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

[Orodruin]

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?

Spoiler

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.

 

[fresh_42]

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?

 

[mfb]

Open the fridge, put the giraffe in, close the fridge.

 

[fresh_42]

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.)

 

[jbriggs444]

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]

 

[fresh_42]

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)

 

[Orodruin]

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.

 

[fresh_42]

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).

 

[jbriggs444]

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?

 

[fresh_42]

Oooh oooh, can I go again?

Yes, time to finish this one.

 

[jbriggs444]

Yes, time to finish this one.

Use the bridge.

 

[fresh_42]

Use the bridge.

Nope. No bridge anywhere.

 

[Orodruin]

Nope.

I know! I know!

 

[jbriggs444]

Oh silly me. The crocs are at the annual get together.

 

[fresh_42]

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.

 

[fresh_42]

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?

 

[mfb]

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?

 

[fresh_42]

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.

 

[mfb]

Correct analysis, wrong count.

Oops, 48 of course. 20+28 is 48, not 49.

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.

 

[lpetrich]

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.

 

[fresh_42]

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?

 

[fresh_42]

Come on, folks, nobody? This is a physics website!

 

[jbriggs444]

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.

 

[fresh_42]

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?

 

[jbriggs444]

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.

 

[fresh_42]

When does my tea get cold faster?

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.

 

[fresh_42]

8. This sequence is arranged according to which rule?
$$
8\quad 5\quad 4 \quad 9 \quad 1\quad 7 \quad 6 \quad 3 \quad 2 \quad 0
$$

 

[Orodruin]

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.

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.

 

[fresh_42]

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.

 

[lpetrich]

8. This sequence is arranged according to which rule?
$$
8\quad 5\quad 4 \quad 9 \quad 1\quad 7 \quad 6 \quad 3 \quad 2 \quad 0
$$

Spoiler

The English names of these numbers in alphabetical order:

eight, five, four, nine, one, seven, six, three, two, zero