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Note: These are via @marcsmith (of unknown urlage) and so for once I don't know the answers either. I'll post my thoughts in the first comment

Quick

A country is preparing for a possible future war. Its tradition is to send only men into battle, so they want to increase the proportion of males to females in the population. A law is passed which requires every married couple to have children and to continue to do so until they have a boy. What effect do you expect this law to have on the balance of the population? (via BBC Radio 4 - More or Less)

Cryptic

There are an infinite number of even integers and an infinite number of odd integers. There are also hence an infinite number of integers overall. The number of even integers is the same as the number of odd integers. By how much does the set of all integers differ from the set of even integers? (via Georg Cantor)

Answer Update (after 8 comments)

Note: Answers from @marcsmith after I answered the questions. Sidenote: I think these questions were a little to complex. If I carry on with PQF (not sure if I will as coming up with Q's is hard), I'll probably look for stuff that can be answered in 1 or 2 paragraphs.

Quick

The proportion will remain the same. At first I considered it as Adrian did and that the number of boys would increase. However he has made the assumption that for the couple with the girl the next child would be a boy. The probability is still 50% in his scenario. That is the next child could be a girl or a boy. However even if there was a predilection for this pattern, i.e. when there is an odd number of couples, the odd couple?s next child will be a boy, the proportion would still remain the same. Let me explain this by increasing the data set. We'll start by simplifying and defining the rules:

1. Every couple can have children,
2. They stop when they have their first boy
3. For every round of babies there is exactly a 50% ratio of boys to girls
4. If there are an odd number of couples the odd couple will have a boy
5. None of the children have their own children.

With these rules it follows that the number of boys, when all the couples in the current generation have had their children, will be the same as the number of couples. I'll use powers of two to keep the numbers simple.

For two couples the first iteration would produce 1 girl and 1 boy. That leaves one couple with a girl and therefore their next child will be a boy. So for 2 couples we have 2 boys and 1 girl.

For four couples the first iteration would produce 2 girls and 2 boys. The next iteration leaves 2 couples. We know from the previous example they produce 2 boys and 1 girl. So for 4 couple we get 4 boys and 3 girls.

For eight couples the first iteration would produce 4 girls and 4 boys. The next iteration leaves 4 couples. We know from the example the example with 4 couple that they produce 4 boys and 3 girls. So for 8 couple we get 8 boys and 7 girls.

Just by examining the examples we can see that for powers of 2 the number of boys is always one more than the number of girls. Let's tabulate this:

Couples  Boys  Girls  Proportion Boys
   2        2      1   66.67%    
   4        4      3   57.14%
   8        8      7   53.33%
  16       16     15   51.61%
  32       32     31   50.79%
4096     4096   4095   50.01%

We can see that as the number of couples of increases the proportion tends towards 50% even though we always finish with boys.

Although we've produced rules that produce deterministic results it holds true if we use a random number generator to predict if a couple is having a boy or a girl with a 50% chance of it being either. Here are the results of a few consecutive runs for two couples from a quick simulation I wrote.

 
Couple 0: B,   Couple 1: GB,    Boys%: 66%
Couple 0: GGB, Couple 1: B,     Boys%: 50%
Couple 0: B,   Couple 1: GGB,   Boys%: 50%
Couple 0: B,   Couple 1: B,     Boys%: 100%
Couple 0: B,   Couple 1: GGGGB, Boys% 33%
Couple 0: GB,  Couple 1: GB,    Boys%: 50%

As for the population itself. I suspect, and I could be wrong, that if everyone had to stop after they had one boy then for a 50% proportion the population would stagnate (that is after an initial growth so that each generation barring the last had all had their children). Anything other than 50% would eventually see a decline.

If there was no restriction on the number of boys then the population would most likely grow as long as the number of the smaller proportioned sex exceeded the number of couples in the previous generation but the proportion would still remain the same.

I could be wrong on the population stuff. I've only just considered it after Adrian's comment.

Cryptic

"I see it, but I don't believe it!" - Georg Cantor

The number of even natural numbers is exactly the same as the number of natural numbers. Every whole number can be mapped to an even number. No exceptions.

So 1=2, 2=4, 3=6, 4=8 ... n=2n

There is no case that exists where n exists and 2n does not. And for every natural number there is exactly one even integer that it maps to.

When there is exactly a one to one mapping between two sets it is called bijection. When one of the sets is the set of infinite natural numbers it is a countable set or a set that is denumerable.

It turns out a lot of things are denumerable:

1. Even numbers
2. Odd numbers
3. Prime numbers
4. Rational numbers (All integers and fractions)

That last one looks even more unlikely than the even numbers. Can there really be the same number of natural numbers as rational numbers. The set of rational numbers seems infinitely larger than the set of natural numbers. For every natural number as a numerator there is an infinite denominators. But looks can be deceiving. It turns out that every rational number can be mapped to a natural number in a one to one mapping.

Not all sets are denumerable. Irrational numbers, any real number that can't be expressed as a fraction, such as PI, are not countable.

This is because rational numbers settle into a pattern (1/2, 1/3, 1/4, 1/n) and can therefore be mapped whereas irrational numbers cannot, consider pi.

For more information and a much clearer explanation check out these links:

Gustavo Duarte: Counting Infinity

Wikipedia: Countable set

Quick

For new builds, why do show apartments normally have no internal doors?

Cryptic

You are receiving lots of complaints about waiting times for your elevator system in the building you manage. Assuming replacing the elevator or the building are not an option, what do you do? And Why?

Answer Update (after 7 comments)

I really wanted to call this "The Perception" edition but I thought that would give it away too much.

You'll pretty much got it.

Quick

Nothing to do with H&S. Can't see not having a door would make a place safer. It's all to do with selling. You perceive a flat / room as bigger without the door. It creates an illusion of space. Makes it an easier sell.

Cryptic

The answer was about how long people think they are waiting. Essentially assuming you can't change the elevator system, you need to change it so people don't find waiting an issue. A few interesting suggesting beyond the one I was looking for (which was also suggested). Which is to put mirrors up so people can look at themselves/other people and don't realise they are waiting.

Ref: Defining the problem of elevator waiting times - 37Signals

Quick

Most things (metal, ice, plastic, cheese) get soft when you heat them. What gets harder when you heat it.

Cryptic

It's oft said that you lose most of your body heat through your head. Why is this both true and false. Or put another way, what is both true and false about this. Bonus points for citing reasons without google it.

In ‘crossword format’

Quick

The government just announced a reduction in VAT from 17.5% to 15%. How much money do you save per £100?

Cryptic

You’re at a magic show. The magician asks you to shuffle the cards (a normal 52 card deck) and hand them back to you. He asks a member of the audience to name a card. The audience member names the Ace of Spades. The magician draws a single card out of the deck. What is the chance (odds?) of the card drawn being the card named?

EOD: Answer Update (after 9 comments)

Quick

The papers/press/media have all been reporting that with a 2.5% reduction in VAT you’ll only be saving £2.50 in every £100 by working out the complex equation of “What’s 2.5% of 100?”. Of course this is clearly wrong.

In £100 payment inclusive 17.5% tax you’re paying £14.89 tax such that the item value would have been £85.11

A 15% tax on £85.11 would result in £12.77 tax.

£14.89 - £12.77 = £2.12 savings.

Of course as Nigel points out, there is actually a more complex answer here too (based on my question), and that even for a specific right answer, there can be multiple right answers.

Also everyone has been writing in to letters pages commenting how pointless the reduction is, as £2.50 is not a lot of money. Which proves my point that most people are idiots who don’t understand anything. I wont explain here why a 2.5% decrease is significant, just to say image how people would have reacted had VAT gone UP by 2.5%.

Cryptic

The question really is about how we evaluate risk. And how we consider the factors involved. There is no real right answer here, the right answer is in how we explain the answer. Of course some answers are more right than others.

Marc was the most correct here, or put another way, had a most correct answer.

A simple way of looking at it is like this

• The random chance of pulling a specific card out of a deck is 1 in 52
• However we know we are at magic show and we know a magician probably knows what we are doing so we could consider the odds are then really 1 in 1.
• However looking at the factors again, we know a magician probably is going to not pull the card out the first time to build suspense, so we could say the odds are 0 in 1.
• Or we can factor together different tricks (as Marc did) and calculate even more precise odds.

So those are the two answers (or two of the answers) to today’s Friday Pop Quiz questions. I’ll be honest I stole the questions from the BBC R4 More or Less podcast. If you want to listen to more detail you can watch it on iPlayer or download here

I quite liked pop quiz Friday. More for the debate and interesting perspectives (including @Jack (2)’s). I might do this every week. If I can think of enough interesting questions.

Me @ 19h28

And people wonder why I don't bother reading the Da Vinci Code: The Dan Brown code

Not me @ 20h11

Dan Brown Reviews

Apparently some people liked it. At least they read it and made their own mind up...

Me @ 20h36

That would imply one should read everything then. Unfortunately I don't have time to read everything in existence, so it's entirely acceptable to base ones decisions on valued advice, good reviews or detailed analyse. In this case the detailed analysis says DB can't write for anything. Reading the quotes from the book was more than enough to know that the worst book in my flat is far more readable.

Not me @ 20h42

For every bad review there are as many, if not more, positive reviews. Apparently it sold "rather well".

Read it, make your own mind up and only then is your view of any interest to anyone else

Me @ 21h41

Popularist does not mean good. The spice girls also sold rather well. That doesn't make them a good band only a popular band.

Why read something I've pretty certain I will find grating. I'm still able to hold a valid opinion based in depth studies of others. Not everything requires 1st hand experience to understand. I can evaluate that I probably would enjoy cocaine without having to try it.