The study of multiples and factors is ripe with opportunities to engage students in intriguing mathematical puzzles. In prior posts (When Factors Put on Their Dancing Shoes, When Factoring Gets Personal, and Open the Safe), I’ve given some examples of what can be done. Now I’d like to introduce you to another puzzle of mine called Mystery Number that focuses on multiples.

Here’s how the puzzle works: Sketchpad has randomly chosen a “mystery” number between 1 and 25, and students must use logical reasoning and their knowledge of multiples to determine it. Students pick and press any of the “Multiple of?” buttons in the table below. Pressing “Multiple of 3?” for example, indicates that the mystery number is a multiple of 3. The goal is to press as few buttons as necessary to determine the mystery number. Sometimes, however, the information in the table may not be enough. In this case, **only** as a last resort, students press the “Show Sum of Digits” button to learn the sum of the digits of the mystery number (For example, 17’s digits sum to 8).

I’ve field tested this puzzle with elementary students, and they become very engaged in the mathematical thinking involved in solving the challenges. The students worked in pairs to solve the puzzles and then came back together as a group to discuss and share strategies. Here are some of the questions that arose:

- Strategically, what is the best button to press first?
- If you learn that the mystery number is a multiple of 3, do you know whether the number is a multiple of 6?
- If you learn that the mystery number is a multiple of 6, do you know whether the number is a multiple of 3?
- If you learn that the mystery number is a multiple of 2 and 5, what else do you now know?
- When is it necessary to press “Show Sum of Digits”?
- Is it always possible to find the mystery number?

And, as with any good problem, the students also proposed some ways to extend it:

- Suppose the list of possible mystery numbers is expanded to include every number between 1 to 30. Will it always be possible to determine its value?
- Can the puzzle be adapted to work when the mystery number is between 1 to 40?

If you have the opportunity to try this activity with your students, let me know what questions they explore!

*An annotated list of all our elementary-themed blog posts is here.*