Another band of loyalist join them in the morning, taking the total to 60 followers. For breakfast, they decide to have bamboo sticks. Unfortunately, he has only 47 bamboo sticks, all of them identical cylinders. Anxious to evenly distribute the 47 sticks amongst them- selves, Tintin decides to cut the sticks with his sword. What is the minimum number of times that Tintin has to use his sword to share the 47 bamboo sticks between all 60 of them, know- ing that each of them must receive identical parts. Note : A sword can be used to break mutliple sticks at a time.
Since we have to divide 47 bamboo sticks to 60 people in equal quantities, every person must end up with `47//60` of the initial piece.
Let us look at, `47//60 = 1//3 + 1//4 + 1//5`. For the first cut, Tintin puts the 47 sticks side by side in a way that he will cut 20 sticks at `1//3` of the length, 15 sticks at `1//4` of the length and 12 sticks at `1//5` of the length. He obtaines
20 parts of length `1//3` (that he keeps for the moment)
20 parts of length `2//3` (to be cut later)
15 parts of length `1//4` (that he keeps for the moment)
15 parts of length `3//4` (to be cut later)
For the 2nd cut, Tintin puts side by side 20 parts of length 2/3 in the middle, 15 parts of length 3/4 at 1/3rd it’s length(i.e., 1/4th of total length) and 12 parts of length 4/5 that he will cut in the middle.
He obtaines
40 additional parts of length `1//3`, that he adds to the 20 parts obtained previously. He can now distribute these 60 pieces of length `1//3`.
15 additional fragments of length `1//4` (which he adds to the existing ones) and 15 fragments of `2//4` length (to be cut later)
24 fragments of length 2/5 (to be cut later)
On the same principle, Tintin needs to cut the rest of the sticks once more to divide the 15 fragments of length `2//4` in 30 fragments of length `1//4` and 24 fragments of length `2//5` into 48 fragments of length `1//5`. He can now distribute all the 60 pieces of `¼` and 60 pieces of `1//5`. Hence the Answer is 3.