March 2016 Problem of the Month

Problem 1: We call an integer below average if each of its interior digits is less than the average of its immediate neighbors. For example, 1248 is below average, because the interior number 2 is less than the average of its neighbors (1 on the left, and 4 on the right), and the interior digit 4 is also less than the average of its neighbors (2 and 8). Find the largest below-average number.

Problem 2: Find, as a function of \(n\), the sum of the digits of the product
\[
9 \times 99 \times 9999 \times \dots \times( 10^{2^n} - 1),
\]
where each factor has twice as many digits as the previous one. (For example, the sum of the digits of 9 is 9 and the sum of the digits of \(9\times 99 = 891\) is \(8+9+1 = 18\).)

No correct student solutions were received to February's Problem 3. We'll offer $5 to the first individual/team to prove that the answer is 14. (Find both an assignment of elevators to floors and a reason why you can't do 15 floors.)

Submit solutions to Mitch Keller (Robinson 206) or Beedle Hinely (Robinson 113A) no later than noon on 31 March 2016. Good luck!