# Ways to write “50” [closed]

A really good friend of mine is an elementary school math teacher. He is turning 50, and we want to put a mathematical expression that equals 50 on his birthday cake but goes beyond the typical "order of operations" problems. Some simple examples are $$e^{\ln{50}}$$ $$100\sin{\frac{\pi}{6}}$$ $$25\sum_{k=0}^\infty \frac{1}{2^k}$$ $$\frac{300}{\pi^2}\sum_{k\in \mathbb{N}}\frac{1}{k^2}$$

What are some other creative ways I can top his cake?

I should note that he is an elementary school teacher. Now he LOVES math, and I can certainly show him a lot of expressions. I don't want them so difficult that it takes a masters degree to solve, but they should certainly be interesting enough to cause him to be wowed. Elementary functions are good, summations are also good, integrals can be explained, so this is the type of expression I'm looking for...

EDIT:: I would make a note that we are talking about a cake here, so use your judgement from here on out. Think of a normal rectangular cake and how big it is. Hence, long strings of numbers, complex integrals, and long summations are not going to work. I appreciate the answers but I need more compact expressions.

## closed as too broad by user642796Dec 11 '14 at 12:37

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• See this question for a different take on $50$ math.stackexchange.com/questions/225518/… – Mark Bennet Dec 10 '14 at 20:14
• $e^{2\pi i}+7^2,$ $\frac{1}{\pi}\int_{-10}^{10}\sqrt{100-x^2}dx.$ – mfl Dec 10 '14 at 20:15
• $25\nabla\cdot\nabla|x|^2$ – BaronVT Dec 10 '14 at 20:17
• What a wonderful idea. This is certainly something I'd love to get for my birthday. :) For my mother's 75th I had cake decorated with an epicycloid with 75 cusps. Not that she's into math, but she sees the beauty in geometrical shapes. – user197789 Dec 10 '14 at 20:39
• I voted to reopen. Perhaps this should be Community Wiki. It is tagged as big list - to say that there are lots of possible answers rather validates the list - the number of interesting answers is probably rather smaller, and voting will distinguish the cases. Given that there was a debate about my question on the number 50, I think there should be a proper debate about this one. Maybe there should be a "birthday fun facts" tag related to recreational mathematics. Such fun facts can have serious content, and some of us enjoy them. – Mark Bennet Dec 10 '14 at 21:35

## 20 Answers

We can use only two famous numbers in mathematics, $\large\pi$ and $\large e$, to produce number $50$.

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

Click the box to see Wolfram Alpha's output to confirm the result

$$\frac{2^{\frac{(2\cdot2)!}{2+2}}+22+2^{2^2}-\sqrt 2^2}{2^{2-\frac{2}{2}}}$$

• That is an awesome one. – Eleven-Eleven Dec 10 '14 at 20:50
• I think it is 2 much. – RghtHndSd Dec 10 '14 at 21:39
• @RghtHndSd 2 good 2 be 2rue? – Conor O'Brien Dec 10 '14 at 23:19
• This answer currently has 22 upvotes. Don't ever let that change. – teadawg1337 Dec 11 '14 at 3:05
• @teadawg1337 Don't worry. As of right now, the counter's at $22 + \lim_{n\to \infty}(\varphi(2)+\dfrac {2\pi i}{n})^n$. – user137731 Dec 11 '14 at 4:13

$$50 = 2\cdot(2\varphi - 1)^4$$ where $\varphi$ is the Golden Ratio.

$$50 = \sum_{i=0}^{+\infty} (0.98)^i$$

(Geometric series)

$$50 = \left(\left(\frac{5^5-5}{5}+5^0\right)\cdot\left(5-5^0\right)\right)^{0.5}$$

$$50 = 0.5 \cdot (5+5)^{\frac{5^0}{0.5}}$$

$$50 = 5\cdot\left(\frac{5}{0.5}+5^0\right)-5$$

(Using only the digits "5" and "0")

$$50 = \frac{3^{3!}-3^{3-3^0}}{3^{3-3^0}}-30$$

(Using only the digits "3" and "0")

$$50 = \frac{(10i)^2\log(i^i)}{\pi}$$

(Using imaginary unit $i$)

$$50 = 3 + 47$$ $$50 = 7 + 43$$ $$50 = 13 + 37$$ $$50 = 19 + 31$$

(As sum of two prime numbers)

$$50 = (7+11)\frac{11}{11-7}+\frac{7+11}{11-7}-11+7$$

(Using only the two next prime numbers of $5$)

$$50 = 7+3+ (7-3)\cdot(7+3)$$

(Using only the previous and next prime numbers of $5$)

$$50 = 3\cdot(2^3+3^2)-(2\cdot 3)^{3-2}+3+2$$

(Using only the two previous prime numbers of $5$)

$$50 = (1^6-2^5+3^4-4^3+5^2-6^1)^2\cdot(4^1 - 3^2 + 2^3 - 1^4)$$ $$50 = 3 - (1^9-2^8+3^7-4^6+5^5-6^4+7^3-8^2+9^1)$$ (Using bases/powers in reverse order)

• I love the bases/powers in reverse order and i think he will get a kick out of that. – Eleven-Eleven Dec 11 '14 at 3:15
• I particularly like the ones using only the digits 5 and 0. – David Richerby Dec 11 '14 at 11:27

No fancy math here, but if you want to emphasize how old your friend is getting, nothing says it better than implying he's halfway to the century mark:

$$100\over2$$

• Or 25 x 2, if you want to say: "hey, your body is just like 25 and your mind is better than that, twice." – Ooker Dec 11 '14 at 11:40
• @Ooker Or vice versa :-) – Venus Dec 11 '14 at 13:34
• @Ooker & Venus, When I turned 50, my (older) brother sent me a birthday email with instructions to "party like you're a pair of 25-year-olds." Or maybe it was to party with a pair of 25-year-olds.... – Barry Cipra Dec 11 '14 at 14:14

Quoting Wikipedia

Fifty is the smallest number that is the sum of two non-zero square numbers in two distinct ways: $50 = 1^2 + 7^2 = 5^2 + 5^2$.

So you could write something like \begin{align} 50 = \min_{n\in \mathbb{N}}\{n =p_i^2+p_j^2=p_k^2+p_l^2 \quad | \quad p_i,p_j,p_k,p_l\in\mathbb{N} \quad \wedge\quad p_k \not =p_i \not = p_l \} \end{align}

I like it, because it doesn't involve some sort of scaling and is not obvious (at least not for me).

• He's going to need a long cake for this!! – Phrancis Dec 10 '14 at 20:27
• I have a proof of this theorem, but there is not enough space on this cake. – k1next Dec 10 '14 at 20:29
• Why not just write $1^2 + 7^2 = 5^2 + 5^2$. When people ask "what the heck?" you can give them a math lesson. – RghtHndSd Dec 10 '14 at 21:00

We also have \begin{align*} 50 &= 11+12+13+14 \\ &= (8+4)+(8-4)+(8\cdot 4)+(8/4) \\ &= 4^2 + 4^2 + 3^2 + 3^2\\ &= 6^2 + 3^2 + 2^2 + 1^2\\ &= (7+i)(7-i) \\ &= (10-\color{red}{5})(10-\color{red}{0})\\ &= 10(\color{red}{5}+\color{red}{0})\\ &= \sqrt{30^2+40^2}\\ &= \sqrt{170^2+310^2}\\ &= \sqrt{146^2+322^2}\\ &= \sqrt{50^2+350^2}\\ \end{align*} Finally $50= 2 + 4 + 8 + 12 + 24$ and $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{12} + \frac{1}{24} = 1$.

• This is a great great list. – Eleven-Eleven Dec 10 '14 at 22:45

Whenever one of my friends has a birthday, I find out how old they get and then I visit their number on Wikipedia

For your friend, I would write something like this:

50 is the smallest number that is the sum of two non-zero square numbers in two distinct ways: $1^2 + 7^2$ and $5^2 + 5^2$. It is also the sum of three squares, $3^2 + 4^2 + 5^2$. It is also a Harshad number and a nontotient and a noncototient. 50 is the aliquot sum of 40 and 94. 50 is also the atomic number of tin and fifth magic number in nuclear physics.

While many of the things on Wikipedia are not Mathematical expressions, and some of it is way too long to write on a cake, I am certain that this will brighten his day if you tell him this stuff!

As for a mathematical expression, I'd go with either $3^2 + 4^2 + 5^2$ because I find it simple but elegant, or fill the cake with stuff like "Harshad number, 5th magic number in nuclear physics, nontotient", etc. and see if he can figure out how old he is.

I hope he doesn't calculate it by adding all numbers ;)

\begin{align} 50 = \sum_{k=0}^{100} (-1)^k k \end{align}

And a last one, involving only $4$s and $9$s:

\begin{align} 4^9 \mod 49 + \sqrt{49} \end{align}

binary code : 50 is given by 110010

$\displaystyle \left(4!-5!-4\right)\int_0^1\frac{\operatorname{Li}_{-1}(t) }{t}\mathbb{d}t$

Although this next one may be too big to fit on a cake, it's certainly beautiful:

$\displaystyle \frac{\pi\operatorname{Li}_1(\frac12)(\Gamma(5)+1)}{\left[\Gamma(\frac32)\right]^2\left(\frac{\Gamma'(1)}{\Gamma(1)}-\frac{\Gamma'({\frac12})}{\Gamma({\frac12})}\right)}$

Here's another one that may or may not fit on a cake:

$\displaystyle\frac{\displaystyle\left(1+2\sqrt{4!+5!}\right)\left[\prod_{n=1}^3\Gamma\left(\frac{n}3\right)\right]^2}{\displaystyle{e}^{2\operatorname{Li}_1(\frac12)}\operatorname{Li}_2\left(\frac{\Gamma'(1)}{\Psi(1)}\right)}$

• Ironically, when I am reading this it says "answered 50 seconds ago" – Simon Forsberg Dec 10 '14 at 20:21
• To whoever just downvoted, $\displaystyle \left(\operatorname{Li}_n(1) =\zeta(n)\right) \land \left(\zeta(0)=-\frac{1}{2}\right)\Rightarrow\left(\operatorname{Li}_0(1) =-\frac{1}{2}\right)$. See Equation 86 here for proof that $\displaystyle \zeta(0)=-\frac{1}{2}$ – teadawg1337 Dec 10 '14 at 20:56
• Someone is downvoting everything in here. Don't take it personally. – Cameron Williams Dec 10 '14 at 21:08
• what's $Li$? I've never seen that notation. – Oria Gruber Dec 11 '14 at 0:41
• This looks like stuff Ramanujan would have written. – MackTuesday Dec 11 '14 at 2:05

I went to visit him while he was lying ill at the hospital.I had come in taxi cab number $50$ and remarked that it was a rather dull number. "No" he replied, "it is a very interesting number. It's the smallest number expressible as the product of $25$ and $2$ in two different ways." https://mathoverflow.net/a/2197/15296

• Hilariously, 50 is the smallest number expressible as the sum of two squares in two different ways, as noted in another answer, which is where I thought you were going with this at first! – aes Dec 11 '14 at 6:37

\begin{align} &50 = \frac{5}{24} \zeta(-7)\\ &50 = \frac{ 1600 \sqrt{2} }{3 \pi ^3}\int_0^{\infty } \frac{x^2 \log ^2(x)}{x^4+1} \, dx \end{align}

• Well, OK, but then you can take any real-valued expression, multiply it by the appropriate number so that it gives 50... I mean it does answer the question but it is not as elegant as an expression which would be specific to 50. (this comment also applies to other answers) – anderstood Dec 10 '14 at 22:03

$$-\frac{12!! - 3^{10}}{705 \text{ mod } 101}-\int_0^3 4x^3\;dx = 50$$

Or, if the double factorial is too weird:

$$\frac{9! - 2^{15}}{10000000_2}-\sqrt{243}^2 = 50$$

# The Complex Answers

I'd suggest $4.471527458208!=50$, but this isn't readily solvable. You suggested summations, and I thought "Hey! Why not nested summations?" The product of my "why not" statement: $$\sum _{n=0}^4\sum _{i=n}^7n=50$$

If you're okay with floor functions: $$\left\lfloor\prod _{n=1}^5\frac{\pi +e}{7}n\right\rfloor=50$$

A slightly more complicated one: $$-1\left(\sum _{n=-3}^{11}-n\right)-10=50$$

If you want to complicate that (for a bit of fun), try, using Euler's identity for $-1$ and $n-2n$ for $-n$: $$e^{i\pi}\left(\sum _{n=3e^{i\pi}}^{11}n-2n\right)-10=50$$

# The Somewhat Easy Answers

Using only 5's and 0's: $$5.5\frac{505}{5}-5.55=50$$ Something kind of neat, using a pattern 1...7,1: $$1\cdot 2+3\cdot 4+5\cdot 6+7-1=50$$ Where $x_n$ denotes $x$ in radix $n$: $$302_4=50$$ $$200_5=50$$ $$32_{16}=50$$ $$62_8=50$$

# Set Theory

If $\alpha=\#A$ states that $\alpha$ is the cardinal of $A$, then: $$\#\{x\in\mathbb{N}:5<x\leq55\}=50$$

Prime factorisation

Simple and elegant: $50=2\times5^2$

Compute the smallest positive integer $n$ such that the first two digits of $n^2$ is $\frac{n}{2}$.

How about this

$$50=12+18+20$$

Reference on this page

Now, this would have your teacher scratch his head a little bit initially, but worth a shot! :)

Some solutions: $$\dfrac{50(-1)^{-i}}{i^{i^{2}}}=50$$ $$50=\sum_{n=1}^{10}(2n-1) \mod 10$$ $$50=1212_3$$ $$\left(4+\frac{4}{4}\right)\dfrac{\binom{4}{4-\frac{4}{4}}}{0.4}=50$$

You could go Roman with $$\Huge{L}$$

Or with $$\begin{array}{c} \\ \\ \end{array}$$

$$\begin{array}{ccccccccccccc} \Huge{X}&&&&&&&&&&&&\Huge{X}\\ \\ \\ \\ &&&&&&\Huge{X}\\ \\ \\ \\ \Huge{X}&&&&&&&&&&&&\Huge{X} \end{array}\\$$

$$50 = \int_{\ln 1}^{3!+2^2}\frac{\Gamma(\frac{1}{2})^4}{6\sum_{1}^{\infty}\frac{1}{n^2}}\sqrt{3^2+4^2}dx$$

can make it more complicated but it wont fit on a cake :)

• Well, but the integrand is constant in $x$ (it's just $5$), so it's not very interesting. – wchargin Dec 11 '14 at 1:55
• Where are you going to get the candles? – user532449 Feb 21 '18 at 3:02