Well, 2024 is almost upon us and so it's time to look at some of the mathematical properties of that number. First and foremost is its factorisation which is:$$2024 = 2^3 \times 11 \times 23$$It can be noted that this factorisation involves only the digits 1, 2 and 3. The final day of 2023 can be written in MM-DD-YY format as 12-31-23 or 123123 which also contains only the digits 1, 2 and 3.
FIRST FUN FACT
The first entry in the OEIS is for A000292:
A000292 | Tetrahedral (or triangular pyramidal) numbers:$$\text{a}(n) = \text{C}(n+2,3) = \frac{n \times (n+1) \times (n+2)}{6}$$ |
Figure 1 illustrates the triangular pyramidal numbers as a sum of triangular numbers stacked upon each other. In the case of 2024, \(n=22\) and this number represents the number of balls in the triangular pyramid in which each edge contains 22 balls. The sequence progresses as follows:
0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180
Figure 1: source |
SECOND FUN FACT
The tetrahedron is one of the Platonic Solids and therefore a shape of great significance.
However, 2024 is also connected with the dodecahedron because it is a member of OEIS A006566 with \(n=8\):
A006566 | Dodecahedral numbers: $$ \text{a}(n) = \text{C}(3n,3) =\frac{n \times (3n - 1) \times (3n - 2}{2} $$ |
The initial members of the sequence are:
0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, 5456, 7140, 9139, 11480, 14190, 17296, 20825, 24804, 29260, 34220, 39711
So 2024 represents the number of balls in the triangular pyramid in which each edge contains 8 balls. The following video shows how to construct a dodecahedron from nanodots:
The connection between the tetrahedron and the dodecahedron is visible in the GIF below:
Dodecahedron with five tetrahedra inside (source) |
THIRD FUN FACT
The next sequence for 2024 listed in the OEIS is A003242:
A0032425 | Number of compositions of \(n\) such that no two adjacent parts are equal (Carlitz compositions). |
In the case of 2024, \(n=15\) and here a few examples of the 2024 possible Carlitz compositions (permalink):
- 1, 2, 1, 2, 1, 2, 1, 2, 1, 2
- 1, 4, 2, 1, 4, 3
- 2, 1, 6, 1, 2, 1, 2
- 3, 2, 1, 4, 5
- 6, 2, 1, 2, 3, 1
FOURTH FUN FACT
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