2025 and counting

Posted on by John Berman

What is the sum of all the numbers in the multiplication table?

Did you get 2025? Happy New Year! This problem reflects a very special property of 2025: It is the square of a triangular number, 2025=(1+2+\cdots+8+9)^2=45^2. Squares of triangular numbers are also sums of consecutive cubes, \displaystyle 1^3+\cdots+n^3=(1+\cdots+n)^2, so in this case 2025=1^3+2^3+\cdots+9^3.

We are very lucky to live through such a year. The last time the year was a square of a triangular number was in 1296, not long after Fibonacci’s time, and the next will be 3025.

Remarkably, there is another connection between 2025 and a 9\times 9 grid, and it has nothing to do with 2025 being a perfect square number. Take a look at the square on the right, above. It is a Latin square, which means that each number appears once in each row and once in each column. In yellow is a transversal, a choice of nine cells, one in each row and one in each column, containing one of each number.

How many transversals does this particular Latin square have? Surprisingly, the answer is 2025, though it is much harder to calculate. Welcome to the year of the 9\times 9 grid!

Here are some other counting problems with amusing answers, in roughly increasing order of difficulty. They get hard quickly, and there are a few at the end that I don’t know how to solve without a computer (or a lot of time). To make things easier, I will give you the answers (not in order). Your goal is to match the answers to the questions. But beware – there is one number which is the answer to two of the questions.

Answers: 2015, 2016, 2017, 2018, 2019, 2020, 2021, 2022, 2023, 2024, 2025

Counting problems

1. How many three-element subsets of \{1,\ldots,25\} have greatest common divisor 1?

2.
a) How many ways are there to make a circular necklace with 9 beads so that exactly three colors are used? (That is, each color must appear at least once.) Two necklaces are considered the same if one is a rotation of the other.

b) How many ways are there to color an infinite grid in exactly three colors so that each cell is the same color as those three cells above it and three cells to its right? Each color must be used at least once, and two colorings are considered the same if one is a translation of the other.

3. How many ways are there to rearrange the letters in MISSISSIPPI so that no letter appears twice in a row?

4. How many ways are there to rearrange the numbers 1,2,3,\ldots,15 around a circle so that the difference between any two adjacent numbers is at most 3? Two arrangements are considered the same if one is a translation and/or reflection of the other.

5.
a) An infinite line is drawn through each pair of vertices of a regular 13-gon. How many bounded regions is the plane divided into?

b) An infinite line is drawn through each pair of lattice points on the perimeter of a 4\times 4 square grid. (There are 16 such lattice points.) How many bounded regions is the plane divided into?

6. How many unordered sets \{A,B,C,D\} are there such that each of A,B,C,D is a subset of \{1,2,3,4,5\}, but none is a subset of another?

7. How many ways are there to tile a 4\times 4 grid with monominoes and L-shaped triominoes? An example is shown below.

8.
a) How many multigraphs are there with five labeled vertices such that each vertex has degree 4?

b) How many graphs are there with nine unlabeled vertices which admit an Eulerian cycle?

9. Up to congruence, how many right triangles are there which can be placed on a 40\times 40 grid in such a way that each vertex lies on a lattice point?

1 thought on “2025 and counting”

  1. James Stewart's avatarJames Stewart
    January 5, 2025 at 9:05 pm

    The sum of the elements in the multiplication table is $$(1+2+dots+9)(1+2+dots+9)=1cdot 1+1cdot 2+1cdot 3+dots+9cdot 9 = (1+2+dots+9)^2=45^2=2025.$$

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