You are correct, but you can trisect that piece of wood an infinite number of ways. The logical equilibrium point is 3 even pieces.

The student chose a ratio of 1,1,1; which is the logical equilibrium point. your image shows 1.5,0.75,0.75; which is the second most logical ratio because it is in the form x + 2y = 3 (which can be trisected an infinite number of ways while maintaining that ratio). The third form would be x + y + z = 3; which can also be trisected an infinite number of ways and would be the least intuitive.

i am agreeing with you, i am just trying to show that it is illogical for it to be 'open for debate'.

There is a game theory term for this type of equilibrium, but i forgot its name. Its the same type of equilibrium as "there are three colors and a number, which one is different?" type sesame street problems.

I didn't see the picture at first, and reasoned just as you exposed.

However, the teacher corrects it by writing "4 = 20". This is plainly wrong and with no possible explanation, since following the above reasoning, cutting in 4 pieces would require: 10 + 10 / 2 + (10 / 2) / 2 = 17.5 minutes.

I can cut a piece of wood into thousands of pieces in 5 seconds. I just slice it across the top a few times with my saw, and all the sawdust that comes off counts as separate pieces.

As someone said below, it's only open for debate if you want to be pedantic. The Dr. Sheldon Cooper's among us may debate it, but it's pretty obvious what the question was looking for. There is even an illustration showing the cut, which would take an identical amount of time.

The question says "board" while the illustration shows more of a rod. I deduce the question to be at least one of inconsistent or incomplete.

But even with your picture the answer can be 20 seconds. You're assuming the person is starting at the top of the line and cutting all the way through the board to the bottom. But they could just as easily rotate the board 90° and cut across a different axis. Assuming a 1" thick board, this means they're cutting through 1" of wood on each cut, meaning both cuts take the same amount of time.

this was my first thought, but then I realized the teacher gave justification for his answer.

the problem is poorly formulated. The teacher would have been correct if it had said "it took 10 minutes to cut away 2 pieces from a very large board (thus resulting in 2 cuts, 3 pieces total)", whereas the student's answer assumes a single cut, which is more reasonable.