Answer:
The first thing you should notice is that the order of operations is
not "PEMDAS" -- that is, multiplication isn't necessarily done
before addition and subtraction. You can tell this from the first
row. If a product is subtracted from a single-digit number, you can
never get 21.
You should pick a row or column with the fewest combinations to start
with. I picked the first column, because it's result is the highest
number. The only two products that are within single-digit range of
53 are 6x8+5 and 5x9+8 (or the first two can be interchanged, giving 8x6+5
and 9x5+8). It is interesting to note that both the 5 and 8 have to
be used in the first column no matter which of these combinations turns
out right. Moreover, the bottom number in the first column is either
5 or 8. The possibilities are:
The next biggest total is the third row, which totals 31. The
only products in single-digit range of 31 are 8x4, 5x7, 9x4, and
8x5. These combinations (and their reversals) seem like a whole lot
to check, but don't worry -- many can be eliminated right off the
bat. Since both 5 and 8 have to appear in the first column, you can
eliminate 8x5. You can also eliminate 9x4 because the lower-left
number has to be either 5 or 8. This leaves just two possibilities
for the bottom row: 8x4-1 and 5x7-4. Now, the possibilities are:
Turning now to the first row, only 3 x 7 equals 21, so let's look at
the four possibilities, above. In the first possibility, 5-2x7 is
the only way the first row can be filled out.
The second possibility
allows 9-2x3=21, but then the middle column would be 2+3x4, using the 3
twice, so that's no good. The second possibility also allows
9-6x3=21, but then the middle column doesn't work, either. So the
second possibility is out.
The third and fourth possibilities can be ruled out because of the
middle column -- 7 isn't a factor of 20, so it doesn't matter what the
first two numbers of the 2nd column are; the product can't be
21.
That leaves one possibility for filling out the first and third rows:
Now there are only two numbers that haven't been used -- 3 and 6.
9 divided by 3 times 6 equals 18, so this is the final answer.
