## Fun problem

### Fun problem

Problem 4. In front of you are $N$boxes numbered $1$ through $N$,and each contains one object, also numbered 1 through N, but objects and box num- bers don't necessarily match. You know where all the objects are, but your friend knows nothing other than the general rules. You need to make a sequence of statements of this form:

Object___is in box___(or box___ or box____...)

such that if your friend hears every statement they can figure out where all the objects are, but if they miss any statement, they cannot.

What is the maximum number of statements you can make that follows these conditions?

Example: $(N = 4)$

Object $2$ is in box $3$ or $4$

Object $3$ is in box $1$ or $3$

Object $1$ is in box $1$ or $4$

Object $2$ is in box $1$, $2$, or $3$

Object___is in box___(or box___ or box____...)

such that if your friend hears every statement they can figure out where all the objects are, but if they miss any statement, they cannot.

What is the maximum number of statements you can make that follows these conditions?

Example: $(N = 4)$

Object $2$ is in box $3$ or $4$

Object $3$ is in box $1$ or $3$

Object $1$ is in box $1$ or $4$

Object $2$ is in box $1$, $2$, or $3$

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