More than just a random number, 1729 connects a London taxi to a mathematical marvel. It’s a number that can be expressed as the sum of two different pairs of cubes – 10³ + (-1)³ and 9³ + 10³ – a discovery that made it famous in mathematical circles. Today, we explore three puzzles loosely inspired by this historic number.
Square Pair
What is the smallest number that can be expressed as the sum of two squares in two different ways?
The smallest number that can be written as the sum of two squares in two different ways is 50.
- 5² + 7² = 25 + 49 = 50
- 1² + 7² = 1 + 49 = 50
This puzzle highlights interesting patterns in number theory. The ability to express a number as a sum of squares relates to the fundamental theorem of arithmetic, which underpins much of modern mathematics.
Strip Tease
Can you add a seventh strip to five existing ones (1, 2, 7, 17, and 29 cm) without allowing three of them to form a triangle?
The possible lengths for the seventh strip are 3 cm, 4 cm, and 5 cm.
The triangle inequality states that in any triangle, the sum of two sides must exceed the third side. With the existing strips, it’s impossible to form a triangle because the sides are too disparate. Adding a strip of 3, 4, or 5 cm maintains this condition.
For example, with sides 3, 4, and 5, you can form a right-angled triangle. This puzzle demonstrates how geometry and inequalities intersect in surprising ways.
Sick Sixth
Given four numbers a, b, c, and d, and five of their pairwise products are 2, 3, 4, 5, and 6, what is the sixth product?
The sixth product is 2.4.
The product of all six pairwise combinations (ab, cd, ac, bd, ad, bc) equals (abcd)². By analyzing the relationships, we find that 2 × 6 = 3 × 4 = 12, so the sixth product must be 2.4 (since 5 × 2.4 = 12).
This puzzle showcases how algebra and pattern recognition can solve complex problems. It also demonstrates that even seemingly unrelated numbers can be interconnected.
Conclusion
These puzzles reveal how mathematics weaves through everyday phenomena, from taxi numbers to geometric shapes and algebraic patterns. They remind us that numbers aren’t just abstract concepts—they’re the foundation of our understanding of the world
