## More CrossNumber Puzzles

Last fall, I mentioned that while looking at the Puzzle Page of the FOCUS magazine published by the Mathematical Association of America, I thought to myself, “Hey, I write lots of puzzles.  Maybe some of mine can get published!”  So I submitted a few Number Search puzzles to the editor, and to my delight, she included them in the December/January issue.  Here’s the proof….

Incidentally, these puzzles are the same ones I wrote about almost two years ago — hard to believe I’ve been blogging that long!  So if you want to try them, you can look at Number Searches I and Number Searches II.

Since I had success with one round of puzzles, I thought I’d try again.  This time, I wanted to try a few CrossNumber puzzles (which I wrote about on my third blog post).  But as my audience was professional mathematicians and mathematics teachers, I wanted to try to come up with something a little more interesting than the puzzles in that post.

To my delight again, my new trio of puzzles was also accepted for publication!  So I thought I’d share them with you.  (And for those wondering, the editor does know I’m also blogging about these puzzles; very few of my followers are members of the MAA….)

Here is the first puzzle.

Answers are entered in the usual way, with the first digit of the number in the corresponding square, then going across or down as indicated.  In the completed puzzle, every square must be filled.

I thought this was an interesting twist, since every answer is a different power of an integer.  I included this as the “warmup” puzzle.  It is not terribly difficult if you have some software (like Mathematica) where you can just print out all the different powers and see which ones fit.  There are very few options, for example, for 3 Down.

The next puzzle is rather more challenging!

All the answers in this puzzle are perfect cubes with either three or four digits, and there are no empty squares in the completed puzzle.  But you might be wondering — where are the Across and Down clues?  Well, there aren’t any….

In this puzzle, the number of the clue tells you where the first digit of the number goes — or maybe the last digit.  And there’s more — the number can be written either horizontally or vertically — that’s for you to decide!  So, for example, if the answer to Clue 5 were “216,” there would be six different ways you could put it in the grid:  the “2” can go in the square labelled 5, and the number can be written up, down, or to the left.  Or the “6” can go in the square labelled 5, again with the same three options.

This makes for a more challenging puzzle.  If you want to try it, here is some help.  Let me give you a list of all the three- and four-digit cubes, along with their digit sum in parentheses:  125(8), 216(9), 343(10), 512(8), 729(18), 1000(1), 1331(8), 1728(18), 2197(19), 2744(17), 3375(18), 4096(19), 4913(17), 5832(18), 6859(28), 8000(8), 9261(18).  And in case you’re wondering, a number which is a palindrome reads the same forwards and backwards, like 343 or 1331.

The third puzzle is a bit open-ended.

To solve it, you have to fill each square with a digit so that you can circle (word search style) as many two- and three-digit perfect squares as possible. In the example above, you would count both 144 and 441, but you would only count 49 once. You could also count the 25 as well as the 625.

I don’t actually know the solution to this puzzle.  The best I could do was fill in the grid so I could circle 24 out of the 28 eligible perfect squares between 16 and 961.  In my submission to MAA FOCUS, I ask if any solver can do better.  Can you fit more than 24 perfect squares in the five-by-five grid?  I’d like to know!

I’m very excited about my puzzles appearing in a magazine for mathematicians.  I’m hoping to become a regular contributor to the Puzzle Page.  It is fortunate that the editor likes the style of my puzzles — when the magazine gets a new editor, things may change.  But until then, I’ll need to sharpen my wits to keep coming up with new puzzles!