Pancake numbers
Adrian Monk has problems and an assistant, Sharona Fleming. Sharona can deal with most of Adrian's problems except his lack of punctuality paying her rumination. 2 pay checks down and she prepares him pancakes for breakfast. Knowing that he will be unable to eat them unless they are stacked in ascending order of size she leaves him only a skillet which he can insert at any point in the pile and flip all the above pancakes, repeating until the pile is sorted. Sharona has left the pile of n pancakes such that the maximum number of flips is required. Adrian is determined to do this in as few flips as possible. This sequence n->p(n) is known as the Pancake numbers.
The task is to determine p(n) for n = 1 to 9.
Sorting_algorithms/Pancake_sort actually performs the sort some giving the number of flips used. How do these compare with p(n)?
Few people know p(20), generously I shall award an extra credit for anyone doing more than p(16).
- References
F#
<lang fsharp> // Pancake numbers. Nigel Galloway: Octber 28th., 2020 let pKake z=let n=[for n in 1..z-1->Array.ofList([n.. -1..0]@[n+1..z-1])]
let e=let rec fG n g=match g with 0->n |_->fG (n*g) (g-1) in fG 1 z let rec fN i g l=match (Set.count g)-e with 0->i |_->let l=l|>List.collect(fun g->[for n in n->List.permute(fun g->n.[g]) g])|>Set.ofList fN (i+1) (Set.union g l) (Set.difference l g|>Set.toList) fN 0 (set1..z) 1..z
[1..9]|>List.iter(fun n->printf "%d->%d " n (pKake n)); printfn "" </lang>
- Output:
1->0 2->1 3->3 4->4 5->5 6->7 7->8 8->9 9->10
Julia
<lang julia>function pancake(len)
gap, gapsum, adj = 2, 2, -1 while gapsum < len adj += 1 gap = gap * 2 - 1 gapsum += gap end return len + adj
end
for i in 1:25
print("pancake(", lpad(i, 2), ") = ", rpad(pancake(i), 5)) i % 5 == 0 && println()
end
</lang>
- Output:
pancake( 1) = 0 pancake( 2) = 1 pancake( 3) = 3 pancake( 4) = 4 pancake( 5) = 5 pancake( 6) = 7 pancake( 7) = 8 pancake( 8) = 9 pancake( 9) = 10 pancake(10) = 11 pancake(11) = 13 pancake(12) = 14 pancake(13) = 15 pancake(14) = 16 pancake(15) = 17 pancake(16) = 18 pancake(17) = 19 pancake(18) = 20 pancake(19) = 21 pancake(20) = 23 pancake(21) = 24 pancake(22) = 25 pancake(23) = 26 pancake(24) = 27 pancake(25) = 28
Phix
Extra credit to anyone who can prove that this is in any way wrong?
The algorithm is freshly made up today, from scratch, by yours truly.
It agrees with https://oeis.org/A058986/b058986.txt which would put p(20) as either 22 or 23.
Note that several other references/links disagree on p(17) and up.
<lang Phix>function pancake(integer n)
integer gap = 2, sum_gaps = gap, adj = -1 while sum_gaps<n do adj += 1 gap = gap*2-1 sum_gaps += gap end while n += adj return n
end function sequence t = tagset(20),
r = columnize({t,apply(t,pancake)}), p = apply(true,sprintf,{{"p(%2d) = %2d"},r})
printf(1,"%s\n",join_by(p,1,5))</lang>
- Output:
p( 1) = 0 p( 2) = 1 p( 3) = 3 p( 4) = 4 p( 5) = 5 p( 6) = 7 p( 7) = 8 p( 8) = 9 p( 9) = 10 p(10) = 11 p(11) = 13 p(12) = 14 p(13) = 15 p(14) = 16 p(15) = 17 p(16) = 18 p(17) = 19 p(18) = 20 p(19) = 21 p(20) = 23
vs. max() of ten runs each of pancake_sort(shuffle(tagset(n))), modified to return the number of flips it made:
p( 1) = 0 p( 2) = 1 p( 3) = 3 p( 4) = 5 p( 5) = 6 p( 6) = 9 p( 7) = 10 p( 8) = 11 p( 9) = 12 p(10) = 15 p(11) = 16 p(12) = 17 p(13) = 20 p(14) = 22 p(15) = 25 p(16) = 28 p(17) = 28 p(18) = 31 p(19) = 33 p(20) = 34
Obviously the sort focuses on getting one pancake at a time into place, and therefore runs closer to 2n flips.