This week, we’ll look at one of my favorite types of puzzles — CrossNumber Puzzles. These are like crossword puzzles, except that the clues describe numbers instead of words. The only rule is that no entry in a CrossNumber Puzzle can start with a “0.” You can try this one — but don’t worry if you get stuck. We’ll look at different ways you can go about solving it in just a moment.
How would you go about solving this puzzle? Try to look for the clues which give you the most information. For example, look at 1 Across and 3 Down. Now 1 Across is the cube of a two digit number, and its third digit is actually the first digit of the cube root. So we might want to print out a chart of all four-digit cubes of two-digit numbers:
| 10 | 1000 | 16 | 4096 | |
| 11 | 1331 | 17 | 4913 | |
| 12 | 1728 | 18 | 5832 | |
| 13 | 2197 | 19 | 6859 | |
| 14 | 2744 | 20 | 8000 | |
| 15 | 3375 | 21 | 9261 |
You can see that the only possibility is that 1 Across is 4913 and 3 Down is 17.
Looking at 5 Across doesn’t help much, since there are too many possibilities.
But looking at 5 Down is a good next choice. Note that 9 Across has to start with 1 or 3 in order to fit four odd digits in the grid, but no perfect squares end in 3, and so no perfect fourth powers end in 3, either. This means that 9 Across has to start with 1 so that 5 Down ends in 1. To help figure out 5 Down, below is a list of four-digit fourth powers:
| 6 | 1296 | 8 | 4096 | |
| 7 | 2401 | 9 | 6561 |
So 5 Down must be either 2401 or 6561. If it were 2401, then 6 Across would begin with a “0,” so that leaves 6561 as the only option for 5 Down.
I’ll leave it to you to complete the puzzle. I won’t post a solution so that you’re not tempted to peek — but if you add 2 Down and 6 Across when you’re done, you’ll get 157,991.
How can you make your own CrossNumber puzzle? Start by making a grid, and shade in some of the squares. Usually the pattern of shaded squares is symmetric, but it doesn’t have to be. Fill in some of the entries with numbers which have specific properties, like being a perfect square or cube. Or perhaps make one of the entries the product or sum of two others. The only limit is your imagination! It might help to continue reading below, since then you could print different charts and look at the numbers for something interesting. (And get another puzzle to solve, too.)
Creativity in Mathematics 