Riddles and Puzzles: Extend the following to a valid equation

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In summary, the task is to determine the correct labeling of the urns (WW, WB, BB) by drawing balls from each urn without looking and using the information that the urn labels have been switched.
  • #1
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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  • #2
-(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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  • #3
For the three 2's, it ought to be (2 * 2) + 2. The others are correct.
 
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  • #4
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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  • #5
fresh_42 said:
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!
 
  • #6
fbs7 said:
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!
 
  • #7
fresh_42 said:
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.
 
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  • #8
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?
 
  • #9
Orodruin said:
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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  • #10
fbs7 said:
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.
 
  • #11
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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  • #12
fresh_42 said:
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!
 
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  • #13
Nobody with an idea for number 3 in post 11? I know it's one of the kind preschoolers are better than mathematicians :wink:
 
  • #14
fresh_42 said:
Nobody with an idea for number 3 in post 11? I know it's one of the kind preschoolers are better than mathematicians :wink:
@fresh_42 I just looked at it for over 1/2 hour. Apparently the obvious must elude me.
 
  • #15
Charles Link said:
@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".
 
  • #16
fresh_42 said:
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...
 
  • #17
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 :headbang:

@WWGD should be quick at it.
 
  • #18
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
 
  • #19
Is it as simple as "One line is wrong", meaning the line with ones is wrong?
 
  • #20
DavidSnider said:
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.
 
  • #21
Got it after the preschool hint. The wrong line:
The line f(57352)=1 is wrong. f(57352)=0
 
  • #23
Charles Link said:
I tried a google: https://en.wikipedia.org/wiki/Preschool Am I dense? I still don't have clue...
Maybe it is easier mathematically by the following claim (but I haven't checked the details, I only think it's true):

##f## is a homomorphism from the Kleene star over ##\{\,0,1,2,3,4,5,6,7,8,9\,\}## into the half group ##(\mathbb{N}_0,+)##.
 
  • #24
I give up. I wonder if the answer may lie outside my realm of experience.
 
  • #25
Since @mfb has solved number 3, here comes the next one:

4. In one of three urns there are two white balls, in another a white and a black ball, and in the third two black balls. The urns are labeled: one sign says WW, one WB and the third BB. But someone has switched the signs so that none of them specify the contents of the individual urns anymore.

One may take a ball from one of the urns, one after the other (without looking into the urn) until it is clear which urns contain which of the three ball pairs. How many balls do you have to take out at least to reach this goal?
 
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  • #26
Charles Link said:
I give up. I wonder if the answer may lie outside my realm of experience.
Forget numbers. Count circles.
f(0)=1, f(1)=0, ..., f(8)=2.
 
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  • #27
mfb said:
Forget numbers. Count [topological] circles.
f(0)=1, f(1)=0, ..., f(8)=2.
 
  • #28
mfb said:
Forget numbers. Count circles.
f(0)=1, f(1)=0, ..., f(8)=2.
It's no wonder it eluded me=I don't feel so bad. It's quite clever, but somewhat un-mathematical.:smile:
 
  • #29
Charles Link said:
It's no wonder it eluded me=I don't feel so bad. It's quite clever, but somewhat un-mathematical.:smile:
No it is not, just unconventional. It counts the genuses of concatenated topological geometric objects. On the left we have words of a formal language without grammar, and every letter has a weight. I wonder if we could make a baric algebra out of it.
 
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  • #30
mfb said:
Got it after the preschool hint.

Holly Choo-Choo! You are a genius!
 
  • #31
fresh_42 said:
Since @mfb has solved number 3, here comes the next one:

4. In one of three urns there are two white balls, in another a white and a black ball, and in the third two black balls. The urns are labeled: one sign says WW, one WB and the third BB. But someone has switched the signs so that none of them specify the contents of the individual urns anymore.

One may take a ball from one of the urns, one after the other (without looking into the urn) until it is clear which urns contain which of the three ball pairs. How many balls do you have to take out at least to reach this goal?

One ball.

Let small letters denote true content and capital letters labelled content.

The urn WB is not wb, so it must be bb or ww. Taking a ball out of it will therefore tell you which it is. If we get white out of WB, then it is ww. We are now left with the urns WW and BB. We know that those labels are wrong and that one of those urns is bb != BB. Therefore bb = WW and wb = BB. The corresponding argument goes if we get black out of WB, but exchanging the roles of the blacks and whites.

This can also be seen as follows: There are two possibilities for labels:
{WW = wb, WB = bb, BB = ww} or {WW = bb, WB = ww, BB = wb}
Clearly they cannot be discriminated without taking a ball out. If we take a ball out of WB, it will distinguish between the two possible cases.
 
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  • #32
5. Let's have some fun. We will perform a 4 step manager test. I will post 4 questions step by step, i.e. question 2 if the first will be answered correctly etc. It is an old joke, but here I will add a "lessons learnt" comment after each question.

a. How do you get a giraffe into a fridge?
 
  • #33
Open the fridge, put the giraffe in, close the fridge.
 
  • #34
mfb said:
Open the fridge, put the giraffe in, close the fridge.
This question investigated whether you tend to find much too complicated solutions of very easy problems.

5.b. How do you get an elephant into the fridge? (@mfb please pause on this one.)
 
  • #35
fresh_42 said:
This question investigated whether you tend to find much too complicated solutions of very easy problems.

5.b. How do you get an elephant into the fridge? (@mfb please pause on this one.)
Open the fridge, take the giraffe out, put the elephant in, close the fridge.

[Zoning into the apparently expected level of abstraction]
 

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