Препод сказал сделать так чтобы этот метод имел while и for, но я сделал вот так и программа не принимает при проверке.

int i = -1;
while (count[characters[++i]] != max);
return characters[i];

for (char c: characters)
    if (count[c] == max)
        return c;

public static char maxOccuringCharacter(String sentence) throws IllegalArgumentException {
        char[] characters = sentence.replaceAll("[^a-zA-Z]", "").toLowerCase().toCharArray();
        int[] cells = new int[256];
        int character = 0;

        for (int i = 0; i < characters.length; i++)
            cells[characters[i]]++;

        int max = -1;

        for (int i = 0; i < cells.length; i++) {
            if (cells[i] > max) {
                max = cells[i];
                character = i;
            }
        }
        return (char) character;
    }

try {
} catch(IllegalArgumentException e) {
}

public static char maxOccuringCharacter(String sentence) throws IllegalArgumentException {
        char[] characters = sentence.replaceAll("[^a-zA-Z]", "").toLowerCase().toCharArray();
        int[] cells = new int[256];
        int character = 0;
        try {
            for (int i = 0; i < characters.length; i++)
                cells[characters[i]]++;

            int max = -1;

            for (int i = 0; i < cells.length; i++) {
                if (cells[i] > max) {
                    max = cells[i];
                    character = i;
                }
            }
        } catch (IllegalArgumentException e) {

        }
        return (char) character;
    }

char[] characters = sentence.toCharArray();
int[] cells = new int[128];
for (char c: characters)
    if (c < 128)
        cells[c | 32]++;

int max = -1;
for (int i = 97; i < 123; i++)
    if (max < cells[i])
        max = cells[i];

cells[c | 32]++;

char[] characters = sentence.toCharArray();
        int[] cells = new int[128];
        for (char c: characters) {
            if (c < 128) {
                cells[c | 32]++;
            }
        }

        int max = -1;
        int character = ' ';
        for (int i = 97; i < 123; i++) {
            if (max < cells[i]) {
                max = cells[i];
                character = i;

            }
        }

        return (char) character;

if (sentence == null)
    throw new IllegalArgumentException();

if (c < 128)
    cells[c | 32]++;
else
    throw new IllegalArgumentException();

public static char maxOccuringCharacter(String sentence) throws IllegalArgumentException

public static char maxOccuringCharacter(String sentence) throws IllegalArgumentException {
        if (sentence == null) {
            throw new IllegalArgumentException();
        }
        int character = 0;
        char[] characters = sentence.replaceAll("[^a-zA-Z]", "").toLowerCase().toCharArray();
        int[] cells = new int[256];
        for (char c : characters)
            cells[c]++;

        int max = -1;

        for (int i = 0; i < cells.length; i++) {
            if (cells[i] > max) {
                max = cells[i];
                character = i;
            }
        }

        return (char) character;
    }

char[] characters = sentence.toCharArray();
        int[] cells = new int[128];
        for (char c : characters) {
            if (c < 128)
                cells[c | 32]++;
            else
                throw new IllegalArgumentException();
        }

        int max = -1;
        int character = ' ';
        for (int i = 97; i < 123; i++) {
            if (max < cells[i]) {
                max = cells[i];
                character = i;

            }
        }

        return (char) character;

public static char maxOccuringCharacter(String sentence) throws IllegalArgumentException {
        if (sentence.isEmpty()) {
            throw new IllegalArgumentException();
        }
        int character = 0;
        char[] characters = sentence.replaceAll("[^a-zA-Z]", "").toLowerCase().toCharArray();
        int[] cells = new int[256];
        for (char c : characters)
            cells[c]++;

        int max = -1;

        for (int i = 0; i < cells.length; i++) {
            if (cells[i] > max) {
                max = cells[i];
                character = i;
            }
        }

        return (char) character;
    }

if (sentence == null || sentence.isEmpty()) {
    throw new IllegalArgumentException();
}

...
if (max < 1)
    throw new IllegalArgumentException();
return character;