class E1 {
  
  /** Return the sum of the squares from 1 to n.
    Requires: n >= 1. */
  public static int s(int n) {
    return (n == 1) ? 1 : s(n-1) + n*n;
  }
  
}

/* Pedagogical notes:
 * 
 *   The two calculations are:
 *     k^2 + ((k+1)^2 + ((k+2)^2 + (... + (n^2))...))
 *     (...(((k^2) + (k+1)^2) + (k+2)^2) + ...) + n^2
 *
 *   The order of the arguments to + in the code has no effect on the order of operations,
 *    but reflects the order of the numbers k^2, ..., n^2.
 * 
 *   Exercise: rearrange the above and the code so that the only difference is counting
 *    from k to n versus n to k.
 * 
 *   Exercise: simplify the code by extending the domain to k = n+1.
 */

class E2 {
  
  /** Return the sum of the squares from k to n.
    Requires: k <= n. */
  public static int sk(int k, int n) {
    return (k == n) ? k*k : k*k + sk(k+1, n);
  }
  
  /** Return the sum of the squares from k to n.
    Requires: k <= n. */
  public static int sn(int k, int n) {
    return (k == n) ? n*n : sn(k, n-1) + n*n;
  }
  
}

/* Pedagogical notes:
 *
 *   Calculates it as (...(((1^2) + 2^2) + 3^2) + ...) + n^2
 *    by passing along the current subtotal.
 *   Notice syntactic clues to this in the code: sumSoFar on left of +, n*n
 *
 *   Exercise: simplify by extending the domain of s and/or sHelper.
 */

class E3 {
  
  /** Return the sum of the squares from 1 to n.
    Requires: n >= 1. */
  public static int s(int n) {
    return sHelper(0, 1, n);
  }
  
  /** Return sumSoFar + next^2 + (next+1)^2 + ... + n^2.
    Requires: 1 <= next <= n. */
  public static int sHelper(int sumSoFar, int next, int n) {
    return (next == n) ? sumSoFar + n*n : sHelper(sumSoFar + next*next, next+1, n);
  }
  
}

/* Pedagogical notes:
 *
 *   Calculates it as 1^2 + (2^2 + (3^2 + (... + (n^2))...))
 *    by passing along the current subtotal.
 *   Notice syntactic clues to this in the code: sumSoFar on right of +, 1*1
 *
 *   Exercise: do something analagous to the exercises mentioned in E2 and E3.
 */

class E4 {
  
  /** Return the sum of the squares from 1 to n.
    Requires: n >= 1. */
  public static int s(int n) {
    return sHelper(0, n);
  }
  
  /** Return 1^2 + 2^2 + ... + next^2 + sumSoFar.
    Requires: 1 <= next. */
  public static int sHelper(int sumSoFar, int next) {
    return (next == 1) ? 1*1 + sumSoFar : sHelper(next*next + sumSoFar, next-1);
  }
  
}