2. In the list of positive integers $1,3,9,4,11$ the positive differences between each pair of adjacent integers in the list are $3-1=2,9-3=6,9-4=5$, and $11-4=7$. In this example, the smallest positive difference between any two adjacent integers in the list is 2 .
(a) Arrange the integers $1,2,3,4,5$ so that the smallest positive difference between any two adjacent integers is 2 .
(b) Suppose that the twenty integers $1,2,3, \ldots, 18,19,20$ are arranged so that the smallest positive difference between any two adjacent integers is $N$.
(i) Explain why $N$ cannot be 11 or larger.
(ii) Find an arrangement with $N=10$. (Parts (i) and (ii) together prove that the maximum possible value of $N$ is 10 .)
(c) Suppose that the twenty-seven integers $1,2,3, \ldots, 25,26,27$ are arranged so that the smallest positive difference between any two adjacent integers is $N$. What is the maximum possible value of $N$ ? Prove that your answer is correct.
2. (a) One way of arranging these integers is $1,3,5,2,4$.
The positive differences between the pairs of adjacent integers are $2,2,3,2$.
There are many other ways of arranging these integers so that the desired property is true.
(We note that this property is equivalent to arranging the five integers so that no two consecutive integers are adjacent.)
(b) (i) In any arrangement, the integer 10 must be placed next to at least one other integer. From the list $1,2,3, \ldots, 20$, the integers "furthest away" from 10 are 1 and 20 .
The positive differences between 10 and these integers are 9 and 10 .
In other words, the positive difference between 10 and every other integer in the list is less than or equal to 10 .
Therefore, in any arrangement, there must be a positive difference that is at most 10 , and so $N$ (the smallest of these differences) is less than or equal to 10 , which means that $N$ cannot be 11 or larger.
(ii) Consider the following arrangement:
$$
10,20,9,19,8,18,7,17,6,16,5,15,4,14,3,13,2,12,1,11
$$

The positive differences between the pairs of adjacent integers are
$$
10,11,10,11,10,11,10,11,10,11,10,11,10,11,10,11,10,11,10
$$

The smallest of these positive differences is 10 .
(c) Consider the integer 14, which is the middle integer in the list $1,2,3, \ldots, 25,26,27$.
The maximum possible positive difference between 14 and another number from this list is 13 .
Therefore, in any arrangement, there must be a positive difference that is no larger than 13 . This tells us that $N$, the smallest of the positive differences, is less than or equal to 13 . To show that the maximum possible value of $N$ is 13 , we need to show that there actually exists an arrangement with $N=13$, since it is possible that, for some other reason, we could not make a list with $N=13$.
Here is such an arrangement:
$$
14,27,13,26,12,25,11,24,10,23,9,22,8,21,7,20,6,19,5,18,4,17,3,16,2,15,1
$$

The positive differences between the pairs of adjacent integers are
$$
13,14,13,14,13,14,13,14,13,14,13,14,13,14,13,14,13,14,13,14,13,14,13,14,13,14
$$

Therefore, the maximum possible value of $N$ is 13 .