What patterns are(n't) possible?

RingTheBell · February 3, 2021, 2:13am

Eight Balls!

A and B are as before, and C, D, E, F, and G generate patterns of eight balls containing 2, 3, 4, 5, and 6 blue balls, respectively. Also, I used hexadecimal as shorthand for the binary this time since it’s much more convenient than decimal (I really should have been using it all along ).

@pt153 An insight I’ve gained from eight balls is that crossovers may be the limiting resource: there were a few cases where I ran out, but luckily I was able to substitute a bit. This may be a constraint going forward, but on the other hand I think this is the end of arbitrary pattern generators so the demand may be less. (Or maybe I just use more crossovers than are strictly necessary )

Solutions

0x00 - 00000000: A
0x01 - 00000001: A
0x02 - 00000010: A
0x03 - 00000011: A
0x04 - 00000100: A
0x05 - 00000101: A
0x06 - 00000110: A
0x07 - 00000111: A
0x08 - 00001000: A
0x09 - 00001001: A
0x0A - 00001010: C1
0x0B - 00001011: D1
0x0C - 00001100: A
0x0D - 00001101: A
0x0E - 00001110: A
0x0F - 00001111: A

0x10 - 00010000: A
0x11 - 00010001: A
0x12 - 00010010: C2
0x13 - 00010011: D2
0x14 - 00010100: C3
0x15 - 00010101: D3
0x16 - 00010110: D4
0x17 - 00010111: E1
0x18 - 00011000: A
0x19 - 00011001: A
0x1A - 00011010: D5
0x1B - 00011011: E2
0x1C - 00011100: A
0x1D - 00011101: A
0x1E - 00011110: A
0x1F - 00011111: A

0x20 - 00100000: A
0x21 - 00100001: A
0x22 - 00100010: C4
0x23 - 00100011: D6
0x24 - 00100100: B2
0x25 - 00100101: B2
0x26 - 00100110: B2
0x27 - 00100111: E3
0x28 - 00101000: C5
0x29 - 00101001: D7
0x2A - 00101010: D8
0x2B - 00101011: E4
0x2C - 00101100: D9
0x2D - 00101101: E5
0x2E - 00101110: E6
0x2F - 00101111: F1

0x30 - 00110000: A
0x31 - 00110001: A
0x32 - 00110010: B4
0x33 - 00110011: B4
0x34 - 00110100: D10
0x35 - 00110101: E7
0x36 - 00110110: E8
0x37 - 00110111: F2
0x38 - 00111000: A
0x39 - 00111001: A
0x3A - 00111010: E9
0x3B - 00111011: F3
0x3C - 00111100: A
0x3D - 00111101: A
0x3E - 00111110: A
0x3F - 00111111: A

0x40 - 01000000: A
0x41 - 01000001: A
0x42 - 01000010: C6
0x43 - 01000011: D11
0x44 - 01000100: C7
0x45 - 01000101: D12
0x46 - 01000110: D13
0x47 - 01000111: E10
0x48 - 01001000: C8
0x49 - 01001001: B2
0x4A - 01001010: D14
0x4B - 01001011: B2
0x4C - 01001100: B2
0x4D - 01001101: E11
0x4E - 01001110: E12
0x4F - 01001111: F4

0x50 - 01010000: C9
0x51 - 01010001: D15
0x52 - 01010010: D16
0x53 - 01010011: B1
0x54 - 01010100: B1
0x55 - 01010101: B1
0x56 - 01010110: B1
0x57 - 01010111: F5
0x58 - 01011000: D17
0x59 - 01011001: E13
0x5A - 01011010: E14
0x5B - 01011011: F6
0x5C - 01011100: E15
0x5D - 01011101: F7
0x5E - 01011110: F8
0x5F - 01011111: G1

0x60 - 01100000: A
0x61 - 01100001: A
0x62 - 01100010: D18
0x63 - 01100011: E16
0x64 - 01100100: B4
0x65 - 01100101: E17
0x66 - 01100110: B4
0x67 - 01100111: F9
0x68 - 01101000: D19
0x69 - 01101001: E18
0x6A - 01101010: E19
0x6B - 01101011: F10
0x6C - 01101100: B3
0x6D - 01101101: B3
0x6E - 01101110: F11
0x6F - 01101111: G2

0x70 - 01110000: A
0x71 - 01110001: A
0x72 - 01110010: E20
0x73 - 01110011: F12
0x74 - 01110100: E21
0x75 - 01110101: F13
0x76 - 01110110: F14
0x77 - 01110111: G3
0x78 - 01111000: A
0x79 - 01111001: A
0x7A - 01111010: F15
0x7B - 01111011: G4
0x7C - 01111100: A
0x7D - 01111101: A
0x7E - 01111110: A
0x7F - 01111111: A

pt153 · February 9, 2021, 6:01pm

Amazing and disciplined effort. Well done @RingTheBell !
Any 8 ball combination from what is basically 7 “standard” approaches makes me think we are not that close to the limit yet. I would still be surprised if there were any 10 ball combinations that cannot be done. In the “Universal pattern Generator” crossovers were my limiting factor too . I had a solution for any pattern up to 6 balls on the standard board size, but it used more crossovers than the standard set. I think loads of crossovers comes partly from wanting to have fewer standard boards to generate multiple patterns each.
Hex notation works absolutely brilliantly for 8 balls. Notation that works for both 9 and 10 balls might be more of a challenge (but I have an idea for that :- ).

RingTheBell · February 22, 2021, 2:45am

Woo! Nine Balls!

I’ve created a new pattern generator, C, that can do everything A does, but also patterns that are blue, then red, then blue, then red. However, A is not obsolete because it can generate longer patterns than C.

When I did the eight-ball patterns, I noticed that having two counters was somewhat redundant, so for D (2 blue, 7 red, and 3 blue, 6 red), E (4 blue, 5 red), F (5 blue, 4 red), and G (6 blue, 3 red), there is only one counter and then some supporting logic. Also since there are so many versions of these in use here, I didn’t bother numbering the ones that are only used once, and instead marked them with ‘v’.

Solutions 0x00-0x3F

0x00 - 000000000: A
0x01 - 000000001: A
0x02 - 000000010: A
0x03 - 000000011: A
0x04 - 000000100: A
0x05 - 000000101: A
0x06 - 000000110: A
0x07 - 000000111: A
0x08 - 000001000: A
0x09 - 000001001: A
0x0A - 000001010: Dv
0x0B - 000001011: C
0x0C - 000001100: A
0x0D - 000001101: A
0x0E - 000001110: A
0x0F - 000001111: A