Looking for strategies for solving math puzzles

Hi! Occasionally I come across, in various magazines, puzzles like this:

Arrange the digits 1-9 in such a way that their sum is 100.

or

Adding only 2 additions and 2 subtractions to the following sequence of digits, obtain a result of 621:

9 8 7 6 5 4 3 2 1

I am wondering if anyone has strategies - either mental, or computational, for solving such problems.

With the speed of today's personal computers, one can often solve a puzzle like this using 'brute force' by writing a short program in your favorite computer language. One advantage to using a computer program is that all possible solutions can often be found. One disadvantage to relying on a computer, is that sometimes puzzles rely on a trick (like flipping over the puzzle so that a '6' becomes a '9'. By relying too heavily on the computer to find the solution, the solver may never realize that a trick is necessary.

For 'manual' solution, there are several techniques. Here are a few:

1. Using digital roots (expressing a number in modulo 9). 2. Prime factoring. 3. Proving that parts of the expression must be odd or even (or multiples of other numbers). 4. Searching for isomorphisms (finding a puzzle with characteristics that map onto the current one). 5. Using 'Divide-and-Conquer' (dividing the puzzle up into smaller puzzles with simpler goals). 6. Drawing and navigating tree diagrams.

Carl G.

...

I also prefer puzzles that can be solved using insight, and don't require a computer to solve. Here are a few reasons why I believe that many of the newer puzzles (especially those published in the media) seem to require computer solution:

1. It is hard to come up with new puzzles that require insight. It is especially hard for puzzle columnists to come up with a good one every time they publish. It is hard to create a new 'insight puzzle' that wasn't previously published (e.g., in a collection by Sam Loyd or Henry Dudeney). Allowing computers to solve puzzles opens up more possibilities for puzzles, and gives the author more opportunities for an 'original' puzzle.

2. Even if the puzzle was designed to be solved by hand, there are people who will use a computer to solve it. The authors receive comments like, 'Hey, why are your puzzles getting so easy? My Pentium IV solved your latest puzzle in only 10 seconds.'. Pride puts pressure on the authors to make their puzzles harder. Unfortunately, it often makes them too hard to solve without a computer.

3. By using a computer to check his solution, the author can ensure that his solution is best. Before computers were used, a puzzle would only ask for the person to find a solution that was as good as the author's. Sometimes the author was surprised to find out that someone found a better or alternate solution. To prevent someone from finding such a solution, today's authors often use computers to check that their solutions are optimal and/or unique. In many cases, they don't even try to find a solution themselves, but rely on their readers to find the best solutions.

4. Puzzle columnists often rely on their readers for new puzzle ideas. Since many of their readers have discovered that the puzzles getting published require computer solutions, they tend to send in similar puzzles.

I would like to see some comments from newsgroup members that describe the techniques they use to create original puzzles, especially puzzles that require insight. Can 'insight puzzles' be engineered (generated by following documented procedures), or are they by their very nature impossible to create except by a 'flash of insight'?

Carl G.

I happen to be designing a new puzzle in the background, right now. I have a set of 160 convex quadrilaterals I'm working with... I'm trying to make a reasonable problem from a subset of them.

I also dislike puzzles that require a computer to solve. One of my favorite recent puzzles was to draw 5 arcs through the points of a 5x5 grid of dots without lifting the pencil part of the compass off the paper. The idea for the puzzle was by David Eppstein, and Koshi Arai was the first solver.

Computers are good for finding excellent small puzzles. For example, Burr Puzzles are frequently put through a computer analysis when a new variety is thought of. With the aid of a computer, a highly challenging

I am a computer guy, but I don't find much pleasure in solving puzzles with a program that brute-forces all combinations. Some puzzles are very difficult to solve with a computer, either because it requires insight or because a brute force algorithm would take too long time for a computer.

The puzzles that are fun to solve with a computer are the ones that are not trivial. It should take much longer to come up with the algorithm than to type the program into the computer. A puzzle that is equally difficult to solve with or without a computer is a good puzzle (I'm not saying that no other puzzles are 'good'.

An example of problems that is difficult but not impossible to solve with a computer are 3x3 sequence puzzles, like the one I posted last year (now on the web[1]). I've tried to make a program that solves that kind of puzzles, and it is pretty good by now. Making a program that solves small magic squares on the other hand is not very exciting, but it would be a challenge to write one that finds the smallest number of unit cubes required to totally enclose a number of spaces[2].

ObPuzzle: Come up with a puzzle or a family of puzzles that are more or less equally hard to solve with or without a computer.

[1] http://puzzles.zzlevo.net/seq1.html [2] http://groups.google.com/groups?ic=1& th=ec22b32584458cce&seekd=956439117

Andreas

I liked the Theseus and the Minitaur Java puzzle that was posted here awhile back. The author mentioned that the specific mazes were generated by a computer algorithm (though eventual human selection played a part), and yet they all seemed to require a flash of insight to solve.