Q:

Determining if a list numbers are sequential

I'm working in Java.  I have an unordered list of 5 numbers ranging from 0-100 with no repeats.  I'd like to detect if 3 of the numbers are sequential with no gap.  
Examples:  
[9,12,13,11,10] true
[17,1,2,3,5] true
[19,22,23,27,55] false

As for what I've tried, nothing yet.  If I were to write it now, I would probably go with the most naive approach of ordering the numbers, then iteratively checking if a sequence exists.

A:

int sequenceMin(int[] set) {
    int[] arr = Arrays.copy(set);
    Arrays.sort(arr);
    for (int i = 0; i < arr.length - 3 + 1; ++i) {
        if (arr[i] == arr[i + 2] - 2) {
            return arr[i];
        }
    }
    return -1;
}

This sorts the array and looks for the desired sequence using the if-statement above, returning the first value.

Without sorting:
(@Pace mentioned the wish for non-sorting.) A limited range can use an efficient "boolean array", BitSet. The iteration with nextSetBit is fast.
    int[] arr = {9,12,13,11,10};
    BitSet numbers = new BitSet(101);
    for (int no : arr) {
        numbers.set(no);
    }
    int sequenceCount = 0;
    int last = -10;
    for (int i = numbers.nextSetBit(0); i >= 0; i = numbers.nextSetBit(i+1)) {
        if (sequenceCount == 0 || i - last > 1) {
            sequenceCount = 1;
        } else {
            sequenceCount++;
            if (sequenceCount >= 3) {
                System.out.println("Sequence start: " + (last - 1));
                break;
            }
        }
        last = i;
    }
    System.out.println("Done");

A:

Very naive (but faster) algorithm : (your array is input[], assuming it only contains 0-100 numbers as you said)
int[] nums=new int[101];
for(i=0;i<N;i++) 
   {
   int a=input[i];
   nums[a]++;
   if (a>0) { nums[a-1]++; }
   if (a<100) { nums[a+1]++; }
   }

Then look if there is an element of nums[]==3.
Could be faster with some HashMap instead of the array (and removes the 0-100 limitation)

Edit : Alas, this does NOT work if two numbers could be equal in the initial sequence

A:

This code seems to be implementing your requirements:
public class OrderdedList {
    public static void main(String[] args) {
        System.out.println(orderedWithNoGap(Arrays.asList(9, 12, 13, 11, 10))); // true
        System.out.println(orderedWithNoGap(Arrays.asList(17,1,2,3,5))); // true
        System.out.println(orderedWithNoGap(Arrays.asList(19,22,23,27,55))); // false
    }

    private static boolean orderedWithNoGap(List<Integer> list) {       
        Collections.sort(list);
        Integer prev = null;
        int seq = 0;
        for(Integer i : list) {
            if(prev != null && prev+1 == i)
                seq = seq == 0 ? 2 : seq+1;
            prev = i;
        }
        return seq >= 3;
    }

}

