This post was inspired by Robert Euell’s excellent session on Problem Solving with Polya, at The Complete Mathematics Conference, #MathsConf25, March 2021.
Robert discussed teaching problem solving to students and looked at various problems, considering the steps in Polya’s problem solving process. One problem I had not come across before is the above census taker problem, a wonderful problem to illustrate the problem solving process.
This particular version can be seen in these sample pages from The Art and Craft of Problem Solving, the solution is presented here also.
Searching a little further I found further references to this problem and variations – some useful links:
- Census Puzzle (mathsisfun.com) from Assorted Maths Puzzles and quizzes on Mathisfun
- From Cornell University, Revisiting a Number-Theoretic Puzzle: The Census-Taker Problem
- Ages of Three Children puzzle – Wikipedia
From the University of California, Berkeley see this very useful summary of Polya’s problem-solving techniques – including a summary, in the Polya’s own words, on strategies for attacking problems in mathematics class from the book, How To Solve It, by George Polya, 2nd ed., Princeton University Press, 1957, ISBN 0-691-08097-6.paragraph
On mathinaction.org we see problems solving strategies, each is illustrated by various examples for younger students. Video Solutions are available to challenges. Problems are suitable for younger students, grade 1 through 6.
Robert recommended websites that are excellent sources of problems including my own favourites, Underground Mathematics, Open Middle, Jonny Griffith’s RISPS and from The University of Waterloo’s Centre for Education in Mathematics and Computing; note the Problem of the Week and Problem of the Month.
Talking of Open Middle, I really like this from Tim Brzezinski on Building a distributive property expression. A lovely problem. Try this on GeoGebra.
I would add Nrich to the go-to list, see in particular this Starter Problem Selection, ideal for thinking about the problem-solving process as the problems have been chosen because they only make use of simple mathematical concepts, the emphasis in these problems is on thinking rather than knowledge. Also on Nrich you can search on Problem Solving, then filter your search by type, age and challenge level. Note their Guide to Problem Solving and for older students, the Advanced Problem Solving modules for STEP Preparation are excellent. And of course the UK Maths Challenges.
For more problems – try The Math Contest. Students from around the world can submit answers.
See Polya on Let us teach guessing.
And to finish … a final play on Mathigon…thinking about that census problem. So pleased I can embed Mathigon activities! This is more of an illustration than an activity.