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.
  • #36
jbriggs444 said:
Open the fridge, take the giraffe out, put the elephant in, close the fridge.
This question investigated whether you are aware of the consequences of your doings.

5.c. The Lion King is holding up his yearly conference of all animals. However, one is missing. Which one?
(this time @jbriggs444 please pause)
 
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  • #37
fresh_42 said:
5.c. The Lion King is holding up his yearly conference of all animals. However, one is missing. Which one?
The elephant. He is in the fridge.
 
  • #38
Orodruin said:
The elephant. He is in the fridge.
This question checked your memory.

5.d. You have to cross a river where dozens of crocodiles live. How do you resolve this situation?
(@Orodruin to pause).
 
  • #39
fresh_42 said:
This question checked your memory.

5.d. You have to cross a river where dozens of crocodiles live. How do you resolve this situation?
(@Orodruin to pause).
Oooh oooh, can I go again?
 
  • #40
jbriggs444 said:
Oooh oooh, can I go again?
Yes, time to finish this one.
 
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  • #41
fresh_42 said:
Yes, time to finish this one.
Use the bridge.
 
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  • #42
jbriggs444 said:
Use the bridge.
Nope. No bridge anywhere.
 
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  • #43
fresh_42 said:
Nope.
I know! I know! o0)
 
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  • #44
Oh silly me. The crocs are at the annual get together.
 
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  • #45
jbriggs444 said:
Oh silly me. The crocs are at the annual get together.
Yes, you swim.

This question was about whether and how fast you learn from mistakes. (Although we had none.)

According to a study by Andersen Consulting, around 90% of all tested executives worldwide have answered all questions incorrectly.

On the other hand, several preschool children had correct answers.

Andersen claims that this clearly disproves the thesis that executives have the mental faculties of a four-year-old.
 
  • #46
6. How many years of life did somebody complete on the first day of the year 30 AD, if he was born on the last day of the year 20 BC?
 
  • #47
More of a history question - there was no year 0. The last day of 1 BC he was 19 years old, the next day was the first day of 1 AD. The last day of 1 AD he was 20 years old, the last day of 29 AD he was 49 years old -> 49.You find a historic letter dated 5 BC talking about an event that happened 20 BC. What is wrong?
 
  • #48
mfb said:
More of a history question - there was no year 0. The last day of 1 BC he was 19 years old, the next day was the first day of 1 AD. The last day of 1 AD he was 20 years old, the last day of 29 AD he was 49 years old -> 49.
Correct analysis, wrong count.
You find a historic letter dated 5 BC talking about an event that happened 20 BC. What is wrong?
The five is wrong. The description of the twenty is not specific enough to be wrong, since it could also mean fifteen years prior to the letter, i.e. relates the dating to the impossible gauge.
 
  • #49
fresh_42 said:
Correct analysis, wrong count.
Oops, 48 of course. 20+28 is 48, not 49.
fresh_42 said:
The five is wrong.
Right. You can make the same puzzle with a letter e.g. 20 AD, but then you need to figure out when the AD counting was introduced (long after that), so the puzzle is easier with BC.
 
  • #50
mfb said:
You find a historic letter dated 5 BC talking about an event that happened 20 BC. What is wrong?
That dating system did not exist at the time, so that document could not have referred to such dates. So such dates would be a present-day interpretation of whatever dates that it used.
 
  • #51
7. Jennifer likes to drink black tea. One day, a friend tells her that tea tastes very good with a few drops of freshly squeezed lemon. Jennifer decides to test it the next morning. As the water starts to boil, she squeezes out a lemon and puts a teabag in the cup.

Jennifer has the following information as a math-ace in her class:
- The cup has a capacity of 10 cl.
- The tea bag has a volume of 1 cl.
- She has 1 cl of lemon, which she wants to give in the tea (temperature: 20 ° C).
- She has 8 cl of hot water (temperature: 100 ° C).
- She knows she needs exactly 5 minutes to make her school sandwiches.

Jennifer now faces the following question:

When does my tea get cold faster? If I put the lemon in the cup right away, or if I do that after I made the school sandwiches?
 
  • #52
Come on, folks, nobody? This is a physics website!
 
  • #53
fresh_42 said:
Come on, folks, nobody? This is a physics website!
It's a mathematics forum in a physics website. There is no mathematical solution because the problem is not completely specified.

Bringing physics into the situation... Primary heat loss in 100 degree hot water is likely to be evaporation. Early mixing to bring that temperature down is going to be a win.
 
  • #54
jbriggs444 said:
Early mixing to bring that temperature down is going to be a win.
Do you mean a win in the sense that early mixing results in a cooler tea?
 
  • #55
fresh_42 said:
Do you mean a win in the sense that early mixing results in a cooler tea?
Rather, early mixing would lead to a warmer tea since the net heat loss rate is lowered.

I was assuming that the goal was warmer tea, not colder. But perhaps the opposite is desired.
 
  • #56
fresh_42 said:
When does my tea get cold faster?
jbriggs444 said:
Rather, early mixing would lead to a warmer tea since the net heat loss rate is lowered.
Correct. Early mixing lowers the temperature difference of tea and air, so the tea will not cool down as fast as if it was when boiling. So a late mixing is cooler.
 
  • #57
8. This sequence is arranged according to which rule?
$$
8\quad 5\quad 4 \quad 9 \quad 1\quad 7 \quad 6 \quad 3 \quad 2 \quad 0
$$
 
  • #58
fresh_42 said:
Correct. Early mixing lowers the temperature difference of tea and air, so the tea will not cool down as fast as if it was when boiling. So a late mixing is cooler.
I think I agree with @jbriggs444 that this is not really a math problem. As specified, it requires a non-zero amount of physical modelling to have a solution.

fresh_42 said:
8. This sequence is arranged according to which rule?
$$
8\quad 5\quad 4 \quad 9 \quad 1\quad 7 \quad 6 \quad 3 \quad 2 \quad 0
$$
We cannot really say. There could be many valid rules that would produce that exact sequence. Which one of those you used is not possible to determine.
 
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  • #59
Orodruin said:
We cannot really say. There could be many valid rules that would produce that exact sequence. Which one of those you used is not possible to determine.
Sure. So find one. Of course there is an easy solution, but maybe we get some funny complicated stuff here. And please do not use ##\prod_k (x-k)^{n_k}=0## or similar trivial but high order solutions.
 
  • #60
fresh_42 said:
8. This sequence is arranged according to which rule?
$$
8\quad 5\quad 4 \quad 9 \quad 1\quad 7 \quad 6 \quad 3 \quad 2 \quad 0
$$
The English names of these numbers in alphabetical order:

eight, five, four, nine, one, seven, six, three, two, zero
 
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  • #61
9. We are looking for a pair of numbers. Mr. P knows the product of them and Mr. S their sum. The numbers are one of the pairs: ##(3,5)\, , \,(8,11)\, , \,(2,7)\, , \,(4,13)##. We heard the following dialogue of the two:
  • Mr. P: I don't know the numbers.
  • Mr. S: I don't know them either, but I knew you couldn't know them.
  • Mr. P: Then I know the two numbers by now.
  • Mr. S: Me, too.
 
  • #62
Orodruin said:
We cannot really say. There could be many valid rules that would produce that exact sequence. Which one of those you used is not possible to determine.
As I was searching for new puzzles I stumbled upon one which was quite easy to solve, i.e. it took me only a few seconds and was too easy to qualify for here. It was in a newspaper and they titled: Who can solve this has an IQ of 150.

My first reaction was: never ever. Now I think, what if they are right? 100 is said to be the average. What does this tell us about the state of our societies?
 
  • #63
These newspapers don’t want to tell their readers they are stupid - not even the stupid readers. Some readers will try and give up. If you tell them “everyone with average intelligence should be able to solve this” you might lose some readers. If you tell them “you need [high number] IQ then they don’t feel stupid.
 
  • #64
fresh_42 said:
9. We are looking for a pair of numbers. Mr. P knows the product of them and Mr. S their sum. The numbers are one of the pairs: ##(3,5)\, , \,(8,11)\, , \,(2,7)\, , \,(4,13)##. We heard the following dialogue of the two:
  • Mr. P: I don't know the numbers.
  • Mr. S: I don't know them either, but I knew you couldn't know them.
  • Mr. P: Then I know the two numbers by now.
  • Mr. S: Me, too.
Some big problems are that the sums and products both have ambiguous decompositions, and also that none of these number pairs' sums and products overlap -- they are all distinct.
 
  • #65
I assume P and S don't know that these four are the only options, otherwise P should know the numbers (the products are 15, 88, 14, 52, all unique).

P can know the numbers if and only if the product P is a prime, otherwise (1,P) and (n,P/n) are options for a non-trivial divisor n. None of the products is a prime, so the first line doesn't help us. Let's look at S' reply:
If the sum is 8 then the product could be 7 (1,7) and P could know the numbers. S knows this cannot be the case, therefore the sum cannot be 8 and the numbers can't be (3,5).
If the sum is 19 then the numbers are (n,19-n) and n(19-n) is not a prime for any n (for n=1 we get 18). Similarly for 9 and 17 as sum. In all three cases S knows P cannot know the numbers. In fact p+1 for some prime p is the only sum where the product might be unique: For all other numbers S knows P cannot know the numbers.

Let's go back to P.
What could the numbers be if the product was 88? (1,88), (2,44), (4,22), (8,11) are all options, leading to sums 89, 46, 26, 19. None of these is one larger than a prime. If the product is 88 then P knows from the beginning on that S knows that they cannot determine the numbers.
What could the numbers be if the product was 14? (1,14) and (2,7) are the only options, leading to sums 15 and 9. Again none of these is one larger than a prime.
What could the numbers be if the product was 52? We have (1,52), (2,26), (4,13) with sums 53,28,17. Same here.
All three options looks possible. I don't see how P could continue to rule out some of them based on what S said.
 
  • #66
lpetrich said:
Some big problems are that the sums and products both have ambiguous decompositions, and also that none of these number pairs' sums and products overlap -- they are all distinct.
Assume they don't know the split.
 
  • #67
The solution to number 9. I was looking for was the pair ##(4,13)##. The clue was to rule out all possibilities of the form ##pq## which allowed a unique factorization by comparison to the according level of information.
 
  • #68
10. Once upon a time there were two good friends, Tim and Jim. One day, they came up with the idea of making a small bet. It was supposed to be a small horse race organized by John. He set the rules as follows:

The race starts at the church, from there the two horses have to cross a hill, past a bus stop, then along the lake and finally back to the church! Of course the way had to be kept exactly! In addition, whose horse is the last to arrive again at the church, whose master is the winner and has won the bet!

But after John arranged and organized everything, something very strange happened. At first Tim and Jim looked at each other questioningly and wondered how they should proceed best, but then Tim suddenly stormed off, ran like a wolf, jumped on the horse, swept off and rode over the hill past the bus stop, along the lake and back to the church. Jim stood there and did not know what happened to him, started way too late and could not catch up.

John watched the race and when Tim arrived at the finish, he congratulated him and said that he had won the race!

How could this happen?
 
  • #69
Tim is riding Jim’s horse.
 
  • #70
11.
242560

How did Carlsen (B) to draw win and why?
 
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