Pancake Riddle

Relevant wiki: Egyptian Fractions

From the question, to divide $11$ pancakes among $12$ people, each person would get $\dfrac{11}{12}$ of the whole pancake.

Intuitively, one could divide each pancake into $12$ parts, and each person would take $11$ slices as an equal share. However, that would require too much time and too many cuts, which wouldn't be our desired solution.

By limiting the cuts along diameters only, that means each slice would have an even denominator, and such denominator would need to be as least as possible to avoid too many cuts. As such, the fraction would need to be in its reduced lowest terms (reduced denominator as possible), and by approximation with the small integer (preferably $2$ ), we are approaching the desired fraction with the highest possible value as called Greedy Algorithm .

In order to perform fewest cuts, we could adopt a primitive method used by the Egyptians, the Egyptian Fractions , or fractions with numerator $1$ as shown:

$\dfrac{11}{12} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6}$

Thereby, after starting with even denominator of $2$ , the next least possible one is $4$ , according to Greedy Algorithm , resulting in the summation above. Note that by skipping $4$ to $6$ , other variation is possible, such as:

$\dfrac{11}{12} = \dfrac{1}{2} + \dfrac{2}{6} + \dfrac{1}{12}$

But note that the sum of the three denominators in the latter case is higher, resulting in more cuts. Thus, the optimal summation is:

$\dfrac{11}{12} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6}$

In other words, we wouldn't need to divide any pancake into $12$ pieces. Instead, each person would get a share of one-half plus one-fourth and one-sixth.

That is, first, we would take $6$ pancakes and divide each in halves (one cut each). That would result in $12$ one-half slices, and so each person would get that one-half. (Totally $6$ cuts)

Then we would take $3$ pancakes and divide each in quarters (2 cuts each). That would result in $12$ one-fourth slices, and so each person would get that one-fourth. (Totally $3\times 2 = 6$ cuts)

Finally, we would take the remaining $2$ pancakes and divide each in six portions (3 cuts each). That would result in $12$ one-sixth slices, and so each person would get that one-sixth. (Totally $2\times 3 = 6$ cuts)

Overall, we could divide $11$ pancakes among $12$ people by cutting only $\boxed{18}$ times.