16. (2001 World Inter-City Competition) B has to guess A's two-digit number. If both digits are correct, or one is correct and the other differs by 1, A says “near”; otherwise, A says “far”. For example, if A's number is 65, and B guesses 65, 64, 66, 55, or 75, A says “near”; otherwise, A says “far”.
(a) Prove that B cannot guarantee to know A's number in just 18 guesses;
(b) Find a method for B to guarantee knowing A's number in 24 guesses;
(c) Is there a method for B to guarantee knowing A's number in 22 guesses?
16. (a) The first 17 guesses can exclude at most \(17 \times 5 = 85\) numbers, leaving 5 numbers; the 18th guess cannot determine the number from the remaining 5 numbers.
(c) Construct a \(10 \times 9\) table, where the cell in the \(i\)-th column and \(j\)-th row contains \((10i + j - 1)\), where \(1 \leqslant i \leqslant 9, 1 \leqslant j \leqslant 10\).
Divide this table by bold lines, as shown in the figure.
There are 11 cells of the form "克", containing 55 numbers, so B first guesses \(21, 26, 34, 42, 47, 55, 63, 68, 71, 76, 84\) for a total of 11 times. If A says "near," for example, 21 is "near," B only needs to guess \(11, 20, 31\) to determine A's number; if "far," then B guesses 11 times to exclude 55 numbers.

There are 7 cells of the form "士ㅗ", containing 28 numbers, so B then guesses \(13, 18, 39, 50, 89, 92, 97\) for a total of 7 times. If A says "near," for example, 13 is "near," B only needs to guess \(12, 14\) to determine A's number; if "far," then B guesses 7 times to exclude 28 numbers.
At this point, there are 7 numbers left: \(10, 30, 80, 90, 15, 59, 95\).
B then guesses \(20, 80\) for a total of 2 times. If 20 is "near," A's number must be among \(10, 30\), and guessing 10 will determine A's number; if 80 is "near," A's number must be among \(80, 90\), and guessing 70 will determine A's number; if "far," then B guesses 2 more times to exclude 4 numbers.
Finally, only 3 numbers remain: \(15, 59, 95\), and guessing 2 more times will determine A's number.

Thus, it only takes \(11 + 7 + 2 + 2 = 22\) guesses to determine A's number.
At this point, (b) is clearly satisfied.