> I get the impression it is a set number of turns. In that case,
> you'll need
> to be a bit canny, doing something like biasing the roll according to the
> distance to the goal.
I enjoyed this problem way too much, so I've written an implementation (in
Lingo, I'm afraid - it's my first language. But it should be easy enough to
follow)
on getRolls tSquares, tTurns, tList
if voidP(tList) then tList = []
if tTurns = 1 then
tList.add(tSquares)
return tList
end if
tMean = 1.*tSquares / tTurns
-- Find the minimum value needed: min q>=1 such that (S-q)/(n-1)<=6
tMin = tSquares - 6*(tTurns - 1)
if tMin<1 then tMin = 1
if tMin>6 then return #no
-- Find the maximum value needed: max r<=6 such that (S-r)/(n-1)>=1
tMax = tSquares - (tTurns - 1)
if tMax>6 then tMax = 6
if tMax<1 then return #no
if tMax = tMin then
-- Only one possible answer here
tRoll = tMin
else
tOptions = tMax - tMin
-- Set probability distribution
tWeight = 1 - 1.0/tTurns
tProbabilities = []
tProbMean = tMean - tMin
-- ensure p lies between 0 and 1
tSkew = min(tProbMean, tOptions - tProbMean)
if tWeight> 2*tSkew / tOptions then
tWeight = 2*tSkew / tOptions
end if
tBinomialProb = (tProbMean - tWeight*tOptions/2) / ((1 -
tWeight)*tOptions)
repeat with i = 0 to tOptions
tProbabilities.add(tWeight/(tOptions+1) + (1 -
tWeight)*choose(tOptions,i)*power(tBinomialProb,i)*power(1-tBinomialProb,
tOptions-i))
end repeat
-- choose a random number according to this distribution
r = random(1000)
repeat with i = 1 to tProbabilities.count
tProb = tProbabilities[i]*1000
if r<tProb then
tRoll = i-1+tMin
exit repeat
end if
r = r-tProb
end repeat
end if
tList.add(tRoll)
return getRolls(tSquares - tRoll, tTurns-1, tList)
end
on choose n, k
if k>n/2 then return choose(n,(n-k))
tRes = 1
repeat with i=0 to k-1
tRes = tRes * (n-i)
end repeat
repeat with i=1 to k
tRes = tRes / i
end repeat
return tRes
end
on getLotsOfRolls s, n, m
repeat with i=1 to m
put getRolls(s,n)
end repeat
end
I asked for getLotsOfRolls(50,15,20) and got this:
-- [6, 4, 3, 6, 6, 4, 1, 5, 1, 1, 5, 1, 1, 1, 5]
-- [5, 6, 1, 5, 5, 2, 2, 1, 1, 2, 2, 6, 4, 4, 4]
-- [3, 6, 5, 2, 4, 2, 1, 1, 6, 5, 6, 1, 1, 3, 4]
-- [5, 4, 1, 3, 5, 6, 1, 1, 3, 2, 3, 1, 4, 6, 5]
-- [1, 3, 2, 1, 1, 1, 5, 3, 2, 6, 6, 6, 2, 5, 6]
-- [1, 5, 2, 2, 2, 4, 1, 6, 4, 1, 6, 6, 1, 4, 5]
-- [4, 1, 6, 6, 2, 1, 1, 3, 2, 4, 3, 6, 3, 5, 3]
-- [3, 6, 1, 3, 4, 2, 1, 1, 6, 2, 3, 1, 6, 6, 5]
-- [5, 3, 1, 4, 6, 1, 2, 3, 2, 3, 3, 6, 1, 4, 6]
-- [1, 5, 4, 1, 4, 3, 1, 5, 5, 5, 3, 4, 5, 1, 3]
-- [2, 2, 5, 4, 3, 2, 3, 4, 1, 5, 6, 3, 1, 3, 6]
-- [1, 1, 2, 5, 6, 2, 5, 1, 5, 2, 6, 3, 5, 4, 2]
-- [1, 4, 5, 3, 3, 6, 4, 1, 4, 3, 1, 5, 6, 1, 3]
-- [3, 5, 1, 3, 4, 3, 3, 3, 1, 6, 6, 4, 3, 3, 2]
-- [3, 4, 4, 1, 6, 5, 6, 1, 1, 1, 4, 6, 1, 1, 6]
-- [1, 6, 5, 3, 2, 1, 5, 2, 5, 6, 1, 6, 1, 3, 3]
-- [1, 5, 3, 2, 5, 2, 2, 2, 3, 5, 5, 4, 3, 6, 2]
-- [1, 2, 5, 1, 2, 6, 5, 5, 2, 2, 3, 6, 4, 2, 4]
-- [2, 5, 1, 6, 4, 3, 1, 3, 3, 5, 2, 3, 1, 6, 5]
-- [3, 1, 2, 3, 3, 4, 2, 5, 6, 5, 1, 3, 3, 4, 5]
Looks pretty decent to me.
Danny
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