Determining whether two two-digit numbers share a digit in common
You can simplify the boolean parts of your method.

public boolean shareDigit(int a, int b) {
    int lefta = a / 10;
    int righta = a % 10;
    int leftb = b / 10;
    int rightb = b % 10;
    return lefta == leftb || lefta == rightb || righta == leftb
            || righta == rightb;
}