# Engaging Mathematical Riddles for Classroom Enrichment: examples and impact studies references

Hello fellow educators and math enthusiasts,

I am on the hunt for entertaining and thought-provoking mathematical riddles suitable for a classroom setting. My objective is to find riddles that not only captivate the students' interest but also sharpen their problem-solving skills.

Could you please share the most humorous or intriguing math riddle you've encountered? Additionally, I'm interested in any books or resources that compile such mathematical puzzles. If you know of any "What is...?" type riddles that are particularly engaging, I'd love to hear about those as well.

Furthermore, I'm curious about the pedagogical impact of using mathematical riddles as a teaching tool. Are there any studies or research findings on how such riddles can enhance learning or engagement in the classroom?

Thank you for your insights and recommendations!

HERE IS A REPRESENTATIVE EXAMPLE: hat is what is: hat i has a head and it has a tail but no body. Answer: a coin.

being a little bit mor precise: I am searching for clever riddles that would be suitable for use in a stand-up comedy routine.

• Could you give an example of what you are looking for? Commented Nov 5, 2023 at 18:32
• Commented Nov 5, 2023 at 21:05
• Just browsing through my memory: Martin Gardner, Math Olympiads, the Hungarian Köhmal, ... Commented Nov 6, 2023 at 12:07
• @Tommi.(Here it is, It's not a good example, however): what is what is: hat i has a head and it has a tail but no body. Answer: a coin. Commented Nov 8, 2023 at 12:30
• another example slightly better: What weighs more? 1 kilogram of cotton or 1 kilogram of gold? younglings answer 1 kilogram og gold quite often Commented Nov 8, 2023 at 12:32

I like the following puzzle because the answer is unexpected and also because it doesn’t require any advanced mathematics.

What is the next number in the following sequence?

0, 4, 8, 21, 52, 65, 96, ?

The next number is 1. The rule is to add 4 and then reverse the digits to get the next number. For example 21+4=25 and reverse it to get 52.

I gave this puzzle to my daughter’s grade six class and as a group they solved it in about five minutes.

• observe however There is a substantial and heated debate regarding the validity and purpose of such questions about the next element in a sequence. After all, various different rules could be established, in principle... Commented Nov 8, 2023 at 12:40

This one is a mathematics killer:
What's the next line, knowing that the numbers can only be $$1$$, $$2$$ or $$3$$ (you can never have $$4$$ or higher):
$$1$$
$$1 1$$
$$2 1$$
$$1 2 1 1$$
$$1 1 1 2 2 1$$
$$3 1 2 2 1 1$$

You just need to read what it says: $$1$$ reads as "one one" (one time the number one)
$$1 1$$ reads as "two ones" (two times the number one) $$2 1$$ reads as "one two one one (one time the number two and one time the number one)

Why do I call this a mathematics killer? Well, the better you know mathematics, the more you will look for additions, subtractions and other multiplications, powers, primes, ..., and you'll never find it, although this has nothing to do with the actual solution.

• I had a Physics professor give this to the class back in the 1970s. It really annoyed me. It seemed so tricky and stupid. Years later, I heard how John Conway found interest in the sequence so I realized I had better rethink my bad attitude about the "Look and Say" sequence. youtu.be/ea7lJkEhytA?si=7QpCnSGplbcaaqBA Commented Nov 14, 2023 at 0:22

Here's a puzzle that I used to give to all grades Algebra through Calculus on the first day of class as something fun to do while many students were waiting to sort out schedule and laptop issues.

Draw an $$10 \times 10$$ square grid. How many squares are there in total? Not just $$1 \times 1$$ squares, but also $$2 \times 2$$ squares, $$3 \times 3$$ squares, and so on.

I liked this puzzle in particular because the solution was not immediately obvious to even the brightest / oldest students, yet all students (even the younger grades / weaker students) were able to understand the goal of the problem and work through concrete examples to notice a pattern that they could extrapolate to find the solution. (The higher grades / brighter students just picked up on the pattern faster.)

Whenever students needed some guidance getting started, I would explicitly walk them through the process of drawing up a $$3 \times 3$$ grid and counting the number of $$1 \times 1$$ squares (there are $$9$$), $$2 \times 2$$ squares (there are $$4$$), and $$3 \times 3$$ squares (there is $$1$$), for a total of $$9 + 4 + 1 = 14$$ squares. I would then leave them alone to do this for a $$4 \times 4$$ grid and then a $$5 \times 5$$ grid, after which I would help them notice the following pattern:

\begin{align*} 3 \times 3 \textrm{ grid} \quad \to & \quad 9 + 4 + 1 \\ & \quad 3^2 + 2^2 + 1^2 \\[5pt] 4 \times 4 \textrm{ grid} \quad \to & \quad 16 + 9 + 4 + 1 \\ & \quad 4^2 + 3^2 + 2^2 + 1^2 \\[5pt] 5 \times 5 \textrm{ grid} \quad \to & \quad 25 + 16 + 9 + 4 + 1 \\ & \quad 5^2 + 4^2 + 3^2 + 2^2 + 1^2 \\[5pt] \end{align*}

This pattern is almost a "punch line":

What's the total number of squares (square shapes)? Well, it's the total sum of squares (square numbers).

Extrapolating the pattern to solve the puzzle:

\begin{align*} 10 \times 10 \textrm{ grid} \quad \to & \quad 10^2 + 9^2 + 8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1^2 \\ & \quad = 385 \textrm{ squares in total} \\[5pt] \end{align*}

• ... and how many rectangles do you have? (While calculating that, keep in mind that $4 \times 1$ and $1 \times 4$ rectangles must be seen as different, although they look similar, which might be a real challenge for your students :-) ) Commented Nov 13, 2023 at 8:21