New Year: Fun And Fascinating Mathematical Expressions Of '2025'

Happy New Year, 2025!

Each year brings a unique number to celebrate, and with every number comes the opportunity to explore its mathematical expressions.

This article dives into the many creative ways to represent the number 2025. Perfect for students, teachers, and anyone who loves maths, these examples aim to spark curiosity and encourage a deeper appreciation for numbers.

Let’s uncover what makes '2025' uniquely interesting in the world of mathematics:

A Perfect Square

2025 = 45²

The year 2025 is a perfect square, formed by squaring 45. The last time this occurred was in 1936, and the next will be in 2116.

Sum of Cubes of 1 through 9

2025 = 1³ + 2³ + 3³ + ··· + 9³

This demonstrates the beauty of cubic numbers summing up to form 2025.

Sum of the First 45 Odd Numbers

2025 = 1 + 3 + 5 + ··· + 89

Adding the first 45 odd numbers yields 2025.

Harshad Number

2 + 0 + 2 + 5 = 9 and 2025 ÷ 9 = 225

Since 2025 is divisible by the sum of its digits (9), it qualifies as a Harshad number.

Triangular Square

45 = (9 x 10)/2

2025 is the square of 45, which is also the 9th triangular number.

Sum of Three Squares

2025 = 40² + 20² + 5²

2025 can be expressed as the sum of three perfect squares.

Kaprekar Link

2025 = 45² and 20 + 25 = 45

Splitting 2025 into 20 and 25, their sum is 45, linking it to the Kaprekar property.

Power Sum Representation

2025 = 2¹⁰ + 2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2⁴ − 2³ + 2² − 2¹ − 2⁰

A creative use of powers of 2 with alternating additions and subtractions to reach 2025.

Creative Arithmetic Representations

2025 = 12 × 3 + 45 × 6 × 7 + 89 + 10

2025 = 10 + (9 + 8 × 76 + 54) × 3 + 2 × 1

2025 = 12 × 3 + (4 + 5) × (6 + 7) × (8 + 9)

2025 = 9 × 8 + 76 + 5⁴ × 3 + 2 × 1

Playful Mathematical Expressions

2025 = (1 + 1)¹¹ − 11 − 11 − 1

2025 = (2 × 22 + 2/2)²

2025 = 3 × (3 × (3 + 3)³ + 3³)

2025 = (44 + 4/4)^(4+4)/4

2025 = 5 × (5 × (5 × 5 + 55) + 5)

2025 = 6 × 6 × 6 × 6 + (6 × 6 / (6 + 6))⁶

2025 = (7 + 7 + 7/7) × ((7 + 7)⁷ / 7 + 7)

2025 = 88 × (8 + 8 + 8 − 8/8) + 8/8

2025 = 9 × ((9 + 9) × (9 + 9) − 99)

Magic Squares

A magic square is a square grid filled with distinct positive integers such that the sum of the numbers in each row, column, and diagonal is the same. This constant sum, 2025, is called the magic constant.

Here are some unique representations of fourth-order, fifth-order, sixth-order, ninth-order, tenth-order, and fifteenth-order magic squares whose magic sum is 2025.

As we step into 2025, let's keep exploring the magic of numbers — there's always more to discover in the world of mathematics.