# Challenge Riddles and Puzzles: Extend the following to a valid equation

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#### fresh_42

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1. Extend the following to a valid equation, using only mathematical symbols!

Example: $1\; 2\; 3 \;=\; 1 \longrightarrow - (1 \cdot 2) + 3 = 1$. Solutions are of course not unique.

$9\;9\;9\;=\;6$
$8\;8\;8\;=\;6$
$7\;7\;7\;=\;6$
$6\;6\;6\;=\;6$
$5\;5\;5\;=\;6$
$4\;4\;4\;=\;6$
$3\;3\;3\;=\;6$
$2\;2\;2\;=\;6$
$1\;1\;1\;=\;6$
$0\;0\;0\;=\;6$

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#### DavidSnider

Gold Member
-(9 / sqrt(9)) + 9
8 - sqrt(sqrt(8+8))
7-(7/7)
(6-6) + 6
5+(5/5)
sqrt(4)+sqrt(4) + sqrt(4)
3 * 3 - 3
(2 * 2) + 2
(1 + 1 + 1)!
((0!) + (0!) + (0!))!

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#### lpetrich

For the three 2's, it ought to be (2 * 2) + 2. The others are correct.

#### fresh_42

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2. On a round playing field with radius $R$, two players place coins of the same radius $r < R$ without moving coins. The first who doesn't find a place to position his coin has lost. For which ratios $R/r$ is the game fair?

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#### fbs7

On a round playing field with radius $R$, two players place coins of the same radius $r < R$ without moving coins. The first who doesn't find a place to position his coin has lost. For which ratios $R/r$ is the game fair?
Wow, what an exquisite puzzle!!! It beats my little brain by a long, long distance - I look forward to reading the solution!!!

#### fresh_42

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Wow, what an exquisite puzzle!!! It beats my little brain by a long, long distance - I look forward to reading the solution!!!
You are able to find the solution!

#### Orodruin

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On a round playing field with radius $R$, two players place coins of the same radius $r < R$ without moving coins. The first who doesn't find a place to position his coin has lost. For which ratios $R/r$ is the game fair?
Trick question. It is never fair. The first player can always win by starting to play dead center and then playing to maintain a $C_2$ symmetry (i.e., playing diametrically opposed to the other player). By doing so the first player can always play.

#### SSequence

Nice ...... Also, I think, strictly speaking, one also has to show that the game always terminates. It is somewhat trivial though (pigeonhole principle etc.).

But I think a good variation of the question might be that we aren't allowed to play the center position as first move. Or is that easy too?

#### fbs7

Trick question.
Brilliantly thought!!!! Holy Choo-Choo - many times the simplest arguments are the most eloquent ones!!! People here are really exceptional - it's like waiting for a lightning, it comes sudden and when it comes it amazes you!

Even the nature of the question befuddled me - how can a non-probabilistic game that can't draw be fair? If a game can't draw (as the one above), then there must be at least one path starting of the 1st players 1st turn that ends in his win whatever the 2nd does, or there must be multiple paths for the 2nd player to always win whatever the 1st player does. So I couldn't reconcile the ideas of "deterministic", "non-drawing" and "fair".

Anyone has an example of a deterministic game that can't draw and is proven fair?

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#### fresh_42

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Anyone has an example of a deterministic game that can't draw and is proven fair?
No, but you assumed finiteness in your argument. If it is infinite at prior, and I think it has to be, i.e. of arbitrary finite length to be exact, then the argument for the existence of a strategy isn't obvious anymore. And another hidden assumption was, that we only consider two player games. You might say of course, I say: needs to be stated.

#### fresh_42

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2018 Award
3. One line is wrong.

\begin{align*}
f(87956)&=4\\
f(82658)&=5\\
f(11111)&=0\\
f(46169)&=4\\
f(84217)&=3\\
f(57352)&=1\\
f(18848)&=7\\
f(27956)&=\;?
\end{align*}

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#### fbs7

No, but you assumed finiteness in your argument. If it is infinite at prior, and I think it has to be, i.e. of arbitrary finite length to be exact, then the argument for the existence of a strategy isn't obvious anymore. And another hidden assumption was, that we only consider two player games. You might say of course, I say: needs to be stated.
Ah! I see!! Thank you!! Fbs7 = learned something today!!!

#### fresh_42

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Nobody with an idea for number 3 in post 11? I know it's one of the kind preschoolers are better than mathematicians

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Nobody with an idea for number 3 in post 11? I know it's one of the kind preschoolers are better than mathematicians
@fresh_42 I just looked at it for over 1/2 hour. Apparently the obvious must elude me.

#### fresh_42

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@fresh_42 I just looked at it for over 1/2 hour. Apparently the obvious must elude me.
I found the clue as I read "preschoolers".

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I found the clue as I read "preschoolers".
The preschoolers must be super-smart. I still don't see anything that stands out... I trust the function isn't random...

#### fresh_42

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No, not random. However, formal languages might be of greater help than calculus. The puzzle was posed without the function $f$. I have added it in order to avoid endless discussions about sloppiness.

And don't make any nonsense if you will have figured it out. I seriously warn of

@WWGD should be quick at it.

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Here is a joke I once heard, but I don't think it will help in solving this: "Why is 6 afraid of 7?" Answer: "Because 7 comes after 6, and 7 ate 9." LOL

#### DavidSnider

Gold Member
Is it as simple as "One line is wrong", meaning the line with ones is wrong?

#### fresh_42

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2018 Award
Is it as simple as "One line is wrong", meaning the line with ones is wrong?
The question has two parts: which result has the last line and which line is the wrong one. It is simple, but there is a logical explanation.

"Riddles and Puzzles: Extend the following to a valid equation"

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