maps to 2 Dice index 12 = 2-6 point = 8
index 161: 5-3-5 point = 13 ways = 21 Maps to 2 Dice index 34 = 6-4 point = 10
index 162: 5-3-6 point = 14 ways = 15 Maps to 2 Dice index 21 = 4-3 point = 7
index 163: 5-4-1 point = 10 ways = 27 Maps to 2 Dice index 10 = 2-4 point = 6
index 164: 5-4-2 point = 11 ways = 27 Maps to 2 Dice index 22 = 4-4 point = 8
index 165: 5-4-3 point = 12 ways = 25 Maps to 2 Dice index 12 = 2-6 point = 8
index 166: 5-4-4 point = 13 ways = 21 Maps to 2 Dice index 33 = 6-3 point = 9
index 167: 5-4-5 point = 14 ways = 15 Maps to 2 Dice index 26 = 5-2 point = 7
index 168: 5-4-6 point = 15 ways = 10 Maps to 2 Dice index 30 = 5-6 point = 11
index 169: 5-5-1 point = 11 ways = 27 Maps to 2 Dice index 18 = 3-6 point = 9
index 170: 5-5-2 point = 12 ways = 25 Maps to 2 Dice index 33 = 6-3 point = 9
index 171: 5-5-3 point = 13 ways = 21 Maps to 2 Dice index 34 = 6-4 point = 10
index 172: 5-5-4 point = 14 ways = 15 Maps to 2 Dice index 26 = 5-2 point = 7
index 173: 5-5-5 point = 15 ways = 10 Maps to 2 Dice index 31 = 6-1 point = 7
index 174: 5-5-6 point = 16 ways = 6 Maps to 2 Dice index 35 = 6-5 point = 11
index 175: 5-6-1 point = 12 ways = 25 Maps to 2 Dice index 28 = 5-4 point = 9
index 176: 5-6-2 point = 13 ways = 21 Maps to 2 Dice index 29 = 5-5 point = 10
index 177: 5-6-3 point = 14 ways = 15 Maps to 2 Dice index 21 = 4-3 point = 7
index 178: 5-6-4 point = 15 ways = 10 Maps to 2 Dice index 30 = 5-6 point = 11
index 179: 5-6-5 point = 16 ways = 6 Maps to 2 Dice index 35 = 6-5 point = 11
index 180: 5-6-6 point = 17 ways = 3 Maps to 2 Dice index 36 = 6-6 point = 12
index 181: 6-1-1 point = 8 ways = 21 Maps to 2 Dice index 3 = 1-3 point = 4
index 182: 6-1-2 point = 9 ways = 25 Maps to 2 Dice index 9 = 2-3 point = 5
index 183: 6-1-3 point = 10 ways = 27 Maps to 2 Dice index 5 = 1-5 point = 6
index 184: 6-1-4 point = 11 ways = 27 Maps to 2 Dice index 32 = 6-2 point = 8
index 185: 6-1-5 point = 12 ways = 25 Maps to 2 Dice index 28 = 5-4 point = 9
index 186: 6-1-6 point = 13 ways = 21 Maps to 2 Dice index 34 = 6-4 point = 10
index 187: 6-2-1 point = 9 ways = 25 Maps to 2 Dice index 9 = 2-3 point = 5
index 188: 6-2-2 point = 10 ways = 27 Maps to 2 Dice index 19 = 4-1 point = 5
index 189: 6-2-3 point = 11 ways = 27 Maps to 2 Dice index 27 = 5-3 point = 8
index 190: 6-2-4 point = 12 ways = 25 Maps to 2 Dice index 23 = 4-5 point = 9
index 191: 6-2-5 point = 13 ways = 21 Maps to 2 Dice index 29 = 5-5 point = 10
index 192: 6-2-6 point = 14 ways = 15 Maps to 2 Dice index 31 = 6-1 point = 7
index 193: 6-3-1 point = 10 ways = 27 Maps to 2 Dice index 5 = 1-5 point = 6
index 194: 6-3-2 point = 11 ways = 27 Maps to 2 Dice index 27 = 5-3 point = 8
index 195: 6-3-3 point = 12 ways = 25 Maps to 2 Dice index 18 = 3-6 point = 9
index 196: 6-3-4 point = 13 ways = 21 Maps to 2 Dice index 24 = 4-6 point = 10
index 197: 6-3-5 point = 14 ways = 15 Maps to 2 Dice index 21 = 4-3 point = 7
index 198: 6-3-6 point = 15 ways = 10 Maps to 2 Dice index 35 = 6-5 point = 11
index 199: 6-4-1 point = 11 ways = 27 Maps to 2 Dice index 32 = 6-2 point = 8
index 200: 6-4-2 point = 12 ways = 25 Maps to 2 Dice index 23 = 4-5 point = 9
index 201: 6-4-3 point = 13 ways = 21 Maps to 2 Dice index 24 = 4-6 point = 10
index 202: 6-4-4 point = 14 ways = 15 Maps to 2 Dice index 26 = 5-2 point = 7
index 203: 6-4-5 point = 15 ways = 10 Maps to 2 Dice index 30 = 5-6 point = 11
index 204: 6-4-6 point = 16 ways = 6 Maps to 2 Dice index 36 = 6-6 point = 12
index 205: 6-5-1 point = 12 ways = 25 Maps to 2 Dice index 28 = 5-4 point = 9
index 206: 6-5-2 point = 13 ways = 21 Maps to 2 Dice index 29 = 5-5 point = 10
index 207: 6-5-3 point = 14 ways = 15 Maps to 2 Dice index 21 = 4-3 point = 7
index 208: 6-5-4 point = 15 ways = 10 Maps to 2 Dice index 30 = 5-6 point = 11
index 209: 6-5-5 point = 16 ways = 6 Maps to 2 Dice index 35 = 6-5 point = 11
index 210: 6-5-6 point = 17 ways = 3 Maps to 2 Dice index 36 = 6-6 point = 12
index 211: 6-6-1 point = 13 ways = 21 Maps to 2 Dice index 34 = 6-4 point = 10
index 212: 6-6-2 point = 14 ways = 15 Maps to 2 Dice index 31 = 6-1 point = 7
index 213: 6-6-3 point = 15 ways = 10 Maps to 2 Dice index 35 = 6-5 point = 11
index 214: 6-6-4 point = 16 ways = 6 Maps to 2 Dice index 36 = 6-6 point = 12
index 215: 6-6-5 point = 17 ways = 3 Maps to 2 Dice index 36 = 6-6 point = 12
index 216: 6-6-6 point = 18 ways = 1 Maps to 2 Dice index 31 = 6-1 point = 7
3 7
4 2 2 2
5 2 3 2 3 3 2
6 3 3 3 3 3 7 3 3 3 3
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
8 4 4 4 4 4 4 4 4 5 4 4 4 5 5 4 4 4 4 4 4 4
9 5 5 5 5 5 5 5 6 6 5 5 5 6 7 6 5 5 6 6 5 5 5 5 5 5
10 6 6 6 6 5 6 6 6 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 5 6
11 8 9 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 8 8 8 9 8 8 8 8
12 9 9 9 9 9 9 8 8 9 9 8 7 8 9 9 9 8 8 9 9 9 9 9 9 9
13 10 10 10 10 10 10 10 9 9 10 10 10 9 10 10 10 10 10 10 10 10
14 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
15 11 11 11 11 7 11 11 11 11 11
16 12 11 11 12 11 12
17 12 12 12
18 7
NEXT: the reverse mapping table...
18) Reverse mapping of 2 dice roll from 3 dice roll
The reverse mapping shows which 3d rolls result from d2 rolls...
1) 1-1 1-1-2 1-1-3 1-2-1 1-3-1 2-1-1 3-1-1
2) 1-2 1-1-4 1-2-2 1-2-3 1-3-2 1-4-1 2-1-2 2-1-3 2-2-1 2-3-1 3-1-2 3-2-1 4-1-1
3) 1-3 1-1-6 1-3-4 1-4-3 1-6-1 2-2-4 2-4-2 3-1-4 3-4-1 4-1-3 4-2-2 4-3-1 6-1-1
4) 1-4 1-4-4 2-2-5 2-2-6 2-3-3 2-5-2 2-6-2 3-2-3 3-3-2 4-1-4 4-4-1 5-2-2 6-2-2
5) 1-5 1-3-6 1-6-3 2-3-4 2-4-3 3-1-6 3-2-4 3-4-2 3-6-1 4-2-3 4-3-2 6-1-3 6-3-1
6) 1-6 1-1-1 1-1-5 1-5-1 2-2-2 2-6-6 3-3-3 4-4-4 5-1-1 5-5-5 6-2-6 6-6-2 6-6-6
7) 2-1 1-1-4 1-2-2 1-2-3 1-3-2 1-4-1 2-1-2 2-1-3 2-2-1 2-3-1 3-1-2 3-2-1 4-1-1
8) 2-2 1-2-5 1-5-2 2-1-5 2-5-1 5-1-2 5-2-1
9) 2-3 1-2-6 1-3-5 1-5-3 1-6-2 2-1-6 2-6-1 3-1-5 3-5-1 5-1-3 5-3-1 6-1-2 6-2-1
10) 2-4 1-4-5 1-5-4 2-4-4 3-3-4 3-4-3 4-1-5 4-2-4 4-3-3 4-4-2 4-5-1 5-1-4 5-4-1
11) 2-5 1-2-4 1-4-2 2-1-4 2-4-1 4-1-2 4-2-1 4-4-6 4-5-5 4-6-4 5-4-5 5-5-4 6-4-4
12) 2-6 1-4-6 1-6-4 3-4-5 3-5-4 4-1-6 4-3-5 4-5-3 4-6-1 5-3-4 5-4-3 6-1-4 6-4-1
13) 3-1 1-1-6 1-3-4 1-4-3 1-6-1 2-2-4 2-4-2 3-1-4 3-4-1 4-1-3 4-2-2 4-3-1 6-1-1
14) 3-2 1-2-6 1-3-5 1-5-3 1-6-2 2-1-6 2-6-1 3-1-5 3-5-1 5-1-3 5-3-1 6-1-2 6-2-1
15) 3-3 2-3-5 2-5-3 3-2-5 3-5-2 5-2-3 5-3-2
16) 3-4 1-3-3 2-2-3 2-3-2 3-1-3 3-2-2 3-3-1 3-5-6 3-6-5 5-3-6 5-6-3 6-3-5 6-5-3
17) 3-5 2-3-6 2-6-3 3-2-6 3-3-5 3-4-4 3-5-3 3-6-2 4-3-4 4-4-3 5-3-3 6-2-3 6-3-2
18) 3-6 1-5-5 2-5-5 3-3-6 3-6-3 4-4-5 4-5-4 5-1-5 5-2-5 5-4-4 5-5-1 5-5-2 6-3-3
19) 4-1 1-4-4 2-2-5 2-2-6 2-3-3 2-5-2 2-6-2 3-2-3 3-3-2 4-1-4 4-4-1 5-2-2 6-2-2
20) 4-2 1-4-5 1-5-4 2-4-4 3-3-4 3-4-3 4-1-5 4-2-4 4-3-3 4-4-2 4-5-1 5-1-4 5-4-1
21) 4-3 1-3-3 2-2-3 2-3-2 3-1-3 3-2-2 3-3-1 3-5-6 3-6-5 5-3-6 5-6-3 6-3-5 6-5-3
22) 4-4 2-4-5 2-5-4 4-2-5 4-5-2 5-2-4 5-4-2
23) 4-5 1-5-6 1-6-5 2-4-6 2-6-4 4-2-6 4-6-2 5-1-6 5-6-1 6-1-5 6-2-4 6-4-2 6-5-1
24) 4-6 1-6-6 3-4-6 3-5-5 3-6-4 4-3-6 4-6-3 5-3-5 5-5-3 6-1-6 6-3-4 6-4-3 6-6-1
25) 5-1 1-3-6 1-6-3 2-3-4 2-4-3 3-1-6 3-2-4 3-4-2 3-6-1 4-2-3 4-3-2 6-1-3 6-3-1
26) 5-2 1-2-4 1-4-2 2-1-4 2-4-1 4-1-2 4-2-1 4-4-6 4-5-5 4-6-4 5-4-5 5-5-4 6-4-4
27) 5-3 2-3-6 2-6-3 3-2-6 3-3-5 3-4-4 3-5-3 3-6-2 4-3-4 4-4-3 5-3-3 6-2-3 6-3-2
28) 5-4 1-5-6 1-6-5 2-4-6 2-6-4 4-2-6 4-6-2 5-1-6 5-6-1 6-1-5 6-2-4 6-4-2 6-5-1
29) 5-5 2-5-6 2-6-5 5-2-6 5-6-2 6-2-5 6-5-2
30) 5-6 3-6-6 4-5-6 4-6-5 5-4-6 5-5-6 5-6-4 5-6-5 6-3-6 6-4-5 6-5-4 6-5-5 6-6-3
31) 6-1 1-1-1 1-1-5 1-5-1 2-2-2 2-6-6 3-3-3 4-4-4 5-1-1 5-5-5 6-2-6 6-6-2 6-6-6
32) 6-2 1-4-6 1-6-4 3-4-5 3-5-4 4-1-6 4-3-5 4-5-3 4-6-1 5-3-4 5-4-3 6-1-4 6-4-1
33) 6-3 1-5-5 2-5-5 3-3-6 3-6-3 4-4-5 4-5-4 5-1-5 5-2-5 5-4-4 5-5-1 5-5-2 6-3-3
34) 6-4 1-6-6 3-4-6 3-5-5 3-6-4 4-3-6 4-6-3 5-3-5 5-5-3 6-1-6 6-3-4 6-4-3 6-6-1
35) 6-5 3-6-6 4-5-6 4-6-5 5-4-6 5-5-6 5-6-4 5-6-5 6-3-6 6-4-5 6-5-4 6-5-5 6-6-3
36) 6-6 4-6-6 5-6-6 6-4-6 6-5-6 6-6-4 6-6-5
The first distinct reverse mapping shows which distinct 3d rolls result from all d2 rolls...
1) 1-1 1-1-2 1-1-3
2) 1-2 1-1-4 1-2-2 1-2-3
3) 1-3 1-1-6 1-3-4 2-2-4
4) 1-4 1-4-4 2-2-5 2-2-6 2-3-3
5) 1-5 1-3-6 2-3-4
6) 1-6 1-1-1 1-1-5 2-2-2 2-6-6 3-3-3 4-4-4 5-5-5 6-6-6
7) 2-1 1-1-4 1-2-2 1-2-3
8) 2-2 1-2-5
9) 2-3 1-2-6 1-3-5
10) 2-4 1-4-5 2-4-4 3-3-4
11) 2-5 1-2-4 4-4-6 4-5-5
12) 2-6 1-4-6 3-4-5
13) 3-1 1-1-6 1-3-4 2-2-4
14) 3-2 1-2-6 1-3-5
15) 3-3 2-3-5
16) 3-4 1-3-3 2-2-3 3-5-6
17) 3-5 2-3-6 3-3-5 3-4-4
18) 3-6 1-5-5 2-5-5 3-3-6 4-4-5
19) 4-1 1-4-4 2-2-5 2-2-6 2-3-3
20) 4-2 1-4-5 2-4-4 3-3-4
21) 4-3 1-3-3 2-2-3 3-5-6
22) 4-4 2-4-5
23) 4-5 1-5-6 2-4-6
24) 4-6 1-6-6 3-4-6 3-5-5
25) 5-1 1-3-6 2-3-4
26) 5-2 1-2-4 4-4-6 4-5-5
27) 5-3 2-3-6 3-3-5 3-4-4
28) 5-4 1-5-6 2-4-6
29) 5-5 2-5-6
30) 5-6 3-6-6 4-5-6 5-5-6
31) 6-1 1-1-1 1-1-5 2-2-2 2-6-6 3-3-3 4-4-4 5-5-5 6-6-6
32) 6-2 1-4-6 3-4-5
33) 6-3 1-5-5 2-5-5 3-3-6 4-4-5
34) 6-4 1-6-6 3-4-6 3-5-5
35) 6-5 3-6-6 4-5-6 5-5-6
36) 6-6 4-6-6 5-6-6
The next distinct reverse mapping shows which 3d rolls result from distinct d2 rolls...
1-1 1-1-2 1-1-3 1-2-1 1-3-1 2-1-1 3-1-1
1-2 1-1-4 1-2-2 1-2-3 1-3-2 1-4-1 2-1-2 2-1-3 2-2-1 2-3-1 3-1-2 3-2-1 4-1-1
1-3 1-1-6 1-3-4 1-4-3 1-6-1 2-2-4 2-4-2 3-1-4 3-4-1 4-1-3 4-2-2 4-3-1 6-1-1
1-4 1-4-4 2-2-5 2-2-6 2-3-3 2-5-2 2-6-2 3-2-3 3-3-2 4-1-4 4-4-1 5-2-2 6-2-2
1-5 1-3-6 1-6-3 2-3-4 2-4-3 3-1-6 3-2-4 3-4-2 3-6-1 4-2-3 4-3-2 6-1-3 6-3-1
1-6 1-1-1 1-1-5 1-5-1 2-2-2 2-6-6 3-3-3 4-4-4 5-1-1 5-5-5 6-2-6 6-6-2 6-6-6
2-2 1-2-5 1-5-2 2-1-5 2-5-1 5-1-2 5-2-1
2-3 1-2-6 1-3-5 1-5-3 1-6-2 2-1-6 2-6-1 3-1-5 3-5-1 5-1-3 5-3-1 6-1-2 6-2-1
2-4 1-4-5 1-5-4 2-4-4 3-3-4 3-4-3 4-1-5 4-2-4 4-3-3 4-4-2 4-5-1 5-1-4 5-4-1
2-5 1-2-4 1-4-2 2-1-4 2-4-1 4-1-2 4-2-1 4-4-6 4-5-5 4-6-4 5-4-5 5-5-4 6-4-4
2-6 1-4-6 1-6-4 3-4-5 3-5-4 4-1-6 4-3-5 4-5-3 4-6-1 5-3-4 5-4-3 6-1-4 6-4-1
3-3 2-3-5 2-5-3 3-2-5 3-5-2 5-2-3 5-3-2
3-4 1-3-3 2-2-3 2-3-2 3-1-3 3-2-2 3-3-1 3-5-6 3-6-5 5-3-6 5-6-3 6-3-5 6-5-3
3-5 2-3-6 2-6-3 3-2-6 3-3-5 3-4-4 3-5-3 3-6-2 4-3-4 4-4-3 5-3-3 6-2-3 6-3-2
3-6 1-5-5 2-5-5 3-3-6 3-6-3 4-4-5 4-5-4 5-1-5 5-2-5 5-4-4 5-5-1 5-5-2 6-3-3
4-4 2-4-5 2-5-4 4-2-5 4-5-2 5-2-4 5-4-2
4-5 1-5-6 1-6-5 2-4-6 2-6-4 4-2-6 4-6-2 5-1-6 5-6-1 6-1-5 6-2-4 6-4-2 6-5-1
4-6 1-6-6 3-4-6 3-5-5 3-6-4 4-3-6 4-6-3 5-3-5 5-5-3 6-1-6 6-3-4 6-4-3 6-6-1
5-5 2-5-6 2-6-5 5-2-6 5-6-2 6-2-5 6-5-2
5-6 3-6-6 4-5-6 4-6-5 5-4-6 5-5-6 5-6-4 5-6-5 6-3-6 6-4-5 6-5-4 6-5-5 6-6-3
6-6 4-6-6 5-6-6 6-4-6 6-5-6 6-6-4 6-6-5
This 'clear' mapping shows which distinct 3d rolls result from distinct d2 rolls...
1-1 1-1-2 1-1-3
1-2 1-1-4 1-2-2 1-2-3
1-3 1-1-6 1-3-4 2-2-4
1-4 1-4-4 2-2-5 2-2-6 2-3-3
1-5 1-3-6 2-3-4
1-6 1-1-1 1-1-5 2-2-2 2-6-6 3-3-3 4-4-4 5-5-5 6-6-6
2-2 1-2-5
2-3 1-2-6 1-3-5
2-4 1-4-5 2-4-4 3-3-4
2-5 1-2-4 4-4-6 4-5-5
2-6 1-4-6 3-4-5
3-3 2-3-5
3-4 1-3-3 2-2-3 3-5-6
3-5 2-3-6 3-3-5 3-4-4
3-6 1-5-5 2-5-5 3-3-6 4-4-5
4-4 2-4-5
4-5 1-5-6 2-4-6
4-6 1-6-6 3-4-6 3-5-5
5-5 2-5-6
5-6 3-6-6 4-5-6 5-5-6
6-6 4-6-6 5-6-6
The point map show how the points, and parts of points, are mapped from distinct 2D rolls...
1-1: 2 1-1-2: 4 1-1-3: 5
1-2: 3 1-1-4: 6 1-2-2: 5 1-2-3: 6
1-3: 4 1-1-6: 8 1-3-4: 8 2-2-4: 8
1-4: 5 1-4-4: 9 2-2-5: 9 2-2-6: 10 2-3-3: 8
1-5: 6 1-3-6: 10 2-3-4: 9
1-6: 7 1-1-1: 3 1-1-5: 7 2-2-2: 6 2-6-6: 14 3-3-3: 9 4-4-4: 12 5-5-5: 15 6-6-6: 18
2-2: 4 1-2-5: 8
2-3: 5 1-2-6: 9 1-3-5: 9
2-4: 6 1-4-5: 10 2-4-4: 10 3-3-4: 10
2-5: 7 1-2-4: 7 4-4-6: 14 4-5-5: 14
2-6: 8 1-4-6: 11 3-4-5: 12
3-3: 6 2-3-5: 10
3-4: 7 1-3-3: 7 2-2-3: 7 3-5-6: 14
3-5: 8 2-3-6: 11 3-3-5: 11 3-4-4: 11
3-6: 9 1-5-5: 11 2-5-5: 12 3-3-6: 12 4-4-5: 13
4-4: 8 2-4-5: 11
4-5: 9 1-5-6: 12 2-4-6: 12
4-6: 10 1-6-6: 13 3-4-6: 13 3-5-5: 13
5-5: 10 2-5-6: 13
5-6: 11 3-6-6: 15 4-5-6: 15 5-5-6: 16
6-6: 12 4-6-6: 16 5-6-6: 17
The point map show how the points, and parts of points, are mapped from 2D points...
point 2: 1-1 1-1-2: 4 1-1-3: 5
point 3: 1-2 1-1-4: 6 1-2-2: 5 1-2-3: 6 2-1 1-1-4: 6 1-2-2: 5 1-2-3: 6
point 4: 1-3 1-1-6: 8 1-3-4: 8 2-2-4: 8 2-2 1-2-5: 8 3-1 1-1-6: 8 1-3-4: 8 2-2-4: 8
point 5: 1-4 1-4-4: 9 2-2-5: 9 2-2-6: 10 2-3-3: 8 2-3 1-2-6: 9 1-3-5: 9 3-2 1-2-6: 9 1-3-5: 9 4-1 1-4-4: 9 2-2-5: 9 2-2-6: 10 2-3-3: 8
point 6: 1-5 1-3-6: 10 2-3-4: 9 2-4 1-4-5: 10 2-4-4: 10 3-3-4: 10 3-3 2-3-5: 10 4-2 1-4-5: 10 2-4-4: 10 3-3-4: 10 5-1 1-3-6: 10 2-3-4: 9
point 7: 1-6 1-1-1: 3 1-1-5: 7 2-2-2: 6 2-6-6: 14 3-3-3: 9 4-4-4: 12 5-5-5: 15 6-6-6: 18 2-5 1-2-4: 7 4-4-6: 14 4-5-5: 14 3-4 1-3-3: 7 2-2-3: 7 3-5-6: 14 4-3 1-3-3: 7 2-2-3: 7 3-5-6: 14 5-2 1-2-4: 7 4-4-6: 14 4-5-5: 14 6-1 1-1-1: 3 1-1-5: 7 2-2-2: 6 2-6-6: 14 3-3-3: 9 4-4-4: 12 5-5-5: 15 6-6-6: 18
point 8: 2-6 1-4-6: 11 3-4-5: 12 3-5 2-3-6: 11 3-3-5: 11 3-4-4: 11 4-4 2-4-5: 11 5-3 2-3-6: 11 3-3-5: 11 3-4-4: 11 6-2 1-4-6: 11 3-4-5: 12
point 9: 3-6 1-5-5: 11 2-5-5: 12 3-3-6: 12 4-4-5: 13 4-5 1-5-6: 12 2-4-6: 12 5-4 1-5-6: 12 2-4-6: 12 6-3 1-5-5: 11 2-5-5: 12 3-3-6: 12 4-4-5: 13
point 10: 4-6 1-6-6: 13 3-4-6: 13 3-5-5: 13 5-5 2-5-6: 13 6-4 1-6-6: 13 3-4-6: 13 3-5-5: 13
point 11: 5-6 3-6-6: 15 4-5-6: 15 5-5-6: 16 6-5 3-6-6: 15 4-5-6: 15 5-5-6: 16
point 12: 6-6 4-6-6: 16 5-6-6: 17
The 'cleaner' point map is...
point 2: 4 5
point 3: 6 5 6 6 5 6
point 4: 8 8 8 8 8 8 8
point 5: 9 9 10 8 9 9 9 9 9 9 10 8
point 6: 10 9 10 10 10 10 10 10 10 10 9
point 7: 3 7 6 14 9 12 15 18 7 14 14 7 7 14 7 7 14 7 14 14 3 7 6 14 9 12 15 18
point 8: 11 12 11 11 11 11 11 11 11 11 12
point 9: 11 12 12 13 12 12 12 12 11 12 12 13
point 10: 13 13 13 13 13 13 13
point 11: 15 15 16 15 15 16
point 12: 16 17
The 'cleanest' point map is...
point 2: 4 5
point 3: 6 5
point 4: 8
point 5: 9 10 8
point 6: 10 9
point 7: 3 7 6 14 9 12 15 18
point 8: 11 12
point 9: 11 12 13
point 10: 13
point 11: 15 16
point 12: 16 17
NEXT: the reverse ways table mapping...
19) Reverse ways table mapping of 2 dice roll from 3 dice roll
The two dice ways table derived from this maping is as expected...
point 2: 1-1 ways = 1
point 3: 1-2 2-1 ways = 2
point 4: 1-3 2-2 3-1 ways = 3
point 5: 1-4 2-3 3-2 4-1 ways = 4
point 6: 1-5 2-4 3-3 4-2 5-1 ways = 5
point 7: 1-6 2-5 3-4 4-3 5-2 6-1 ways = 6
point 8: 2-6 3-5 4-4 5-3 6-2 ways = 5
point 9: 3-6 4-5 5-4 6-3 ways = 4
point 10: 4-6 5-5 6-4 ways = 3
point 11: 5-6 6-5 ways = 2
point 12: 6-6 ways = 1
The full reverse ways table map is...
point 2: 1-1 1-1-2 1-1-3 1-2-1 1-3-1 2-1-1 3-1-1
point 3: 1-2 1-1-4 1-2-2 1-2-3 1-3-2 1-4-1 2-1-2 2-1-3 2-2-1 2-3-1 3-1-2 3-2-1 4-1-1 2-1 1-1-4 1-2-2 1-2-3 1-3-2 1-4-1 2-1-2 2-1-3 2-2-1 2-3-1 3-1-2 3-2-1 4-1-1
point 4: 1-3 1-1-6 1-3-4 1-4-3 1-6-1 2-2-4 2-4-2 3-1-4 3-4-1 4-1-3 4-2-2 4-3-1 6-1-1 2-2 1-2-5 1-5-2 2-1-5 2-5-1 5-1-2 5-2-1 3-1 1-1-6 1-3-4 1-4-3 1-6-1 2-2-4 2-4-2 3-1-4 3-4-1 4-1-3 4-2-2 4-3-1 6-1-1
point 5: 1-4 1-4-4 2-2-5 2-2-6 2-3-3 2-5-2 2-6-2 3-2-3 3-3-2 4-1-4 4-4-1 5-2-2 6-2-2 2-3 1-2-6 1-3-5 1-5-3 1-6-2 2-1-6 2-6-1 3-1-5 3-5-1 5-1-3 5-3-1 6-1-2 6-2-1 3-2 1-2-6 1-3-5 1-5-3 1-6-2 2-1-6 2-6-1 3-1-5 3-5-1 5-1-3 5-3-1 6-1-2 6-2-1 4-1 1-4-4 2-2-5 2-2-6 2-3-3 2-5-2 2-6-2 3-2-3 3-3-2 4-1-4 4-4-1 5-2-2 6-2-2
point 6: 1-5 1-3-6 1-6-3 2-3-4 2-4-3 3-1-6 3-2-4 3-4-2 3-6-1 4-2-3 4-3-2 6-1-3 6-3-1 2-4 1-4-5 1-5-4 2-4-4 3-3-4 3-4-3 4-1-5 4-2-4 4-3-3 4-4-2 4-5-1 5-1-4 5-4-1 3-3 2-3-5 2-5-3 3-2-5 3-5-2 5-2-3 5-3-2 4-2 1-4-5 1-5-4 2-4-4 3-3-4 3-4-3 4-1-5 4-2-4 4-3-3 4-4-2 4-5-1 5-1-4 5-4-1 5-1 1-3-6 1-6-3 2-3-4 2-4-3 3-1-6 3-2-4 3-4-2 3-6-1 4-2-3 4-3-2 6-1-3 6-3-1
point 7: 1-6 1-1-1 1-1-5 1-5-1 2-2-2 2-6-6 3-3-3 4-4-4 5-1-1 5-5-5 6-2-6 6-6-2 6-6-6 2-5 1-2-4 1-4-2 2-1-4 2-4-1 4-1-2 4-2-1 4-4-6 4-5-5 4-6-4 5-4-5 5-5-4 6-4-4 3-4 1-3-3 2-2-3 2-3-2 3-1-3 3-2-2 3-3-1 3-5-6 3-6-5 5-3-6 5-6-3 6-3-5 6-5-3 4-3 1-3-3 2-2-3 2-3-2 3-1-3 3-2-2 3-3-1 3-5-6 3-6-5 5-3-6 5-6-3 6-3-5 6-5-3 5-2 1-2-4 1-4-2 2-1-4 2-4-1 4-1-2 4-2-1 4-4-6 4-5-5 4-6-4 5-4-5 5-5-4 6-4-4 6-1 1-1-1 1-1-5 1-5-1 2-2-2 2-6-6 3-3-3 4-4-4 5-1-1 5-5-5 6-2-6 6-6-2 6-6-6
point 8: 2-6 1-4-6 1-6-4 3-4-5 3-5-4 4-1-6 4-3-5 4-5-3 4-6-1 5-3-4 5-4-3 6-1-4 6-4-1 3-5 2-3-6 2-6-3 3-2-6 3-3-5 3-4-4 3-5-3 3-6-2 4-3-4 4-4-3 5-3-3 6-2-3 6-3-2 4-4 2-4-5 2-5-4 4-2-5 4-5-2 5-2-4 5-4-2 5-3 2-3-6 2-6-3 3-2-6 3-3-5 3-4-4 3-5-3 3-6-2 4-3-4 4-4-3 5-3-3 6-2-3 6-3-2 6-2 1-4-6 1-6-4 3-4-5 3-5-4 4-1-6 4-3-5 4-5-3 4-6-1 5-3-4 5-4-3 6-1-4 6-4-1
point 9: 3-6 1-5-5 2-5-5 3-3-6 3-6-3 4-4-5 4-5-4 5-1-5 5-2-5 5-4-4 5-5-1 5-5-2 6-3-3 4-5 1-5-6 1-6-5 2-4-6 2-6-4 4-2-6 4-6-2 5-1-6 5-6-1 6-1-5 6-2-4 6-4-2 6-5-1 5-4 1-5-6 1-6-5 2-4-6 2-6-4 4-2-6 4-6-2 5-1-6 5-6-1 6-1-5 6-2-4 6-4-2 6-5-1 6-3 1-5-5 2-5-5 3-3-6 3-6-3 4-4-5 4-5-4 5-1-5 5-2-5 5-4-4 5-5-1 5-5-2 6-3-3
point 10: 4-6 1-6-6 3-4-6 3-5-5 3-6-4 4-3-6 4-6-3 5-3-5 5-5-3 6-1-6 6-3-4 6-4-3 6-6-1 5-5 2-5-6 2-6-5 5-2-6 5-6-2 6-2-5 6-5-2 6-4 1-6-6 3-4-6 3-5-5 3-6-4 4-3-6 4-6-3 5-3-5 5-5-3 6-1-6 6-3-4 6-4-3 6-6-1
point 11: 5-6 3-6-6 4-5-6 4-6-5 5-4-6 5-5-6 5-6-4 5-6-5 6-3-6 6-4-5 6-5-4 6-5-5 6-6-3 6-5 3-6-6 4-5-6 4-6-5 5-4-6 5-5-6 5-6-4 5-6-5 6-3-6 6-4-5 6-5-4 6-5-5 6-6-3
point 12: 6-6 4-6-6 5-6-6 6-4-6 6-5-6 6-6-4 6-6-5
The 'clean' reverse ways table map shows only the distinct three dice rolls...
point 2: 1-1 1-1-2 1-1-3
point 3: 1-2 1-1-4 1-2-2 1-2-3 2-1 1-1-4 1-2-2 1-2-3
point 4: 1-3 1-1-6 1-3-4 2-2-4 2-2 1-2-5 3-1 1-1-6 1-3-4 2-2-4
point 5: 1-4 1-4-4 2-2-5 2-2-6 2-3-3 2-3 1-2-6 1-3-5 3-2 1-2-6 1-3-5 4-1 1-4-4 2-2-5 2-2-6 2-3-3
point 6: 1-5 1-3-6 2-3-4 2-4 1-4-5 2-4-4 3-3-4 3-3 2-3-5 4-2 1-4-5 2-4-4 3-3-4 5-1 1-3-6 2-3-4
point 7: 1-6 1-1-1 1-1-5 2-2-2 2-6-6 3-3-3 4-4-4 5-5-5 6-6-6 2-5 1-2-4 4-4-6 4-5-5 3-4 1-3-3 2-2-3 3-5-6 4-3 1-3-3 2-2-3 3-5-6 5-2 1-2-4 4-4-6 4-5-5 6-1 1-1-1 1-1-5 2-2-2 2-6-6 3-3-3 4-4-4 5-5-5 6-6-6
point 8: 2-6 1-4-6 3-4-5 3-5 2-3-6 3-3-5 3-4-4 4-4 2-4-5 5-3 2-3-6 3-3-5 3-4-4 6-2 1-4-6 3-4-5
point 9: 3-6 1-5-5 2-5-5 3-3-6 4-4-5 4-5 1-5-6 2-4-6 5-4 1-5-6 2-4-6 6-3 1-5-5 2-5-5 3-3-6 4-4-5
point 10: 4-6 1-6-6 3-4-6 3-5-5 5-5 2-5-6 6-4 1-6-6 3-4-6 3-5-5
point 11: 5-6 3-6-6 4-5-6 5-5-6 6-5 3-6-6 4-5-6 5-5-6
point 12: 6-6 4-6-6 5-6-6
NEXT: The Master Map
20) The Master Map starting from an indistinguishable 3-dice roll
The Master Map starting from an indistinguishable 3-dice roll
Starting from the 56 distinct combinations from an indistinguishable 3-dice roll, we map to a 1 to 36 range.
From the 1 to 36 range we map to the following ranges:
a 2-dice roll
a 1 to 18 range
a 1 to 12 range
a 1 to 9 range
a 1 to 6 range, (I.E. a 1-die roll)
a 1 to 4 range
a 1 to 3 range
a 1 to 2 range
3-Dice (3-D Point) 1 to 36 2-dice (2-D Point) 1 to 18 1 to 12 1 to 9 1 to 6 1 to 4 1 to 3 1 to 2
------ ----------- ------- ------ ----------- ------- ------- ------ ------ ------ ------ ------
1 1-1-1 ( 3) 6 1-6 ( 7) 6 6 5 4 2 2 2
2 1-1-2 ( 4) 1 1-1 ( 2) 1 1 1 1 1 1 1
3 1-1-3 ( 5) 1 1-1 ( 2) 1 1 1 1 1 1 1
4 1-1-4 ( 6) 2 1-2 ( 3) 2 2 1 1 1 1 1
5 1-1-5 ( 7) 6 1-6 ( 7) 6 6 5 4 2 2 2
6 1-1-6 ( 8) 3 1-3 ( 4) 3 3 2 1 1 1 1
7 1-2-2 ( 5) 2 1-2 ( 3) 2 2 1 1 1 1 1
8 1-2-3 ( 6) 7 2-1 ( 3) 7 7 1 1 1 1 1
9 1-2-4 ( 7) 11 2-5 ( 7) 11 11 6 4 3 2 2
10 1-2-5 ( 8) 8 2-2 ( 4) 8 8 1 1 2 1 1
11 1-2-6 ( 9) 9 2-3 ( 5) 9 9 3 2 1 1 1
12 1-3-3 ( 7) 16 3-4 ( 7) 16 4 6 4 3 2 2
13 1-3-4 ( 8) 13 3-1 ( 4) 13 1 2 1 1 1 1
14 1-3-5 ( 9) 14 3-2 ( 5) 14 2 3 2 1 1 1
15 1-3-6 (10) 5 1-5 ( 6) 5 5 3 2 2 1 1
16 1-4-4 ( 9) 4 1-4 ( 5) 4 4 2 2 1 1 1
17 1-4-5 (10) 10 2-4 ( 6) 10 10 4 3 2 2 1
18 1-4-6 (11) 32 6-2 ( 8) 14 8 5 3 2 2 1
19 1-5-5 (11) 18 3-6 ( 9) 18 6 8 5 4 3 2
20 1-5-6 (12) 28 5-4 ( 9) 10 4 7 5 4 3 2
21 1-6-6 (13) 34 6-4 (10) 16 10 8 6 4 3 2
22 2-2-2 ( 6) 6 1-6 ( 7) 6 6 5 4 2 2 2
23 2-2-3 ( 7) 16 3-4 ( 7) 16 4 6 4 3 2 2
24 2-2-4 ( 8) 3 1-3 ( 4) 3 3 2 1 1 1 1
25 2-2-5 ( 9) 19 4-1 ( 5) 1 7 2 2 1 1 1
26 2-2-6 (10) 19 4-1 ( 5) 1 7 2 2 1 1 1
27 2-3-3 ( 8) 4 1-4 ( 5) 4 4 2 2 1 1 1
28 2-3-4 ( 9) 25 5-1 ( 6) 7 1 3 2 2 1 1
29 2-3-5 (10) 15 3-3 ( 6) 15 3 4 3 3 2 1
30 2-3-6 (11) 27 5-3 ( 8) 9 3 7 5 3 3 2
31 2-4-4 (10) 20 4-2 ( 6) 2 8 4 3 2 2 1
32 2-4-5 (11) 22 4-4 ( 8) 4 10 4 3 3 2 1
33 2-4-6 (12) 23 4-5 ( 9) 5 11 7 5 4 3 2
34 2-5-5 (12) 33 6-3 ( 9) 15 9 8 5 4 3 2
35 2-5-6 (13) 29 5-5 (10) 11 5 9 6 3 3 2
36 2-6-6 (14) 31 6-1 ( 7) 13 7 5 4 2 2 2
37 3-3-3 ( 9) 6 1-6 ( 7) 6 6 5 4 2 2 2
38 3-3-4 (10) 20 4-2 ( 6) 2 8 4 3 2 2 1
39 3-3-5 (11) 17 3-5 ( 8) 17 5 7 5 3 3 2
40 3-3-6 (12) 18 3-6 ( 9) 18 6 8 5 4 3 2
41 3-4-4 (11) 17 3-5 ( 8) 17 5 7 5 3 3 2
42 3-4-5 (12) 12 2-6 ( 8) 12 12 5 3 2 2 1
43 3-4-6 (13) 24 4-6 (10) 6 12 8 6 4 3 2
44 3-5-5 (13) 34 6-4 (10) 16 10 8 6 4 3 2
45 3-5-6 (14) 21 4-3 ( 7) 3 9 6 4 3 2 2
46 3-6-6 (15) 35 6-5 (11) 17 11 9 6 4 3 2
47 4-4-4 (12) 31 6-1 ( 7) 13 7 5 4 2 2 2
48 4-4-5 (13) 33 6-3 ( 9) 15 9 8 5 4 3 2
49 4-4-6 (14) 26 5-2 ( 7) 8 2 6 4 3 2 2
50 4-5-5 (14) 26 5-2 ( 7) 8 2 6 4 3 2 2
51 4-5-6 (15) 30 5-6 (11) 12 6 9 6 4 3 2
52 4-6-6 (16) 36 6-6 (12) 18 12 9 6 4 3 2
53 5-5-5 (15) 31 6-1 ( 7) 13 7 5 4 2 2 2
54 5-5-6 (16) 35 6-5 (11) 17 11 9 6 4 3 2
55 5-6-6 (17) 36 6-6 (12) 18 12 9 6 4 3 2
56 6-6-6 (18) 31 6-1 ( 7) 13 7 5 4 2 2 2
21) The Master Map starting from a 1 to 38 spin
The Master Map starting from a 1 to 38 spin
From the 1 to 36 range we map to the ranges show above
22) Choice of Maps for a 3 dice roll into a 2 dice outcome
The reason for the map being the way its is, well,...
Although SpikerSystems invented chip less gaming in 1990, it was not the first-to-market and now considers the technology as simply
an automation of widely-used common procedures which cannot be patented.
However, SpikerSystems is the world's foremost authority on the mathematics of the cubes and the pioneer of e-gaming, and has kept itself
three steps ahead of the competition.
By developing an easy-to-use database to define any kind of wager, and a framework upon which to test and modify the games developed,
SpikerSystems has in its pocket a rapid game development platform built
into its premium product name e-Gamer's Trice Suite
Spiker systems has used its own game development platform to rapidly test out new games and gaming strategies, leaving the competition in the dust.
Researchers at SpikerSystems have played many two and three dice games, including Ricochet and sic-bo, and came up with a brilliant innovation.
If 3 dice are thrown, the roll can also be used for 2 dice
and for other games. The patent-pending 3-dice to 2-dice roll map by SpikerSystems provides a brand new and exciting frontier to dice games.
ThriceDice is a new game from SpikerSystems that provides concurrent games of TDC_7_14_trips_outs and 2-Dice Crapless Craps
all generated from the same 3-dice roll. Players can play both games concurrently and because of the point map's joining of the outs in both games,
both three-dice craps and 2-dice craps games will occur in
the same game phases; players of both games often root for the same rolls.
We need to map exactly 36 of the 216 rolls to the 2-D 7 point, because there are six ways to make a 7, the table-out in all 2-dice craps games.
The 36/216 = 1/6 just like the 6/36 = 1/6 in 2-D craps. By using the 7 and 14 and all tripples as the table-out, we have a game that has reasonable payback,
(461075766338924, slightly better than 2-D crapless-craps), and provides the 36 outs we need to map to the 2-D 7.
Also, the 3-D 7 is somewhat familiar to the 2-D player; he then only has to consider the 14 and all trips; easy to learn and remember.
There are several maps that can be used, but the one above is really sweet because it is both perfectly symmetrical, and it maps the 3-D point to the
2_dice points in order, as a best possible ordering, meaning that the 3-d numbers are all placed directly above the center of the 2-d counterparts
as best as possible.
This mapping took a long time for SpikerSystems to develop, using both pre-mapped 3-D outs to the 2-D 7, and then using a very complex algorithm
to order the various 3-d rolls and map them to the remaining 30 2-dice rolls.
But why go to all that trouble? Well,... Please Visit SpikerSystems.com for the answer...