Return to my Puzzle pages
Go to my home page

Solving Cross Sums

© Copyright 1996, Jim Loy

Notice that the name "Cross Sums" is used in Dell puzzle magazines. A Cross Sum looks superficially like a crossword puzzle. But the numbers, that appear in the black margins, represent the sums of the digits that you will insert into the empty squares. A number above a diagonal line refers to the empty squares to the right of that number, and a number below a diagonal line refers to the squares below that number. No zeros appear in the puzzle, and no digit is repeated in a particular number.

This diagram is just the upper left corner of such a puzzle. Notice the horizontal sequence of three empty squares with the number 8 to the left of it. A three-digit number will fit in there, with no zeros and no repeated digit. You can deduce that each of these digits must be in the range 1 through 5 (if one of them is 6, then the other two would have to be 1 and 1, which is illegal). The vertical column, with 21 above it, is a three-digit number, all of which must be 4 or greater. Well one of these three digits (the middle digit) is 5 or less (by the previous clue), so that digit must be either a 4 or a 5. If that digit is a 4, then the other two must be 9 and 8. But the top digit is 7 or less (if it were an 8, then the digit to its left would be a zero, which is illegal). So the middle digit is a 5 and the top digit is a 7 (and the bottom digit is a 9). This is the kind of reasoning that eventually leads to the solution of the entire puzzle.

There is another method to determine some digits. In the above diagram, we can determine the digit marked by the red arrow, not by reasoning, but by simple calculation. You might want to deduce how this can be done, before I tell you. It is fairly satisfying to deduce this on your own.

OK, the digit marked by the red arrow is the sum of the horizontal numbers (above the diagonal lines) minus the sum of the vertical numbers (below the diagonal lines). In this example, the sum of the horizontal numbers is 8+8+27+5+14=62. The sum of the vertical numbers is 15+23+21=59. (Horizontal) - (vertical)=3. That number is a 3.

Not only have we calculated one of the digits of the puzzle, but we have partitioned the puzzle into two puzzles (this upper-left corner, and the rest of the puzzle). These two puzzles do not affect each other anymore, no further number in the rest of the puzzle will affect any of the numbers in this upper-left corner of the puzzle. So, you can go ahead and finish this corner of the puzzle, because you won't get any more clues toward its solution from the rest of the puzzle.

Further logical deductions lead to the solution of this part of the puzzle:

Return to my Puzzle pages
Go to my home page