This Is Just Plain Counting
I and many others think it's a good idea to start a math class with a simple non-traditional problem to get the students into the right mood for the class. My main source for the problems below is a Russian booklet by E. G. Kozlova intended for early and middle grades math circles.
The problems are a variation on one of my favorites: breaking chocolate bars. The focus is on the essence of plain counting with a little twist.
A log is sawed into 10 pieces - How many cuts have been made?
In the beginning there was just one piece (the log itself). Every cut adds 1 piece to the count. To get 10 pieces one has to make 9 cuts.
There were several logs to start with. After 10 cuts, there were 16 pieces. How many logs there were at the beginning?
Logic is exactly the same. Every cut adds 1 piece to the count. 10 cuts add 10 pieces. You had to start with 6 logs to end up with 16 pieces after 10 cuts.
For both problems it may be observed that it does not matter which logs or pieces get cut. The only parameter of consequence is the number of cuts.
One of the ends of a log is fixed in a mechanical claw. After 10 cuts, how many pieces will fall to the ground?
1 cut will reduce the problem to that of a free log. The remaining 9 cuts will leave 10 pieces.
Both ends of a log are fixed in mechanical claws. After 10 cuts, how many pieces will fall to the ground?
2 cuts will reduce the problem to that of a free log. The remaining 8 cuts will leave 9 pieces.
The log is rigidly fixed in the middle. After 10 cuts, how many pieces will fall to the ground?
As with a free log, the total number of pieces does not depend on the sequence of cuts: you may cut from either end in any order; the result is always the same. Thus, it is also irrelevant at what point the log is actually fixed - the middle or the end. It follows that the number of pieces that fall on the ground after 10 cuts is 10, with one piece being held fixed.
A sponge cake (with a hole in the middle) was cut with 10 radial cuts, how many pieces are there?
The first cut creates a shape topologically equivalent to a log: one can imagine unbending the cake into one straight piece with two ends. The remaining 9 cuts will result in 10 pieces.