Facts and Factors for the Year 2024

/ ivasallay
Contents hide
1 A Countdown to 2024:
2 What Kind of Factors Will the Year 2024 Bring Us?
3 Powerful Facts About the Number 2024:
4 Today’s Puzzle and Some Other 2024-Themed Puzzles:
5 2024 in Pythagorean Triple Triangles
6 2024 in Pascal’s Triangle:
7 2024 Consecutive Number Sums
8 2024 Magic Sums
9 More About the Number 2024:
10 Share this:

A Countdown to 2024:

2024 Countdown

make science GIFs like this at MakeaGif
Here are some other countdowns:

What Kind of Factors Will the Year 2024 Bring Us?

Here’s a factor cake to celebrate 2024’s arrival:

And its factor pairs are outlined on this chart:

The sum of all the factors of a number (excluding itself) determines if a number is deficient, perfect, or abundant. Which of those describes 2024?

Powerful Facts About the Number 2024:

2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ = 2024, as illustrated below

2¹º + 10³ =2024.

2024 is the sum of eleven consecutive even square numbers:

2024 is the difference of two squares in FOUR different ways:

507² – 505² = 2024,
255² – 251² = 2024,
57² – 35² = 2024, and
45² – 1² = 2024.

One of those four equations brings us to…

Today’s Puzzle and Some Other 2024-Themed Puzzles:

Here are some other 2024-themed puzzles:

2024 in Pythagorean Triple Triangles

2024 is a leg in quite a few Pythagorean triple triangles. Here are a few:

I didn’t even include all the triangles listed on the left in the illustration because some of the points were too close together with the scale I used in Desmos.

I also didn’t include
2024² + 512070² = 512074 or
2024² + 128010² = 128026
because those triangles would have made the scale even worse.

There are more triangles, but I think this is a good enough representation.

Here’s a more complete list:

2024 is NOT the hypotenuse of any Pythagorean triple because none of its prime factors leave a remainder of 1 when divided by 4.

2024 in Pascal’s Triangle:

2024 is in the 24th row of Pascal’s triangle. Click on the image to see the numbers in the image better.

Since it is in the third column of that 24th row, 2024 is a Tetrahedral number. That means it is the sum of the first 22 triangular numbers. That fact can be illustrated as I have here in this vertically rotating Desmos 3D image. Go ahead and click on the image below to see this tetrahedron rotating more horizontally. Also, notice that I made the image with 2024 quarter-unit spheres.

As I stated before, the image is made from the first 22 triangular numbers stacked on top of each other. The sum of the first 22 triangular numbers is given below:

This tetrahedral number can also be expressed mathematically in this way:

This way:

The second way listed here:

Or this way:

2024 Consecutive Number Sums

2024 is the sum of consecutive counting numbers in three different ways:

2024 is the sum of consecutive odd numbers in four different ways: (It’s because 2024 is the difference of two squares in four different ways.)

2024 is the sum of consecutive even numbers:

8 consecutive even numbers:
246+248+250+252+254+256+258+260=2024.

11 consecutive even numbers:
174+176+178+180+182+184+186+188+190
+192+194=2024.

23 consecutive even numbers:
66+68+70+72+74+76+78+80+82+84+86+88+90+92
+94+96+98+100+102+104+106+108+110 = 2024.

2024 Magic Sums

2-0+2-4 = 0, so 2024 is divisible by 11. That means it is the magic sum of an 11 × 11 Magic Square. Here is one way that Magic Square can be completed. I followed the directions given in this post. You can see the 11 consecutive numbers listed above along the lower left to upper right diagonal.

If you would like to try completing the magic square yourself, here’s an Excel template that will automatically add the sums while you enter the numbers: 1766-1772 and 2024 Magic Squares

2024 is divisible by 8 but not by 16, so it is the Magic Sum of a 16 × 16 Magic Square. Here are two examples:

For this first one, I wrote the numbers from -1 to 14 across the top of the puzzle and continued in like manner. After the numbers were in place, I started flipping diagonals. How many diagonals did I flip?  34: The sixteen green diagonals, the sixteen blue diagonals, the pink diagonal, and the brown diagonal.

Here is how it looked when I finished:

For this second one, I wrote the numbers from -1 to 14 in the first 4 × 4 square, the numbers from 15 to 30 in the second 4 × 4 square, etc. Then I began flipping diagonals.

The Excel sheet, 1766-1772 and 2024 Magic Squares, also includes a template that will allow you to complete just a 4 × 4 magic square in the lower right corner for the numbers from -1 to 14. The whole 16 × 16 magic square will populate if you just complete that 4 × 4 magic square! But there’s a template if you want to do the whole thing from scratch as well.

More About the Number 2024:

Because 2024 is the sum of the 16 numbers from 119 to 134,
and 16 is even, it follows that
134²-133²+132²-131²+130²-129²+128²-127²+126²
-125²+124²-123²+122²-121²+120²-119²=2024.

Here is a way to make 2024 using only the digits 2, 0, 2, and 4:

(2+0+2+4)×
(2+0+2+4-(2+0)/2+4)×
(((2+0+2+4)×(-(2+0)/2+4))-(2+0+2)/4) = 2024.

The tan²(88.7266556386°) ≈ 2024, so it is the solution to this next problem:

 

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