7. Player B has to guess a two-digit number from Player A. If both digits are correct, or one is correct and the other is off by 1, Player A says “near,” otherwise, Player A says “far.” For example, if Player A’s number is 65, and Player B guesses 65, 64, 66, 55, or 75, Player A says “near,” otherwise, Player A says “far.”
(a) Prove that Player B cannot guarantee knowing Player A’s number in just 18 guesses:
(b) Find a method for Player B to guarantee knowing Player A’s number in 24 guesses:
(c) Is there a method for Player B to guarantee knowing Player A’s number in 22 guesses?
7. (a) The first 17 guesses can exclude at most $17 \times 5=85$ numbers. There are 5 numbers left; 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 $(10 i+j-1)$, where $1 \leqslant i \leqslant 9,1 \leqslant j \leqslant 10$.
Divide this table by bold lines (as shown in Figure 4).
There are 11 cells of the form. It contains 55 numbers, so B first guesses 21, 26, 
$34,42,47,55,63,68,71,76,84$
a total of 11 times. If A says “close,” for example, 21 is “close,” 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 ㄱ․, containing 28 numbers. So B guesses $13,18,39,50,89,92,97$ a total of 7 times. If A says “close,” for example, 13 is “close,” 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 $10,30,80,90,15,59,95$ a total of 7 numbers left.
B then guesses 20,80 a total of 2 times. If 20 is “close,” then A's number must be among 10,30, guessing 10 can determine A's number; if 80 is “close,” then A's number must be among 80,90, guessing 70 can determine A's number; if “far,” then B guesses 2 times to exclude 4 numbers.

Finally, only $15,59,95$ a total of 3 numbers are left. Guessing 2 more times can determine A's number.
Therefore, it only takes $11+7+2+2=22$ guesses to determine A's number.
At this point, (b) is clearly satisfied.