GENERAL PROBLEMS AND PUZZLES (ordered very roughly by likely age of interest, youngest to oldest)
- Numeracy Tasks (All Grade Bands) | Peter Liljedahl
- NRICH enriching mathematics
British site with lots of cool problems.
- 1001 Math Problems | Sian Zelbo
Fun, fairly complicated visual and logic math problems that don't require more than elementary school math. Same author has very fun site 1001visualpuzzles.blogspot.com, but that's now removed from public view.
- Play With Your Math
Most of these puzzles require only elementary-level math content knowledge, but they are not easy to solve! Fun challenges! Many are classics. Recommended by NCTM's Teaching Children Mathematics.
- Math Tasks | youcubed
- Brain Teasers | NCTM Illuminations
Various cool problems aimed at different age levels. Good to browse!
- 30 Math Starters (Quick Puzzles)
Recommended on a UK site. Quick but interesting puzzle problems. Look mainly geared for upper elementary.
- Inside Mathematics
Includes Common Core resources, problems of the month, performance tasks and other assessments, etc.
- Open Middle: Challenging math problems worth solving
Problems with a defined beginning and end, but an "open middle": different solution paths and problem solving techniques could be used.
- Would You Rather?
Asks students to choose their own path and justify it.
- Problems for Developing Mathematical Habits of Mind | NRICH
Problems designed to encourage certain mathematical habits of mind: being curious, collaborative, thoughtful, or determined.
- Math Explorers' Club | Cornell Dept. of Mathematics
Advanced math "modules" for middle school & high school students. Wow!!
- Five Triangles
Terrific, complicated, fun problems, especially for geometry and for math-lovers.
- Brilliant (Math & Science Problems)
Free. Graphite's recommendation says Brilliant has excellent problems and adapts to students' performance, but is probably best for high-end students. Some problems have a test-prep orientation.
- Math Teacher Resources from I Speak Math
Big list of math teaching resources, with pointers to lesson plans, problems, professional development...
- Sunday Puzzle | io9
Aimed at adults, but likely to have good kid puzzles sometimes too.
PROBLEMS AND PUZZLES FOR PARTICULAR TOPICS AND SKILLS (ordered roughly by age, youngest to oldest)
- Proof (Elem. & up): Numbers of Squares Making a Square | artofmathstudio
- Factor Craze | NCTM Problems to Ponder
Currently available only to NCTM members. "Which numbers have exactly three factors? Exactly four factors? Exactly five factors? Exactly six factors? Given a positive integer n, how can we tell exactly how many factors it has?"
- Car Talk Gone Fishing Puzzler | A Recursive Process
- Stained Glass Circle/Area/Perim/Modeling Problem | Illustrative Mathematics
Very cool and realistic scenario in which students need to choose a reasonable model and use knowledge of area (especially of circles), perimeter, and numerical calculations. Requires knowledge of circle area (7th grade standard).
- Conway’s Rational Tangles (of Rope) | Fawn Nguyen
Group activity with rope in which students need to analyze rotating & twisting mathematically. See https://hilbertshotel.wordpress.com/2015/05/06/conways-ropes/ for more on doing this.
- Great Circle/Square Problem with Multiple Solutions | Chris Harrow, NCTM
Multiple solution methods for a @Five_Triangles geometry problem.
- World's Hardest Easy Geometry Problem | Keith Enevoldsen
Requires only elementary geometry (triangle angle sum theorem, etc.) but is very challenging. I have not solved this yet. The hints are not spoilers at all, at least for me!