“The curse of knowledge is a term used to describe a cognitive bias that prevents people from taking into account the fact that others don’t know the same things they do.” (Pailhes & Kuhn, 2023). This egocentric bias explains exactly how I feel about this trick. I have been doing this trick with students since I was a student teacher. My students have been solving this puzzle for decades and every year I questioned whether I should do this again because “They’re gonna know.” They don’t. They never have. Even when I’ve taught siblings one year apart! I’ve done this in workshops with teachers thinking, “Someone is gonna know.” They almost never do. The number of teachers I expect to know this trick is over 50%. It’s usually less than 5%. What I’m trying to say is, even if you know this trick, most people don’t. And even fewer students do. And even if they do, they don’t know why, which is the main part of this investigation – especially if you’re doing this with grades 6+.
I hope this does gain some traction now, because it is a great trick and a nice problem for students to solve, since it involves patterns, doubles (powers of 2), and some practice with addition (the main goal for younger grades). This is accessible for all students – even if they are beginning their understanding of addition (see the file below for different sets of number cards).
The Trick:
How the Magic Number Cards Work:
These are the cards used in the video:
If you look in the upper left corner of each card, you will see the numbers 1, 2, 4, 8, 16. These numbers are the key to the trick. When a student selects a number, then tells you on which cards they see their number, you mentally add the numbers in the upper left corners of those cards.
For example, if the student chooses the number 22, their number would be on the “16” card, the “4” card, and the “2” card. 16 + 4 + 2 = 22. The numbers have been arranged on the cards so that this will work for any number between and including 1-31.
The Lesson:
Perform the trick several times with students. When they ask, “How did you do that?” tell them that is the problem they are going to solve. How can you “read someone’s mind” with number cards?
After introducing this trick/problem, display all of the cards you used for students to see (you may wish to use the image above). Ask students what they notice and wonder. Students should share with an elbow partner before sharing with the whole group.
Possible notices:
- The numbers on the first card are all odd.
- None of the cards has all even numbers.
- There are 5 cards.
- The last card, the one that starts with 16, has the numbers in order.
- The card that starts with 8 has two sets of numbers in order 8 – 15 and 24 – 30.
- The card that starts with 4 has 4 rows of numbers in order 4 – 7, 12 – 15, 20 – 23, and 28 – 31.
- The first numbers of the cards all double, 1, 2, 4, 8, 16.
- They all have 31 on them.
- The number 3 is only on the cards that start with 1 and 2.
- There are 16 numbers on each card.
Facilitate a discussion about what they noticed, highlighting patterns within cards (like strings of consecutive numbers) and between cards (like the doubles).
Next, display another set of similar cards with fewer numbers and ask students, again, what they notice:
Possible notices:
- The same numbers are in the upper right corner (1, 2, 4, 8). We’re just missing 16.
- There are 4 cards.
- This looks easier because there aren’t as many numbers.
- There are 8 numbers on each card.
- I notice that 6 is only on the middle two cards (the ones with 2 and 4 in the top left corner).
- I think it has something to do with the top numbers because 3 is on the 1 and 2 cards again and 6 is on the 4 and 2 cards.
Facilitate a discussion about what they noticed, highlighting the patterns between the two sets of cards as well as other patterns noticed. Students may have some theories about how this works and this is a good time for them to test them with their groups.
As students work to figure this out, some student groups may need some additional support. If so, you can ask them some questions like:
- What if there was a smaller set of cards that only had three cards? How many numbers do you think might be on each of those cards? Students may need to see the other two sets of cards to determine this. Show them this smaller set of cards.
- What do you notice about the numbers on all three sets of cards?
- How can you use the numbers on the cards to determine a thought of number if you know which cards that number is on?
At this point, most groups have a solution and can then practice performing for people in their group. For younger students, we can close the lesson here, with students sharing the patterns they discovered and how they discovered it. They can then practice their mental addition to perform for administrators, other teachers, other classes, or even for a parent math night. A nice extension for younger students (maybe for the next day, is to use the patterns on the cards they have seen to try to make the next set of cards.
- Which numbers would belong on each card?
- How do you know?
- What patterns can you use to efficiently solve this problem?
For middle school students, the closing will probably go a little deeper into why the magic number cards work. What is so important about the numbers 1, 2, 4, 8, and 16? Why are they so special? If students haven’t worked with exponents, yet. This could be a nice introduction.
One way to do this is to investigate doubling patterns (powers of 2) with another activity such as paper folding, where students connect the powers of 2 to the number of folds in a piece of paper to the number of sections in the paper when unfolded. Briefly, 0 folds = 2^0 = 1 section, 1 fold = 2^1 = 2 sections, 2 folds = 2^2 = 4 sections, etc.
After investigating this, come back to the magic number cards and discuss the powers of 2 in the upper left corners of the cards. Why powers of 2? Make a conjecture for students to try to find a counterexample to disprove: I claim that prime numbers cannot be expressed as the sum of powers of 2. When they disprove it, make a new claim like, “I claim that powers of 2 cannot be expressed as a sum of powers of 2.” When they disprove this, see if some students have a conjecture to state, or you can give another claim. The mathematical proof for this is beyond grades 6-8, but that’s not the goal here. For middle school, this is an investigation into exponents (specifically powers of 2, here) and one of the unique characteristics of powers of 2.
If you have introduced the idea of “What if…” into your classroom, students may have some questions to dive into, such as:
- What if we tried powers of 4?
- What if we only wanted to use certain numbers on the cards, could we use another power?
- What if we used words instead of numbers. Could we code the words with powers of 2?
When kids ask questions like this, they are asking to do more math and dive deeper into the content. Spend some time answering students questions. It pays off in depth of understanding, but more importantly it pays off in the positive shift in attitude about mathematics and growth mindset.