Can You Win at “Rock, Paper, Scissors, Lizard, Spock?”

This Week's Fiddler

To calculate the scenarios that result in decisive victory, we multiply $3$ possible winners, $5$ possible elements chosen by the winner, $2$ losing element options for one of the losing players, and $2$ losing element options for the other losing player.

This makes $60$ scenarios out of $5^3 = 125$ in all.

$\frac{60}{125} = \frac{12}{25} = 48\%$

Brute force check

import itertools

edges = {
    "RS",
    "RL",
    "PR",
    "PV",
    "SP",
    "SL",
    "LP",
    "LV",
    "VR",
    "VS",
}

wins = 0
for a, b, c in itertools.product("RPSLV", repeat=3):
    wins += (
        ((a + b) in edges and (a + c) in edges)
        or ((b + a) in edges and (b + c) in edges)
        or ((c + a) in edges and (c + b) in edges)
    )
print(wins)

Output:

Which matches the 60/125 result we got.

Answer: 48%

Extra Credit

We can make a recursive function which traverses all the directed edges.

Each edge represents one element that beats another. Letters stand for their respective element, except for V, which stands for Vulcan/Spock (S already stands for scissors) :)

edges = {
    "RS",
    "RL",
    "PR",
    "PV",
    "SP",
    "SL",
    "LP",
    "LV",
    "VR",
    "VS",
}


def find_paths(edges: set[str], last: str | None = None):
    if not edges:
        return [[]]
    paths = []
    for edge in edges:
        if last and not edge.startswith(last):
            continue
        for path in find_paths(edges - {edge}, edge[-1]):
            paths.append([edge, *path])
    return paths


paths = sorted(
    path[0][0] + "".join(node[1] for node in path)
    for path in find_paths(edges)
)
print('\n'.join(paths))
print(len(paths))

Output:

... (truncated for brevity)
110

Full output (including all valid paths)

LPRLVRSPVSL
LPRLVSPVRSL
LPRSLVSPVRL
LPRSPVRLVSL
LPRSPVSLVRL
LPVRLVSPRSL
LPVRSLVSPRL
LPVRSPRLVSL
LPVSLVRSPRL
LPVSPRLVRSL
LPVSPRSLVRL
LVRLPRSPVSL
LVRLPVSPRSL
LVRSLPVSPRL
LVRSPRLPVSL
LVRSPVSLPRL
LVSLPRSPVRL
LVSLPVRSPRL
LVSPRLPVRSL
LVSPRSLPVRL
LVSPVRLPRSL
LVSPVRSLPRL
PRLPVRSLVSP
PRLPVSLVRSP
PRLVRSLPVSP
PRLVRSPVSLP
PRLVSLPVRSP
PRLVSPVRSLP
PRSLPVRLVSP
PRSLVRLPVSP
PRSLVSPVRLP
PRSPVRLVSLP
PRSPVSLVRLP
PVRLPRSLVSP
PVRLVSLPRSP
PVRLVSPRSLP
PVRSLPRLVSP
PVRSLVSPRLP
PVRSPRLVSLP
PVSLPRLVRSP
PVSLVRLPRSP
PVSLVRSPRLP
PVSPRLVRSLP
PVSPRSLVRLP
RLPRSLVSPVR
RLPRSPVSLVR
RLPVRSLVSPR
RLPVSLVRSPR
RLPVSPRSLVR
RLVRSLPVSPR
RLVRSPVSLPR
RLVSLPRSPVR
RLVSLPVRSPR
RLVSPRSLPVR
RLVSPVRSLPR
RSLPRLVSPVR
RSLPVRLVSPR
RSLPVSPRLVR
RSLVRLPVSPR
RSLVSPRLPVR
RSLVSPVRLPR
RSPRLPVSLVR
RSPRLVSLPVR
RSPVRLVSLPR
RSPVSLPRLVR
RSPVSLVRLPR
SLPRLVRSPVS
SLPRLVSPVRS
SLPRSPVRLVS
SLPVRLVSPRS
SLPVRSPRLVS
SLPVSPRLVRS
SLVRLPRSPVS
SLVRLPVSPRS
SLVRSPRLPVS
SLVSPRLPVRS
SLVSPVRLPRS
SPRLPVRSLVS
SPRLPVSLVRS
SPRLVRSLPVS
SPRLVSLPVRS
SPRSLPVRLVS
SPRSLVRLPVS
SPVRLPRSLVS
SPVRLVSLPRS
SPVRSLPRLVS
SPVSLPRLVRS
SPVSLVRLPRS
VRLPRSLVSPV
VRLPRSPVSLV
VRLPVSPRSLV
VRLVSLPRSPV
VRLVSPRSLPV
VRSLPRLVSPV
VRSLPVSPRLV
VRSLVSPRLPV
VRSPRLPVSLV
VRSPRLVSLPV
VRSPVSLPRLV
VSLPRLVRSPV
VSLPRSPVRLV
VSLPVRSPRLV
VSLVRLPRSPV
VSLVRSPRLPV
VSPRLPVRSLV
VSPRLVRSLPV
VSPRSLPVRLV
VSPRSLVRLPV
VSPVRLPRSLV
VSPVRSLPRLV
110

Answer: 110