Welcome to the 178th edition of the Playful Math Education Blog Carnival — a smorgasbord of delectable tidbits of mathy fun. It’s like a free online magazine devoted to learning, teaching, and playing around with math from preschool to high school.
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There’s so much playful math to enjoy!
By tradition, we start the carnival with a puzzle/activity in honor of our 178th edition. But if you’d rather jump straight to our featured blog posts, click here to see the Table of Contents.
Activity: Nicomachus’s Theorem
Welcome to 2025, a perfectly square year — and the only one this century!
2025 = (20 + 25)2
- When is the next time we’ll have a perfect-square year?
- Can you find the only perfect square less than 2025 that works by this pattern? When you split the number’s digits into two smaller numbers and square their sum, you get back to that number.
2025 = the sum of all the numbers in the multiplication table, from 1×1 to 9×9
2025 = the sum of the first 9 perfect cubes
- When is the next time this will happen, that the year is the sum of the first n perfect cubes?
And by Nicomachus’s theorem:
2025 = 13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93
so it must also = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)2
Try it for yourself with small numbers: Get some blocks, and build the first few perfect cubes. Then see if you can rearrange the block to form the sum of those numbers squared.
Can you show that…
- 13 = 12
- 13 + 23 = (1 + 2)2
- 13 + 23 + 33 = (1 + 2 + 3)2
- 13 + 23 + 33 + 43 = (1 + 2 + 3 + 4)2
- 13 + 23 + 33 + 43 + 53 = (1 + 2 + 3 + 4 + 5)2
Older Students: Can you see that the pattern would continue as long as you want? How might you prove that?
Here’s the formula for triangular numbers, to get you started:
(1 + 2 + 3 + … + n) = n(n + 1)/2
Denise Gaskins' Let's Play Math 