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A wonderful friend who is a maths professor, is turning 80 next month. He's just published his 22nd math textbook! [Prof John Vince.... his books are about math for computer graphics] I would love to put some clever mathematical expression for "80" onto his cake. Could be an equation, a series, an integral, a summation.... anything mathematical really.

I got lots of inspiration from a question posed here 6 years ago with a similar request for a 50th birthday. My math is good enough to adulterate some of these to apply to 80, rather than 50....

$$e^{\ln{80}}$$ $$160\sin{\frac{\pi}{6}}$$ $$40\sum_{k=0}^\infty \frac{1}{2^k}$$ $$\frac{48s0}{\pi^2}\sum_{k\in \mathbb{N}}\frac{1}{k^2}$$

.... but i'm sure there must be some elegant/beautiful 80-specific equations out there, such as this one that someone suggested for 50:

$$\bbox[8pt,border:3px #FF69B4 solid]{\color{red}{\Large \lfloor e^\pi \rfloor + \lfloor \pi^e \rfloor + \lfloor \pi \rfloor + \lfloor e \rfloor = 50}} $$

Multiplying this by 8/5 doesn't feel very elegant to me!!

I got so much wonderful inspiration from the previous question that I shall make a birthday card with all the different ideas, but the most beautiful will make it onto the cake.

Thank you for any and all ideas!

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    $\begingroup$ fun fact : your friend's age is actually 50 in hexadecimal $\endgroup$ – Laassila souhayl Apr 18 at 14:23
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    $\begingroup$ "Clever", "interesting", and "beautiful" are not objective qualities. This isn't a puzzle, it's an open-ended, subjective, popularity contest. Voted to close. $\endgroup$ – bobble Apr 18 at 16:08
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    $\begingroup$ We also have $80 = (3+1)(3-1)(3+i)(3-i)$. $\endgroup$ – Torsten Schoeneberg Apr 18 at 16:09
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    $\begingroup$ Why was this question migrated? It feels like it would fit better on Math SE, ironically enough, as it's a open-ended math question with no pre-defined solution (unless there's a similar restriction on open-ended questions on Math SE, of course) $\endgroup$ – samm82 Apr 18 at 16:35
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    $\begingroup$ @samm82 It seems to be a unilateral decision of one of the moderators (Xander Henderson) $\endgroup$ – Ben Grossmann Apr 18 at 16:42
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Taking things in a different direction: $$ 80 = 8!!! $$ Where $n!!!$ denotes a triple factorial.

This is perhaps more comically presented as "I can't believe you're $8^{0!}!!!$"


The Wikipedia page has some interesting ones. For instance,

  • $80 = \varphi(1) + \varphi(2) + \cdots + \varphi(16)$
  • $80 = 2222_3$
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  • $\begingroup$ What's also cool about 2222 in base 3 is that it's just one short of 10000. $\endgroup$ – loopy walt Apr 18 at 16:44
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$\require{begingroup} \begingroup$ $\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}\def\Catalan{\mathsf{Catalan}}$

This is a simple one:

\begin{align} (\sqrt[\e]{(\pi+\pi+\pi)^{\e+\e+\e+\e}}/\pi/\pi/\pi-\pi)/\pi &=80 \end{align}

\begin{align} \left\lfloor \frac{\e^{\e\cdot\e}}{\e\cdot\e\cdot\e}\right\rfloor &=80 ,\\ \left\lceil\frac{\pi\cdot\pi^\pi}{\sqrt[\pi]{\pi}}\right\rceil &=80 ,\\ \left\lceil\sqrt[\gamma]{\frac{\e\cdot\e^{\e\cdot\e}}{\e^\e\cdot\e^\pi}}\right\rceil &=80 ,\\ \left\lfloor \sqrt{\e^{\textstyle\e\cdot\left(\pi^{\pi-\e}+\sqrt[\e]{\pi\cdot\frac{\pi}{\e}}\right) }} \right\rfloor &=80 . \end{align}

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If you're willing to use the golden ratio $\varphi$ and the Euler-Mascheroni constant $\gamma$, I was able to come up with $$\left\lfloor e^\varphi+\varphi^e+e^\pi+\pi+\pi e+e+\pi^e+\varphi^\pi+\pi^\varphi\right\rfloor+\lceil\gamma\rceil=80$$

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    $\begingroup$ There are plenty of ways to express $1$ besides $\lceil \gamma \rceil$, if one doesn't want to invoke the Euler-Mascheroni constant. For instance, $0!$ is a fun one. $\endgroup$ – Ben Grossmann Apr 18 at 14:39
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    $\begingroup$ If you’re adding 1 to a floored non-integer real number, why not just take the ceiling? $\endgroup$ – Lawrence Apr 18 at 16:41
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    $\begingroup$ @Lawrence I just like gamma and I suppose I like the symmetry of having a floor and ceiling operation. $\endgroup$ – K.defaoite Apr 18 at 16:45
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Some more suggestions:

$$80 = 2^6 + 4^2 = 4^2 + 4^3$$

$$80 = \lceil 22^{\sqrt2} \rceil$$ $$80 = (1+2+3+4)\cdot5+(6+7+8+9)$$

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    $\begingroup$ At last one without floor or ceiling! Especially the 123456789 version (I plan to remember it for my 80th (in 2028)) $\endgroup$ – NL_Derek Apr 18 at 21:24
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How about:

$$\lfloor e + \pi \rfloor \cdot \lfloor e \rfloor^{\lfloor e \rfloor^{\lfloor e \rfloor}}$$

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Well, not an answer, but rather an addition (i.e. a phrase which can be written on a birthday card for example) which is too large for a simple comment:

Now it's time to start from a new line!

Of course, this is a reference to the ancient (but still existing in some contexts) limit of 80 character per line. If your friend writes about computer graphics (and is 80 years old), he has likely seen and remembered those good old times when the programs were written on punched cards.

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Surprised no-one posted this one yet:

$$4e^\pi-4\pi = 80$$

(unless you make rounding errors.)

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Consider:

$\lceil{e^{\pi}-\pi}\rceil \lceil{\pi}\rceil = (20)(4) = 80$

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Here's one that doesn't involve integer rounding

$$\frac{\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(\pi n)^2}}{\displaystyle\sum_{m=1}^{\infty} \frac{1}{((2m-1)\pi)^6}} = 80$$

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