Comments on: Why I Love Mathematics (the real thing)

By: fadhilah

fadhilah — Fri, 30 Jul 2010 14:14:04 +0000

i love math…i start learn math when i was 5 years old…now i learn precalculus…i feel so excited to know more about math

By: shAdOwArt

shAdOwArt — Thu, 01 Oct 2009 15:49:56 +0000

A lovely riddle, that ‘Blue Eyes’ one. Somehow I always find it much more rewarding to solve riddles that have been asked by someone rather than riddles found among a list of 100 other riddles.

Here’s two for future readers chew on!

Take the ball riddle, but modify it to n balls and k allowed weightings, when can the odd ball be found and it’s nature identified?

What is the product of all the natural numbers? All the whole numbers? All the rational numbers? All the real numbers? All the complex numbers? Always excluding zero of course.

By: Aurora

Aurora — Tue, 01 Sep 2009 14:46:47 +0000

Discussed it with a friend, and we finally got it. Still working on the balls one, though. I’m not as good at this sort of thinking as I’d like to be…

By: Vlad Dolezal

Vlad Dolezal — Tue, 01 Sep 2009 11:18:23 +0000

@Aurora:

I like the way you’re thinking :p. I guess this should teach me to start riddles with “assuming an idealized spherical mother in vacuum…”

Or maybe not.

As for the blue eyes, have you tried starting from the basic cases and then working up the numbers? That always helps with riddles involving “1000 prisoners” etc.

By: Aurora

Aurora — Mon, 31 Aug 2009 20:55:02 +0000

This is more psychological than mathematical, but for the two kids problem…if both of her children were boys, wouldn’t she most likely have said that was her oldest/youngest son, not just her son? So it’s most likely that her other child is a girl, since she just says the boy is her son, with no clarifying adjectives.
Still completely stumped on the blue eyes riddle, though.

By: Vlad Dolezal

Vlad Dolezal — Sun, 02 Aug 2009 08:19:38 +0000

@Katie:

I don’t think you’re missing anything. It’s just that we all need a different way of coming at the right answer (and avoiding some mistakes along the way).

The reason we’re messing with the whole boygirl vs. girlboy difference… let’s say you toss two coins. Quick, what’s the chance of getting one heads and one tails, as opposed to getting two heads?

Hopefully you answered that one heads and one tails is twice as likely. Because it is.

And the reason we were arguing about the girlboy vs. boygirl is the semantic difference between saying “given that at least one coin lands heads” and “given that the first coin lands heads”. Because that makes the difference between your chance of getting two heads being 1/3 and 1/2.

(I think I’m not explaining it very well. Oh well. As an excuse, here’s a picture of a kitten: http://tinyurl.com/dcfwga )

By: Katie

Katie — Sun, 02 Aug 2009 00:15:23 +0000

As far as the two kids riddle is concerned, I don’t think whether it’s boygirl or girlboy matters at all. The chance in either of those cases that both are boys is the same, zero. So you can just pick one.

Her children are either boyboy or boygirl, meaning the chance that both are boys is 1/2.

Her children are either boyboy or girlboy, meaning the change that both are boys is 1/2.

I don’t see the difference logically in these two statements, or the reasoning behind numbering or ordering the kids, for me that just makes it too complicated, but maybe I’m missing something important there.

By: Vlad Dolezal

Vlad Dolezal — Fri, 05 Jun 2009 08:13:01 +0000

@Sky:

Ahhhh, I finally got it!

Damn that was an evil riddle.

By: Vlad Dolezal

Vlad Dolezal — Mon, 01 Jun 2009 23:03:10 +0000

@Sky:

Ya, that’s what I got for the two-boy riddle too, after a reader corrected me! (I also originally thought it was 1/3.)

You have a whiteboard at home? Damn you!

And about the blue eyes… it completely didn’t occur to me to start with the basic cases. Even though that’s my usual approach with these riddles involving big numbers. I’ll get right on it!

By: Sky

Sky — Mon, 01 Jun 2009 21:51:04 +0000

@Vlad: Ah, I think I see where you’re coming from now. The probability of *seeing* one boy at the store is actually different from the probability that she has (at least) one boy, so the way I originally rephrased the problem was wrong. So I think we are really looking for is this: P(she has two boys | we *saw* one boy).
And that’s equal to this:

P(she has two boys and we saw one boy)
———————————————-
P(we saw one boy)

The top is (1/4 * 1), since the probability of her having two boys is 1/4, and the probability of us seeing a boy in that case is 1.

The bottom is P(she has two boys and we saw a boy) + P(her youngest is a girl and we saw a boy) + P(her oldest is a girl and we saw a boy) + P(she has two girls and we saw a boy) = (1/4 * 1) + (1/4 * 1/2) + (1/4 * 1/2) + (1/4 * 0) = 1/4 * (1+.5+.5+0) = 2/4 = 1/2

So the total probability we’re looking for is (1/4) / (1/2) = 1/2.

Man, I don’t think I’ve made use of my whiteboard this much since I bought it.

And as for the blue eyes riddle, a good hint is to try to think inductively. Build it up from base cases.