Riddles and puzzles

[The Last Outlaw]

Nice one . I'm sure I've heard that somewhere before, but I can't remember what the answer was!

I kind of want to say 'music', but I figure that would be too obvious, given the lines about 'rock', 'alternative', 'hip-hop', and 'always at the top of the chart' .

You know that saying, "Go with your first instinct, it's usually right"?  This time... IT IS!

[Kyng]

Here's a problem adapted from an Oxford University admissions test. Here, you're asked to estimate the value of log₂3 without using a calculator. You don't need the exact value: you're given a choice of intervals in which it might lie, and you need to figure out which is the correct one:



I was able to solve it (in much the same way that it's presented in the video), but still, I am glad this wasn't my admissions question !

[Kyng]

Here is a fun little one, that one of my friends sent me:

An orchestra of 120 players takes 40 minutes to play Beethoven's 9th symphony. How long would it take 60 players to play the same symphony?

Let P be the number of players and T be the time playing.

[Kyng]

Now we have another "How long does it take X people to complete a job?" type problem, but this time, it involves three people:

Spoiler

This makes me feel bad for Charlie. His presence only speeds it up by nine minutes, compared to if it had just been Alice and Bob .

[Kyng]

Here's a little maths puzzle:



The diameter of the circle is 10cm. What is the area of the blue square?

(Bonus points if you can do it without first calculating the area of the larger square!)

[Jarkko]

Okay, I only just relearned that the diagonal of the largest possible square inside a circle is equal to the diameter of that circle.

The Pythagorean theorem dictates that a²+b²=c², and since the two other sides of the triangle are equal and the hypoteneuse is 10, it means that the two sides of the triangle have to be √50 (a tiny bit more than 7) each. One has to split them in half to find the hypoteneuse made by the internal square, and each side turns out to be √12.5. 

So that means the hypoteneuse has to be 5 (12.5+12.5=25), meaning the area of the inside triangle is 25 cm².

[Kyng]

Indeed it is 25cm² .

Another way to see that is to flip one side of the square, as follows:



Now that I've done that, it's obvious that one side of the square is the radius of the circle - which is 5cm. So, the area is 5 x 5 = 25cm² .

[Kyng]

Here's a fun one (well, two fun ones). Here, you're given two lists of statements, and you have to figure out which statements (if any) are true, and which are false:

Spoiler

I got the 'exactly' version quite easily . It was clear that all ten statements were mutually exclusive (so at most one was true), and it didn't take long to find the one that was.

The 'at least' version required a little more thought, but I got there in the end .