Snooker scoring

Question

I was watching the world snooker championship and it got me wondering..

Snooker scoring

In the game of snooker there are certain rules that you must adhere too:

• When there are red balls on the table, during your turn you must first pot a red ball
• After potting each red ball, you must pot a colored (not red) ball (the potted colored ball is then replaced onto the table)
• After all the red balls are up (there are 15) you can first choose a colored ball and then you start with the lowest scoring ball and work your way up to the highest scoring ball (these are not replaced)
• Not potting at any point ends your turn.
• Points per ball
  • Red ball: 1 point
  • Yellow ball: 2 points
  • Green ball: 3 points
  • Brown ball: 4 points
  • Blue ball: 5 points
  • Pink ball: 6 points
  • Black ball: 7 points

The question

You start with a table with all the balls still on it - 15 red and one of each of the other coloured balls - and are given the score of a player in snooker after their first turn, what are the ways that they could have achieved this score?

Input will be a score going from 1 to 147. You can choose if it is an integer or a string. Output should be the different combinations of number of times you potted each ball.

Test cases:

Input: 4
Output: 
1r 1g
2r 1y
Input: 25
Output:
4r 3b
5r 2b 1br 1y
5r 2b 2g
...
9r 8y

Rules:

• You can choose whether you output the possibilities divided by a new line or a separator of some sort (/,;|\ or even others I'm missing)

This is codegolf, so the shortest code wins.

Answer 1

Jelly,  65  25 bytes

7Ḋṗ15Ė€F;ɗ€ƊÄċƇ⁸f€RŻ€IṢ€Q

A monadic Link that accepts an integer, "score", from \$[1,147]\$ and yields a list of lists - each being a potential selection of ball scores used in reaching the given score disregarding the order in which they were actually potted.

Try it with only six reds (15 replaced with 6; again, fifteen reds would take an age).

How?

7Ḋṗ15Ė€F;ɗ€ƊÄċƇ⁸f€RŻ€IṢ€Q - Link integer, S
7                         - seven
 Ḋ                        - dequeue -> coloured ball scores, [2..7]
           Ɗ              - last three links as a monad - f([2..7]):
   15                     -   fifteen
  ṗ                       -   ([2..7]) Cartesian power (15) -> all length 15 words
                                using the coloured ball scores as letters
          €               -   for each (such Word):
         ɗ                -     last three links as a dyad - f(Word, [2..7])
                                              e.g. Word = [7,...,7]
     Ė€                   -       enumerate each          [[1,7],...,[1,7]]
       F                  -       flatten                 [1,7,...,1,7]
        ;                 -       concatenate ([2..7])    [1,7,...,1,7,2,3,4,5,6,7] 
            Ä             - cumulative sums, i.e. convert each such legal way to
                              clear the table to a list of the running score
              Ƈ           - filter keep those for which this is truthy (non-zero):
               ⁸          -   chain's left argument (S)
             ċ            -   count - i.e. count of S in the running score
                  R       - range (input Score) -> [1..S]
                f€        - for each (running score list) filter keep ([1..S])
                   Ż€     - prefix each with a zero
                     I    - forward differences of each (back to ball scores)
                      Ṣ€  - sort each
                        Q - deduplicate

---

Original \$66\$ byte entry (\$65\$ really, since the trailing G is post-formatting):

L⁼30µÐfµ7Ḋ;\¤;€Ṣ€µ€;/
ċ1<⁴µÐfµ;Ç
7Ḋœċ⁴Ḷ¤;/L€;$€;@þ2B¤;/ḟ€0ÇS⁼¥Ðf⁸G

Well, it's too slow for TIO now!
...so here is a paste of the 2636 ways to make exactly 100 produced offline.
...and here is a version that will run there with just SIX reds (maximum break = 75)

Prints a grid of numbers each line being a space separated list of ball values (e.g. three red and two green would be on a line reading 1 1 1 3 3).

---

For a value-grouped version that prints lines of counts along with the full names of the balls, at 102 bytes:

ŒrÑ€
Ṫ;ị“¡^³ṗ⁼¬wḌ⁼ø÷OẏK¦ẆP»Ḳ¤$K
L⁼30µÐfµ7Ḋ;\¤;€Ṣ€µ€;/
ċ1<⁴µÐfµ;Ç
7Ḋœċ⁴Ḷ¤;/L€;$€;@þ2B¤;/ḟ€0ÇS⁼¥Ðf⁸Ñ€K€Y

---

How?

L⁼30µÐfµ7Ḋ;\¤;€Ṣ€µ€;/ - Link 1, create post-red-ball games: list of all pre-yellow-ball-games
    µÐf               - filter keep if:
L⁼30                  -   length equals 30 (games that get on to the yellow)
       µ         µ€   - for €ach sequence leading to the yellow:
            ¤         -   nilad followed by link(s) as a nilad:
        7Ḋ            -     7 dequeued  = [2,3,4,5,6,7]
          ;\          -     ;\ cumulative reduce with concatenation  = [[2],[2,3],[2,3,4],...]
             ;€       - concatenate the sequence with €ach of those
               Ṣ€     - sort each one
                   ;/ - reduce with concatenation (flatten by one)

ċ1<⁴µÐfµ;Ç - Link 2, filter bogus entries created and append post-yellow-ball games: list of pre-yellow-ball games (along with the bogus ones with 16 reds potted)
    µÐf    - filter keep if:
ċ1         -   count ones
   ⁴       -   literal 16
  <        -   less than?
       µ   - monadic chain separation
         Ç - call the last link (1) as a monad
        ;  - concatenate

7Ḋœċ⁴Ḷ¤;/L€;$€;@þ2B¤;/ḟ€0ÇS⁼¥Ðf⁸G - Main link: score
7Ḋ                                - 7 dequeued = [2,3,4,5,6,7]
      ¤                           - nilad followed by link(s) as a nilad:
     ⁴                            -   literal 16
    Ḷ                             -   lowered range = [0,1,2,...,15]
  œċ                              - combinations with replacement (every possible colour-ball selection that goes with the red pots)
       ;/                         - reduce with concatenation (flatten by one)
            $€                    - last two links as a monad for €ach:
         L€                       -   length of €ach (number of coloured balls potted)
           ;                      -   concatenate
                   ¤              - nilad followed by link(s) as a nilad:
                 2B               -   2 in binary = [1,0]
                þ                 - outer product with:
              ;@                  -   concatenation with reversed @rguments
                    ;/            - reduce with concatenation (flatten by one)
                      ḟ€0         - filter discard zeros from €ach
                         Ç        - call the last link (2) as a monad
                             Ðf   - filter keep:
                            ¥  ⁸  -   last two links as a dyad, with score on the right
                          S⁼      -     sum(ball values) is equal to score?
                                G - format as a grid
                                  - implicit print

Answer 2

JavaScript (ES7), 188 180 178 bytes

Returns an array of arrays (sorted from red to black).

n=>[...Array(17**6)].map((_,x)=>[2,3,4,5,6,p=7].map(v=>(k=a[++j]=x%17|0,x/=17,t+=k,p+=!!(y=y&&k),s-=k*v),y=s=n,a=[j=t=0])&&(s==15|s>=t)&s<16&s<t+2&t<9+p&&(a[0]=s,a)).filter(a=>a)

Commented

Note: This version doesn't include the last optimization on p (now initialized to 7) which makes the logic harder to understand.

n =>                              // given a target score n:
  [...Array(17**6)].map((_, x) => // for each x in [0 .. 17^6 - 1]:
    [2, 3, 4, 5, 6, 7].map(v =>   //   for each v in [2 .. 7] (yellow to black):
      ( k = a[++j] = x % 17 | 0,  //     k = a[j] = number of colored balls of value v
        x /= 17,                  //     update x to extract the next value
        t += k,                   //     update t = total number of colored balls
        p += !!(                  //     update p = number of consecutive colors that were
          y = y && k              //     potted at least once, using y = flag that is zeroed
        ),                        //     as soon as a color is not potted at all
        s -= k * v ),             //     subtract k * v from the current score s
      y = s = n,                  //     initialize y and s
      a = [j = t = p = 0]         //     initialize a, j (pointer in a), t and p
    )                             //   at this point, s is the alleged number of red balls
    &&                            //   this combination is valid if we have:
      (s == 15 | s >= t) &        //     - 15 red balls or more red balls than colored ones
      s < 16 &                    //     - no more than 15 red balls
      s < t + 2 &                 //     - at most one more red ball than colored ones
      t < 16 + p                  //     - no more than 15 + p colored balls
    &&                            //   if valid:
      (a[0] = s, a)               //     update the combination with red balls and return it
  ).filter(a => a)                // end of outer map(): filter out invalid entries

Example output

Below is the output for n = 140:

//  R   Y  G  Br Bl P  Bk 
[ [ 15, 1, 1, 1, 1, 8, 9  ],
  [ 15, 1, 1, 1, 2, 6, 10 ],
  [ 15, 1, 1, 1, 3, 4, 11 ],
  [ 15, 1, 1, 2, 1, 5, 11 ],
  [ 15, 1, 1, 1, 4, 2, 12 ],
  [ 15, 1, 1, 2, 2, 3, 12 ],
  [ 15, 1, 2, 1, 1, 4, 12 ],
  [ 15, 1, 1, 2, 3, 1, 13 ],
  [ 15, 1, 1, 3, 1, 2, 13 ],
  [ 15, 1, 2, 1, 2, 2, 13 ],
  [ 15, 2, 1, 1, 1, 3, 13 ],
  [ 15, 1, 2, 2, 1, 1, 14 ],
  [ 15, 2, 1, 1, 2, 1, 14 ],
  [ 15, 1, 1, 1, 1, 1, 15 ] ]

Demo

This is too slow for a snippet. You can try it here instead. (You may get one or two unresponsive script alerts, but it should eventually complete.)