## My Cross Sums

A cross sums puzzle 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 multi-digit number. Someone sent me email saying that Dell's Cross Sums seem to be getting more difficult. Some of them are. A few are extremely easy. An improvement that I have noticed is that they never contain errors. I assume that a computer is used in some stage of the design process (perhaps just proof reading).

Here is one that I designed. It seems to be somewhat difficult.

Here is how I solved it:

First of all, about half of the puzzle can be solved without much effort.

Now we have some difficulties. The lower right corner looks promising. Let's work on that. The column that totals 20 can be any of these permutations (the top one must be 8 or 9 because of the numbers to its right):

```     8988899999
9875376543
3357945678```

That allows us to fill in permutations to the left of that (the x's are numbers which didn't work out):

```                           8xxxx9xx99
7xxxx8xx88 1xxxx2xx56 9xxxx7xx43
7xxxx6xx32 3xxxx4xx78```

Shortcut #1: As a shortcut, we observe that if the leftmost number is an 8 (and the upper right number is a 9) then this part of the puzzle is ambiguous (see Assume there are No Typos?). So we can assume that the leftmost number is a 7, and the upper right number is an 8. That also allows us to place a 9 at the top of the column labeled 40. As the puzzle designer here, I actually verified that the ambiguous solution is not consistent with the rest of the puzzle, filling twelve sheets of paper in the process.

And to the right of that we have these:

```             ??? 233
677 988
???
???
112 554
211 343 667
888 122 766```

The numbers 1,2,3,4, and 5 can be placed where the question marks are, producing these:

```     xx445522xx55114455 xx222233xx33333333
xx666677xx77777777 xx999988xx88888888
xx252435xx23451514
xx524253xx32545141
xx111111xx11222222 xx555555xx55444444
xx333344xx44333333 xx666666xx66777777
xx777766xx66666666```

That allows us to fill in some of the adjoining permutations, and eliminate some:

```                        xx8xxxxxxxxxxx7xxx xx4xxxxxxxxxxx4xxx xx2xxxxxxxxxxx3xxx
xx2xxxxxxxxxxx1xxx                    xx6xxxxxxxxxxx7xxx xx9xxxxxxxxxxx8xxx
xx6xxxxxxxxxxx7xxx xx3xxxxxxxxxxx3xxx xx2xxxxxxxxxxx1xxx
xx4xxxxxxxxxxx4xxx xx5xxxxxxxxxxx5xxx
xx1xxxxxxxxxxx2xxx xx5xxxxxxxxxxx4xxx
xx2xxxxxxxxxxx1xxx xx3xxxxxxxxxxx3xxx xx6xxxxxxxxxxx7xxx
xx1xxxxxxxxxxx2xxx xx7xxxxxxxxxxx6xxx```

Shortcut #2: That leaves us with only two permutations for the entire lower right area. We could now list all of the permutations for the upper right corner area. But we can use another shortcut (see Solving Cross Sums) first. The sum of the upper right vertical columns is 4+24+16+26+8+4=82. The sum of the upper right horizontal rows is 6+10+4+8+20+13=61. So the three squares not counted in the second sum (the bottom two numbers of the column marked 24, and the bottom number of the column marked 26) add up to 21. Either all three are 7 (not possible as two of them are in the same column) or they are 6, 7, and 8. So we can now fill in the entire lower right corner of the puzzle.

The rest of the puzzle is relatively easy, although several permutations must probably be listed. Notice that the column labeled 41 cannot have a 4 in it, and the column labeled 40 cannot have a 5 in it.

And here is the final solution:

Comments: If we don't use shortcut #1, as I did not when designing and verifying the puzzle, then we should rely heavily upon shortcut #2. That is how I was able to eliminate most of the possible permutations of the lower right corner under those conditions.

Dell Puzzles ignored my email asking for permission to show my solution to one of their most difficult cross sums. And so, I was forced to design one of my own. It has been an educational experience, involving much work and many mistakes. My pages still use corners from two of their puzzles. Despite their silence, I assume that is all right, for the following reasons:

• What I show are fragments, much like a quote from any piece of literature. Such quoting does not violate copyrights.
• I give Dell credit as the source.
• I am not presenting them as puzzles to be solved, but as good examples of solving techniques.
• Hopefully, these articles of mine sell magazines for Dell, and they should not object.