Return to my Puzzle pages
Go to my home page

Assume there are No Typos?

© Copyright 2002, Jim Loy

As I said in Solving Cross Sums, 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.

Here is the lower right corner of a cross sums puzzle by Dell. After a great deal of work, I find that there are six possible solutions:

    786777  879888
    213222  342344  666655
            997797  779979
    555544  111111  222233
    888899  224423  997787
            666656  888898

Well, I suppose we will have to solve other parts of the puzzle, in order to narrow this down. But, assuming that there are no typos in the puzzle, we can reason further. Assume that off to the left of our top row (786777 879888) that there are no 8s or 7s. Then this entire lower right corner of the puzzle is ambiguous; at least five of these six solutions will work, no matter what the rest of the puzzle looks like.

Can we assume that the puzzle is not ambiguous? Dell implies that there is only one solution. If the puzzle has several solutions, then that is a glaring typo. So, assuming there are not typos, this is the solution (on the right):

This may not be a normal way to solve cross sums. But it is a normal way to solve many other puzzles. Logic problems and liar problems (also called "false logic problems," in which one person is lying, or everyone is telling exactly one lie) depend heavily upon the fact that there are no typos, that there are no ambiguities. Whether you think of that or not, that is one of your assumptions as you solve a logic problem. That is also how we answer such "questions" like, "Guess who I beat at golf today." Well, we have to assume that it was a big upset, otherwise the answer could be almost anyone (highly ambiguous). So we correctly guess, "The club pro," or someone of that ilk.

Return to my Puzzle pages
Go to my home page