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Academic Decathlon 2004 State Championship Logic Quiz

Question 11

Fill in the blanks in this mathematical crossword puzzle so that each of the three number sentences (three across, and three down) are true:


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.


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