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Today is my birthday! Please tell me fun maths facts

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@christianp Can’t really think of a *fun* maths fact off the top of my head, but happy birthday to you!!

@christianp You can draw a line segment by rolling a circle inside a circle of twice the radius. (shadertoy.com/view/3tl3RS) Also, may you have a great birthday.

@christianp Happy Birthday.. My current favourite is the Banach Fixed point theorem.. Which (i think!) says if you lay a map of your home on the floor of your home, one point of the map is exactly above the same point in your home.

Also the Curry-Howard correspondence.

@kat ahh, I set a puzzle for New Scientist using that theorem, but it was a bit too wordy so I think it's being saved for a Christmas issue or omnibus book or something

@christianp Students learning their multiplication tables often use the consecutive digits in the products \(12=3\times4\) and \(56=7\times 8\) as a memory aid. Base 10 is the only base that has any "multiplication facts" of the form \(d_1d_2 = d_3\times d_4\) with the d's consecutive digits.

Base 16 does admit a cute additive identity, \(12 = 3+4+5+6\).

@christianp happy birthday!

I read that:

2021 is the smallest natural number that can be expressed as the product of two consecutive primes.

2021 produces a palindrome when multiplied by its reverse, 1202
(= 2429242).

The 2021th digit of π, e and φ all equal 3.

@christianp Consider the following sequence of rational numbers: 7/3, 99/98, 13/49, 39/35, 36/91, 10/143, 49/13, 7/11, 1/2, 91/1.
Starting with n_0=10 let nᵢ₊₁=nᵢ*p/q, with p/q being the first fraction in above sequence, such that nᵢ*p/q is an integer. Then the exponents of the powers of 10 generated by this are successive primes.
(taken from esolangs.org/wiki/Fractran)

@christianp Glad you like that bit of esoteric computation!

@christianp The attached figure shows two pentagons inscribed in each other: each vertex of one pentagon lies on a line through a side of the other pentagon. You can't do this with quadrilaterals in the Euclidean plane, but you can in the complex projective plane: see en.wikipedia.org/wiki/M%C3%B6b

Happy birthday!

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