Doughnut treat

Logic Level 4

Using only straight cuts by a knife, you need to slice a ring-shaped doughnut into 18 pieces of any shape or volume. If $m$ and $M$ are the minimum and maximum possible cuts required, respectively, what is $m+M?$

The answer is 21.

1 solution

Yash Ghaghada
Sep 29, 2017

The maximum cuts can be easily found that it is 17(it also don't depend on the shape of object) as if we do one cut the will be 2 pieces, this implies (n-1) cuts and n pieces.

The minimum cuts is here 4.The logic is there is maximum intersection points inside the body implies more pieces from less cuts. To make this easier, If we see doughnut from top then it looks like disc with hole. It is easy to find the minimum cuts to divide " 2D doughnut" into 9 parts it is 3. so one more cut in direction perpendicular to all these 3 cuts and there will be 18 pieces from 4 cuts. (sorry for bad typing or English (if you think it is)).

Edit: (why 4 is minimum)

well then lets try for 3. Again let's check "2D doughnut", with the same idea, max pieces is 4 (with two cuts(in 2D)) and one cut perpendicular to these two cuts so we get 8 cuts.(maximum pieces that can be achieved from three cuts).

The minimum cuts case could use more explanation. In general, the 2-D annulus and 3-D donut case are trickier than one would originally expect.

Remember that if we want to show something is a minimium, we have to
1. Show that it can be achieved - This has been done.
2. Show that no smaller number will work - This has not been done.

Calvin Lin Staff - 3 years, 8 months ago

i appreciate your suggestion. I cannot do for "2".

Yash Ghaghada - 3 years, 8 months ago

Doughnut treat

The answer is 21.

1 solution

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