Problems & Puzzles: Puzzles Puzzle 63. Another (3rd) Mike Keith's little puzzle. Take an ndigit integer and construct all the possible expressions you can that use the individual digits of the number combined with parentheses and the operators +, , *, and /. (Note: just to be clear, because people sometimes get confused with this kind of problem, NO other operators are allowed  no exponentiation, no concatenation of digits, nothing.) If you calculate the values of these expressions, some will be nonintegers, some will be negative, etc, but just take those that are (positive) prime numbers. Then find which range of consecutive primes starting with 2 is included in the set. Call the last prime in the range P(n). Now...define a(n) as the largest value of P(n) for any ndigit integer. What are the values of a(n), and which primes produce them? Here are the answers for the first few n. Note that without loss of generality, we can examine just ndigit integers whose digits in order are nondecreasing. So that's the way I've listed them below. n=2: 12, 13, or 36 generate 2...3;
so a(n) = 36 Here is n = 3 in detail: 2 = 732 a) Can you obtain the details for n = 4, 5 & 6 b) Can you calculate a(n) for n= 7, 8 & 9? Solution Jim R. Howell sent (10/07/99) the
details for n = 4 (calculated 'by hand'): 




