Burnt pancakes problem was defined by Gates and Papadimitriou in 1979. A stack $S$ of pancakes with a burnt side must be sorted by size, the smallest on top, and each pancake with burnt side down. The only operation allowed is to split stack in two parts and flip upper part. $g(S)$ is the minimal number of flips needed to sort stack $S$. Stack $S$ may be $-I_n$ when pancakes are in right order but upside down or $-f_n$ when all pancakes are right side up but sorted in reverse order. Gates et al. proved that $g(-f_n)\ge 3n/2-1$. In 1995 Cohen and Blum proved that $g(-I_n)=g(-f_n)+1\ge 3n/2$. In 1997 Heydari and Sudborough proved that $g(-I_n)\le 3(n+1)/2$ whenever some fortuitous sequence of flips exists. They gave fortuitous sequences for $n$=3, 15, 27 and 31. They showed that two fortuitous sequences $S_n$ and $S_{n'}$ may combine into another fortuitous sequence $S_{n''}$ with $n''=n+n'-3$. So a fortuitous sequence $S_n$ gives a fortuitous sequence $S_{n+12}$. This proves that $g(-I_n)\le 3(n+1)/2$ if $n$ is congruent to 3 modulo 4 and $n\ge 23$. In 2011 Josef Cibulka enhanced Gates and Papadimitriou's lower bound thanks to a potential function. He got so $g(-I_n)\ge3n/2+1$ if $n > 1$ proving thereby, that $g(-I_n)=3(n+1)/2$ if $n$ is congruent to 3 modulo 4 and $n\ge 23$. This paper explains how to build generalized fortuitous sequences for $n=15, 19, 23$ and every $n\ge 25$, odd or even, proving thereby that $g(-I_n)=\lceil 3n/2\rceil+1$ for these $n$. It gives $g(-I_n)$ for all $n$.

翻译：烧焦薄饼问题由Gates和Papadimitriou于1979年提出。一个带有烧焦面的薄饼堆栈$S$必须按大小排序，最小的在最上面，且每个薄饼的烧焦面朝下。唯一允许的操作是将堆栈分成两部分并翻转上半部分。$g(S)$表示排序堆栈$S$所需的最小翻转次数。当薄饼顺序正确但上下颠倒时，堆栈$S$可能是$-I_n$；当所有薄饼正面朝上但按相反顺序排序时，堆栈$S$可能是$-f_n$。Gates等人证明了$g(-f_n)\ge 3n/2-1$。1995年，Cohen和Blum证明了$g(-I_n)=g(-f_n)+1\ge 3n/2$。1997年，Heydari和Sudborough证明了只要存在幸运翻转序列，就有$g(-I_n)\le 3(n+1)/2$。他们给出了$n$=3、15、27和31的幸运序列。他们表明，两个幸运序列$S_n$和$S_{n'}$可以组合成另一个幸运序列$S_{n''}$，其中$n''=n+n'-3$。因此，一个幸运序列$S_n$可以生成一个幸运序列$S_{n+12}$。这证明了如果$n$模4余3且$n\ge 23$，则$g(-I_n)\le 3(n+1)/2$。2011年，Josef Cibulka通过势函数改进了Gates和Papadimitriou的下界。他得到如果$n > 1$，则$g(-I_n)\ge3n/2+1$，从而证明了如果$n$模4余3且$n\ge 23$，则$g(-I_n)=3(n+1)/2$。本文解释了如何为$n=15、19、23$以及每个$n\ge 25$（无论奇偶）构建广义幸运序列，从而证明对于这些$n$，$g(-I_n)=\lceil 3n/2\rceil+1$。这给出了所有$n$的$g(-I_n)$值。