Can you solve the maths question for Singapore schoolkids that went viral?

[Chinahand]

 

Four flies are each on the four corners of a square (A,B,C,D) with side length S. At the same moment they all take off flying towards the next fly (A towards B, B towards C etc). They all fly at a constant speed and always travel straight towards their target fly. When they meet, how far has each fly flown?

Please show your workings!

[John Wright]

26 minutes ago, Chinahand said:

 

Four flies are each on the four corners of a square (A,B,C,D) with side length S. At the same moment they all take off flying towards the next fly (A towards B, B towards C etc). They all fly at a constant speed and always travel straight towards their target fly. When they meet, how far has each fly flown?

Please show your workings!

Lol

0.785S

S x 3.14 / 4

[Chinahand]

Nice try, but I'm afraid not. Try again!

 

[Chinahand]

Here's my go ... 

[John Wright]

59 minutes ago, Chinahand said:

Here's my go ... 

But the practical reality is that as all flies move at once the perpendicular line between flies become, in effect tangents to a circle. The path of A to the centre ends of being a quarter circle. we know what the radius is, its S, we know what pi is and can work from there? 

[Declan]

My non-mathematical mind suggests they all fly in an ever decreasing spiral until they meet in the centre. At which point the answer to your question is they will all have flown the same distence. 

[Chinahand]

1 hour ago, John Wright said:

But the practical reality is that as all flies move at once the perpendicular line between flies become, in effect tangents to a circle. The path of A to the centre ends of being a quarter circle. we know what the radius is, its S, we know what pi is and can work from there? 

John it is a little difficult when you've been presented with a proof showing you are incorrect in saying the path is a 1/4 of a circumference in length, especially when you say things like in "practical reality".

The flies paths are perpendicular to the other flies, not to a fixed circle. A work colleague modelled it using matlab. Here's his result (twisted 45 degrees from mine):

The path isn't an arc of a fixed circle - as Declan says it is a spiral heading into the centre of the square such that he path lengths are of length S when they meet in the centre.

The assumptions are unrealistic - constant speed no matter the radius of turn, instant path correction - but tweaking these, ie slowing the flies down as they corner more sharply and adding in reaction times, won't transform the answer into what you are visualising. At the very start, your intuition is exactly right, each fly is flying tangential to a circle radius S, but as the fly chases its target fly that circle's radius decreases.

Ah, thinking about it, that is the issue - they fly tangentially to an ever decreasing circle, not to a fixed one.  Does that help explaining why your initial answer was incorrect?

Edited July 4, 2022 by Chinahand

[John Wright]

2 hours ago, Chinahand said:

John it is a little difficult when you've been presented with a proof showing you are incorrect in saying the path is a 1/4 of a circumference in length, especially when you say things like in "practical reality".

The flies paths are perpendicular to the other flies, not to a fixed circle. A work colleague modelled it using matlab. Here's his result (twisted 45 degrees from mine):

The path isn't an arc of a fixed circle - as Declan says it is a spiral heading into the centre of the square such that he path lengths are of length S when they meet in the centre.

The assumptions are unrealistic - constant speed no matter the radius of turn, instant path correction - but tweaking these, ie slowing the flies down as they corner more sharply and adding in reaction times, won't transform the answer into what you are visualising. At the very start, your intuition is exactly right, each fly is flying tangential to a circle radius S, but as the fly chases its target fly that circle's radius decreases.

Ah, thinking about it, that is the issue - they fly tangentially to an ever decreasing circle, not to a fixed one.  Does that help explaining why your initial answer was incorrect?

Yes

[wrighty]

4 hours ago, John Wright said:

But the practical reality is that as all flies move at once the perpendicular line between flies become, in effect tangents to a circle. The path of A to the centre ends of being a quarter circle. we know what the radius is, its S, we know what pi is and can work from there? 

I don't think the phrase 'practical reality' is relevant in a maths puzzle about 4 flies and a square.  

I'll dust off my Vector Calculus later and see if I can calculate the track, and its arc length.  I may be some time.

[Passing Time]

On 7/1/2022 at 9:50 AM, Chinahand said:

 

Four flies are each on the four corners of a square (A,B,C,D) with side length S. At the same moment they all take off flying towards the next fly (A towards B, B towards C etc). They all fly at a constant speed and always travel straight towards their target fly. When they meet, how far has each fly flown?

Please show your workings!

[wrighty]

2 hours ago, Passing Time said:

Those things are ace! 

[wrighty]

10 hours ago, Chinahand said:

Here's my go ... 

I think that’s it. I tried a vector calculus approach and got hopelessly bogged down. A simpler argument would be “consider the relative motion of B to C. Instantaneously, B is always heading directly to C, and by symmetry C is heading in a perpendicular direction towards D. If C’s velocity vector is subtracted from B’s, leaving C in essence stationary relatively, then B must travel a distance S before hitting C”

The locus of the path is another matter.  There’s probably a ‘calculus of variations’ approach to this question that I don’t know how to do. 

[Chinahand]

Have a break ... do some maths!

Always remember to read the question before answering!!

https://www.bbc.co.uk/bitesize/articles/z6wshcw#xtor=CS8-1000-[Promo_Box]-[News_Promo]-[News_Promo]-[PS_BITESIZE~N~~A_MathsProblemOfTheWeek_SEG_PNC]

 

[Chinahand]

If you can explain something you understand it ... and I'm trying to fix my understanding of power series and log log plots. So here's a go at creating and setting some problems about them!

I'm pretty certain the following are examples of straight lines on a log log plot ... if anyone has any thoughts do chip in!

Let's assume a 10% increase in bodymass results in a 15% increase in brainsize - I'm pretty sure that results in a linear plot on a log log graph.

Now if a rhino's mass is 500kg and it has a brain size of 1 kg, how much will the brain of a 2000kg elephant weigh?

Bizarrely, the maths in this also work with things like world telecoms pricing.

If a country has a per capita gdp 15% higher than another's then the Average Revenue per Capita (ARPU) on its mobile subscriptions will be 10% higher. If the UK's gdp per capita is $40K per year and mobile phone bills are on average $20 per month how much will mobile phone bills be in the US with a GDP per Capita of $55K per year?

For more details of this sort of thing see Scale by Geoffrey West:

https://www.amazon.co.uk/Scale-Universal-Organisms-Cities-Companies/dp/1780225598

All the numbers are entirely made up! I'm just wanting to ensure I can do the maths given the assumptions. I think I can, but I'm not certain ... off to calculate elephant brain sizes and ARPUs in the US!

My job is very strange!

[Moghrey Mie]

On 4/14/2015 at 12:07 PM, Albert Tatlock said:

The sentence...

 

"Cherryl tells Albert and Bernard separately the month and day of her birthday respectively"

 

...couldn't be more precise and succinct.

Could have said she told Albert the month and Bernard the day of her birthday.