# The Aperiodvent Calendar, 2015

Everyone enjoys counting down to Christmas so much, that it seems to happen earlier and earlier each year. Well, sticking to the standard format of counting from 1st December down to 25th using a specially prepared calendar, we present the Aperiodical’s 2015 Advent Calendar, featuring behind each door not a small disappointing piece of chocolate, but a randomly chosen nugget of mathematical goodness for your enjoyment.

From YouTube videos to websites cataloguing number sequences, we’ve got a nice surprise for you each day. We’ll be adding each door as a post on the site, plus you can find them all collected together below, along with interesting number facts. Enjoy!

The value of Legendre’s Constant

Base of the binary number system, the coolest number system

Closest integer to both $\pi$ and $e$

Order of the Klein Four Group

The number Matt Parker considers to be the first ‘true’ prime number

The number of cards in Maths Bastard

A smoothed octagon, with 8 sides, is the worst shape for filling space in 2D

An exponential factorial, because it can be written $3^{2^1}$

$1+2+3+4$ and also $0!+1!+2!+3!$

Smallest positive integer with three syllables

Faces on a dodecahedron, vertices on an icosahedron and edges on a cube

Number of Archimedean Solids

Square pyramidal number – the sum of the first three squares $1+4+9$

The magic constant of a $3\times3$ magic square using the numbers 1 to 9

The only solution to $X = A^B = B^A, A,B \in \mathbb{Z}$

17-gon (heptadecagon) can be constructed using ruler & compass

The only number that equals twice the sum of its digits

Number of spaces in a magic hexagon

Famously equal to 9+10

Number of people needed in a room before the probability of a birthday coincidence is more than 50%

Every prime number squared is one more than a multiple of 24

Merry Christmas!

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