By Marty Wollner

06-Aug-2023

watch?v=xHh0ui5mi_E (7:00) Matt Parker YouTube Video: “The Three Indistinguishable Dice Puzzle. See below for a solid method of validating that a map is correct!

Hot Hot!! Brand new as 04-Jul-2016: The COMPLETE_SPIKER_INSTALL_105B_202B_Rev_d.zip download, Windows10 tested!!! You can run all of the programs and games in these videos on your own computers. This is the culmination of 20+ man-years that I have personally poured in, along with my blood, sweat and tears. Literally. This is my masterpiece, totally free to experiment with. You can write and test your own custom strategies in my own custom NailsULA programming language for all of the 20 games, and save them to disk. You can deploy simulation testing with dozens of players using different strategies, but all playing at the same time (against the same dice rolls, for example, mapped to multiple games).

“SpikerSystems patent #8246446 solution”: This is VALID.

I have a granted US patent # 8246446 that fully describes the mapping, but none other than Galileo discovered it back in 1623 in a paper called SOPRA LE SCOPERTE DEI DADI.

I HAD a set of YouTube video responses that Matt mentioned, BUT MY YouTube channel was removed! THESE NOW POINT TO A PLAYLIST DESCRIBING THE INSTALLATION OF THE GAMING SYSTEM, WHICH YOU NOW DOWNLOAD FOR FREE (above). PART 1 of 3 WAS

https://www.youtube.com/watch?v=uBYVrkHge_I&list=PLbguTdnDdoe07IEPp7OCbLdU7H811bv05

32:08 Response to the three-dice puzzle, part 1 of 3 … started at 17:50 for the explanation of the map.

PART 2 of 3 WAS

https://www.youtube.com/watch?v=uBYVrkHge_I&list=PLbguTdnDdoe07IEPp7OCbLdU7H811bv05

26:30 Response to the three-dice puzzle, part 2 of 3

PART 3 of 3 WAS

https://www.youtube.com/watch?v=uBYVrkHge_I&list=PLbguTdnDdoe07IEPp7OCbLdU7H811bv05

12:09 Response to the three-dice puzzle, part 3 of 3

See also the Wizard of Vegas forum discussing this problem way back in 2011:

http://wizardofvegas.com/forum/questions-and-answers/math/3917-numeric-fair-mapping-from-a-3-dice-roll-into-a-2-dice-roll/

Summary of QUICK MAP for “SpikerSystems patent #8246446 solution”, describing the mapping verbally:

My solution’s POINT NUMBER MAPPING is bilaterally symmetric around both sides of the center of ways.

The 3d(i) 7 point, 14 point and all triples all map to the 2d(i) 7.

For all of the remaining point numbers, the low side of the 2d(i)’s map to the low side of 3d(i)’s, so you can intuitively map 3d POINT numbers “almost straight across” to the 2d point numbers mapped…

3 | 1-1-1A :1| 7-Out

4 | 1-1-2A :3|

5 | 1-1-3A :3 | 1-2-2A :3| point 2

6 | 1-1-4A :3 | 1-2-3A :6 | 2-2-2A :1| point 3

7 | 1-1-5A :3 | 1-2-4B :6 | 1-3-3C :3 | 2-2-3C :3| 7-out

8 | 1-1-6A :3 | 1-2-5A :6 | 1-3-4B :6 | 2-2-4A :3 | 2-3-3A :3| point 4

9 | 1-2-6B :6 | 1-3-5B :6 | 1-4-4A :3 | 2-2-5A :3 | 2-3-4A :6 | 3-3-3A :1| point 5

10 | 1-3-6A :6 | 1-4-5B :6 | 2-2-6A :3 | 2-3-5C :6 | 2-4-4B :3 | 3-3-4A :3 | point 6

11 | 1-4-6A :6 | 1-5-5A :3 | 2-3-6B :6 | 2-4-5C :6 | 3-3-5B :3 | 3-4-4B :3 | point 8

12 | 1-5-6B :6 | 2-4-6B :6 | 2-5-5A :3 | 3-3-6A :3 | 3-4-5A :6 | 4-4-4A :1| point 9

13 | 1-6-6A :3 | 2-5-6B :6 | 3-4-6A :6 | 3-5-5A :3 | 4-4-5A :3| point 10

14 | 2-6-6A :3 | 3-5-6B :6 | 4-4-6C :3 | 4-5-5C :3| 7-Out

15 | 3-6-6A :3 | 4-5-6A :6 | 5-5-5A :1| point 11

16 | 4-6-6A :3 | 5-5-6A :3| point 12

17 | 5-6-6A :3 |

18 | 6-6-6A :1 | 7-Out

For the “list of exceptions” to “almost straight across”, the following 3d mapped numbers are non-triples:

2d 2 ---- all 4, 5 as 113.

2d 3 ---- 5 as 122 all 6.

2d 4 ---- all 8 except 233. The HARD-4 is the 134.

2d 5 ---- 8 as 233, all 9 except 234, 10 as 226. The 2-3 is mapped to singles.

2d 6 ---- 9 as 234, all 10 except 226. The HARD-WAY is mapped to 235. The 1-5 is mapped to the non-145 singles.

ThreeDiceCraps_7_14_Trips_Outs. The 1-6 is mapped to all triples and 115, 266. The 2-5 is mapped to singles.

2d 8 ---- all 11 except 155, 12 as 345. The hard-way is mapped to 245. The 2-6 is mapped to the non-236 singles.

2d 9 ---- 11 as 155, all 12 except 345, 13 as 445. The 4-5 is mapped to singles.

2d 10 --- all 13 except 445. The HARD-10 is the 256.

2d 11 --- all 15, 16 as 556. (Blue is hard to see through.)

2d 12 --- 16 as 466, all 17.

Here is the solid method of validating that a map is correct::

For a map to be valid, each 2d number (shown as unique colors for each) requires a certain COUNT OF WAYS:

Point Dice WAYS POINT-WAYS GROUPINGS

----- ---------------------- ---- ---------------------

2 1-1A:1 1 1 2A(1)

3 1-2A:2 2 1 3A(2)

4 1-3A:2, 2-2B:1 3 2 4A(2), 4B(1)

5 1-4A:2, 2-3B:2 4 2 5A(2), 5B(2)

6 1-5A:2, 2-4B:2, 3-3C:1 5 3 6A(2), 6B(2), 6C(1)

7 1-6A:2, 2-5B:2, 3-4C:2 6 3 7A(2), 7B(2), 7C(2)

8 2-6A:2, 3-5B:2, 4-4C:1 5 3 8A(2), 8B(2), 8C(1)

9 3-6A:2, 4-5B:2 4 2 9A(2), 9B(2)

10 4-6A:2, 5-5B:1 3 2 10A(2), 10B(1)

11 5-6A:2 2 1 11A(2)

12 6-6A:1 1 1 12A(1)

The 21 distinct Point-Ways of 2-d rolls must be individually mapped to one or more of the 56 distinct ways that can be rolled from 3 indistinguishable dice:

point   2:        1-1A:1  1-1-2:3  1-1-3:3

point   3:        1-2A:2  1-1-4:3 1-2-2:3  1-2-3:6

point   4:        1-3A:2  1-1-6:3  1-3-4:6  2-2-4 :3      2-2B:1  1-2-5 :6

point   5:        1-4A:2  1-4-4:3  2-2-5:3  2-2-6:3  2-3-3:3       2-3B:2  1-2-6:6  1-3-5:6

point   6:        1-5A:2  1-3-6:6  2-3-4:6       2-4B:2  1-4-5:6  2-4-4:3  3-3-4:3       3-3C:1  2-3-5:6

point   7:        1-6A:2 1-1-1:1  1-1-5:3  2-2-2:1  2-6-6:3  3-3-3:1  4-4-4:1  5-5-5:1  6-6-6:1       2-5B:2  1-2-4:6  3-5-6:6      3-4C:2  1-3-3:3  2-2-3:3  4-4-6:3  4-5-5:3

point   8:        2-6A:2  1-4-6:6  3-4-5:6       3-5B:2  2-3-6:6  3-3-5:3  3-4-4:3       4-4C:1  2-4-5:6

point   9:        3-6A:2  1-5-5:3  2-5-5:3  3-3-6:3  4-4-5:3       4-5B:2  1-5-6:6  2-4-6:6

point  10:       4-6A:2  1-6-6:3  3-4-6:6  3-5-5:3       5-5B:1  2-5-6:6

point  11:       5-6A:2  3-6-6:3  4-5-6:6  5-5-6:3

point  12:       6-6A:1  4-6-6:3  5-6-6:3

3 | 1-1-1A :1|

4 | 1-1-2A :3|

5 | 1-1-3A :3 | 1-2-2A :3|

6 | 1-1-4A :3 | 1-2-3A :6 | 2-2-2A :1|

7 | 1-1-5A :3 | 1-2-4B :6 | 1-3-3C :3 | 2-2-3C :3|

8 | 1-1-6A :3 | 1-2-5A :6 | 1-3-4B :6 | 2-2-4A :3 | 2-3-3A :3|

9 | 1-2-6B :6 | 1-3-5B :6 | 1-4-4A :3 | 2-2-5A :3 | 2-3-4A :6 | 3-3-3A :1|

10 | 1-3-6A :6 | 1-4-5B :6 | 2-2-6A :3 | 2-3-5C :6 | 2-4-4B :3 | 3-3-4A :3 |

11 | 1-4-6A :6 | 1-5-5A :3 | 2-3-6B :6 | 2-4-5C :6 | 3-3-5B :3 | 3-4-4B :3 |

12 | 1-5-6B :6 | 2-4-6B :6 | 2-5-5A :3 | 3-3-6A :3 | 3-4-5A :6 | 4-4-4A :1|

13 | 1-6-6A :3 | 2-5-6B :6 | 3-4-6A :6 | 3-5-5A :3 | 4-4-5A :3|

14 | 2-6-6A :3 | 3-5-6B :6 | 4-4-6C :3 | 4-5-5C :3|

15 | 3-6-6A :3 | 4-5-6A :6 | 5-5-5A :1|

16 | 4-6-6A :3 | 5-5-6A :3|

17 | 5-6-6A :3|

18 | 6-6-6A :1|

Each of the 2d Point-Way Groups must map to either 6 or 12 of the 3d ways!!! This is required, or the mapping will be wrong as far as keeping the 2d game fair across all point numbers, and being able to tell the difference between the 2-d Point-Way Group for a given point, for example, being able to distinguish between a hard 4 (2-2) and a soft 4 (1-3).

The chart above makes it easy to verify this because the count of ways to make each of the 56 3d rolls is indicated as :1, :3 OR :6 and the count of ways to make each of the 21 2d rolls is indicated as :1 or :2. Just add these 3d counts up and make sure that 1/6 of the sum maps equally to the 2d point ways.

For example, count all of the yellow 3d numbers mapped: there are 2 of ‘em… them, right near the top of the 3d point ways list:

3 | 1-1-1A :1|

4 | 1-1-2A :3|

5 | 1-1-3A :3 | 1-2-2A :3|

Etc…

We have 1-1-2A :3 and 1-1-3A :3 so we add the :3 + :3 and we get 6 3-d ways that we will map to the 2d Point-Way Group 2A. The Point-Way Group 2A requires 1 way, and that is 1/6 of the 6 ways we are mapping from 3 dice. So the Point-Way Group 2A map is valid.

Next, the 3-point number… count all of the green 3d numbers mapped: there are 3 of ‘em… them, also near the top of the 3d point ways list:

4 | 1-1-2A :3|

5 | 1-1-3A :3 | 1-2-2A :3|

6 | 1-1-4A :3 | 1-2-3A :6 | 2-2-2A :1|

7 | 1-1-5A :3 | 1-2-4B :6 | 1-3-3C :3 | 2-2-3C :3|

We have 1-2-2A :3, 1-1-4A :3, and 1-2-3A :6 so we add the :3 + :3 + :6 and we get 12 3-d ways that we will map to the 2d Point-Way Group 3A. The Point-Way Group 3A requires 2 ways, and that is 1/6 of the 12 ways we are mapping from 3 dice. So the Point-Way Group 3A map is valid.

Next, the 4-point number. This has 2 ways, 4A and 4B. For 4A (the 1-3 or SOFT 4):

Count all of the turquoise 3d numbers mapped that specify A: there are 3 of ‘em… them, also in the upper part of the 3d point ways list:

7 | 1-1-5A :3 | 1-2-4B :6 | 1-3-3C :3 | 2-2-3C :3|

8 | 1-1-6A :3 | 1-2-5A :6 | 1-3-4B :6 | 2-2-4A :3 | 2-3-3A :3|

9 | 1-2-6B :6 | 1-3-5B :6 | 1-4-4A :3 | 2-2-5A :3 | 2-3-4A :6 | 3-3-3A :1|

We have 1-1-6A :3, 1-2-5A :6 and 2-2-4A :3 so we add the :3 + :6 + :3 and we get 12 3-d ways that we will map to the 2d Point-Way Group 4A. The Point-Way Group 4A requires 2 ways, and that is 1/6 of the 12 ways we are mapping from 3 dice. So the Point-Way Group 4A map is valid.

For 4B (the 2-2 or HARD 4):

Count all of the turquoise 3d numbers mapped that specify B: there is only 1 of ‘em… them, also in the upper part of the 3d point ways list:

7 | 1-1-5A :3 | 1-2-4B :6 | 1-3-3C :3 | 2-2-3C :3|

8 | 1-1-6A :3 | 1-2-5A :6 | 1-3-4B :6 | 2-2-4A :3 | 2-3-3A :3|

9 | 1-2-6B :6 | 1-3-5B :6 | 1-4-4A :3 | 2-2-5A :3 | 2-3-4A :6 | 3-3-3A :1|

We have 1-3-4B :6 so we have 6 3-d ways that we will map to the 2d Point-Way Group 4B. The Point-Way Group 4B requires 1 way, and that is 1/6 of the 6 ways we are mapping from 3 dice. So the Point-Way Group 4b map is valid.

The 2d 7-point is the only oddball, as far as locating it by lining up the 2d and 3d point numbers. This map is called the TDC_7_14_Trips_Outs map, meaning that any 3d 7, or 14, or triple maps to the 2d 7. Here are all of the 3d distinct rolls that we will map to the 2d 7-point:

1-1-1A :1, 2-2-2A :1, 1-1-5A :3, 1-2-4B :6, 1-3-3C :3, 2-2-3C :3, 3-3-3A :1,

4-4-4A :1, 2-6-6A :3, 3-5-6B :6, 4-4-6C :3, 4-5-5C :3, 5-5-5A :1, 6-6-6A :1

Group 7A (the 6-1):

1-1-1A :1, 2-2-2A :1, 1-1-5A :3, 3-3-3A :1, 4-4-4A :1, 2-6-6A :3, 5-5-5A :1, 6-6-6A :1

Adds up to :12 ways, satisfying the 2 2-d ways needed for Point-Way Group 7a.

Group 7B (the 5-2):

1-2-4B :6, 3-5-6B :6

Adds up to :12 ways, satisfying the 2 2-d ways needed for Point-Way Group 7b.

Group 7C (the 4-3):

1-3-3C :3, 2-2-3C :3, 4-4-6C :3, 4-5-5C :3

Adds up to :12 ways, satisfying the 2 2-d ways needed for Point-Way Group 7c.

In order for the map to be valid:

The 21 distinct Point-Ways of 2-d rolls must be individually mapped to one or more of the 56 distinct ways that can be rolled from 3 indistinguishable dice.

2-d HARD-WAYS should map to 3-d singles (:6) rolls for easy identification.

No empty spots can appear in the 3d layout.

It is impossible to map backwards from 2d to 3d without ambiguity; the map is one way from 3d to 2d.

Side Note: Mapping 36 of the 216 3d(d) rolls into the 2d(i) 7 fits 1/6^th of 3d(i) rolls into table-outs; the same proportion as played in 2-dice crapless-craps. So, this map allows games of “Three-Dice-Crapless-Craps, 7, 14, Trips-Outs” to be MAPPED INTO “Two-Dice Crapless-Craps”, a well-known variation of craps. These two games are thus played CONCURRENTLY and when done so, both games will run in synch with each other… the win/loss cycles of pass-line bets will be the same. One 3d(i) shooter can thus feed the randomization of both games, concurrently, reducing the overhead of casinos and yet still maintain live randomization. This is mapping of “TDC-7, 14, Trips-Outs” to “Crapless-Craps” is exactly what my patent# 8246446 is about!! That’s why I’m so interested in this puzzle from Matt Parker!!

Marty, 25-Apr-2016