Challenge thread

[EetuK00]

Sometimes I get bored of cubing. So does many other cubers. So lets create a collection of challenges here!

 

[Daniel Lin]

CHALLENGE:
2x2-7x7 relay

BUT

You can only do one move per cube at a time
For example
After you do a move on the 2x2, you must make a move on the 3x3, 4x4,5x5, 6x6, AND 7x7 before you can turn the 2x2 again. And keep going around in a circle. Once you finish 2x2 you don't have to touch it again.

see how fast it takes to solve every cube!

 

[shadowslice e]

I suspect there are some people on here who know how to solve these:
1) Radium has a half-life of 1600 years. Suppose there are initially 10g of radium. Find out how much radium is left after 100 years.
2) A kilogram of a toxic material has been spilt into a tank containing 1000 litres of water and is thoroughly mixed. There is only one way to get rid of it; allowing the mixture to drain out of the bottom while refilling the tank with fresh water at the same time. The mixture runs out and is replaced at a rate of 20 litres per second. Find how much time has passed when only 0.1 kilograms of the toxin remain.
3) You are floating on an airbed on a swimming pool checking Facebook on your phone. Because you are a clumsy idiot, you accidentally drop your phone into the water and it sinks to the bottom of the pool. Does the water level go up, down or stay the same (and can you explain why)?
4) Let's say the distance from London to Liverpool is 200 miles. At 10:00, a train sets off from London to Liverpool travelling at 50 mph. At the exact same instant, a fly sets off from Liverpool going to London along the train tracks. When the fly hits the windscreen of the train, instead of getting crushed as we would expect, it simply travels at the same speed it started with back to Liverpool before starting the journey to London again. If the fly travels at 100 mph, how far does it travel by the time the train reaches Liverpool? (this has two methods; one is easier calculationwise but requires some lateral thinking, the other one is brute force- I would be interested to see anyones method if they solve it).
5) Same context as above but this time we know the fly travels 600 miles by the time the train reaches Liverpool. Find its speed.
6) A farmer shares a small circular island with a sheep that is 10 m in diameter. Ten farmer wants to make sure the sheep can only graze on exactly half of the island but the only way he can restrict the sheep is by tying a rope to it and attaching the other end to a stake. He decides he does not want the inconvenience of halving to walk around the sheep's territory so decides to put the stake on the edge of the island. How long must the rope be?
7) At the north pole, you can walk a mile south, a mile east and a mile north and end up in the same spot. Is this the only place on the planet where you can follow those exact directions and end up in the same spot? If not, where else can you do this?
8) If you fire a cannonball up a hill, what is the angle you can fire at relative to flat ground that means the cannonball travels furthest up the hill if the hill is at an incline of x rad (drag is negligible)?
9) What angle should you fire a cannonball at so that the distance travelled in the air (not where it lands) is the greatest when on flat ground?
10) Show that any number which is a multiple of four or an odd number can be written as the difference of two squares. Also show that no number of the form 4n+2 can be written as the difference of two squares.
11) Finally, probably the hardest problem: show any even number can be written as the sum of 2 primes and any odd number can be written as the sum of 3 primes.

 

[sqAree]

For 11, assuming we're only talking about natural numbers, the latter conjecture follows directly from the former, because any odd n can be written as n = 3 + m with m even, 3 being a prime?

 

[shadowslice e]

For 11, assuming we're only talking about natural numbers, the latter conjecture follows directly from the former, because any odd n can be written as n = 3 + m with m even, 3 being a prime?

Yes, if you get first conjecture you can prove the second for free but the reverse is not necesarily true.

 

[Daniel Lin]

1) Radium has a half-life of 1600 years. Suppose there are initially 10g of radium. Find out how much radium is left after 100 years.

Radium left=10g*(1/2)^(100yr/1600yr)
=10 g
sig figs

should this be moved to off topic discussion? are we doing cube related challenges or random stuff?

 

[sqAree]

10) Show that any number which is a multiple of four or an odd number can be written as the difference of two squares. Also show that no number of the form 2n+2 can be written as the difference of two squares.

Let n be odd, so n = 2m+1. We have (m+1)² - m² = 2m+1.
Let n be a multiple of 4, so n = 4m. We have (m+1)² - (m-1)² = 4m.
For 2n+2 there is the simple counter example of n = 1.

EDIT: But we can prove that no number of the form 4n+2 can be written as the difference of two squares. Just calculate x² - (x - 2y)² and x² - (x - 2y - 1)² to find out that a difference of two squares is either odd or a multiple of 4 as all cases are covered.

 

[Cale S]

7) At the north pole, you can walk a mile south, a mile east and a mile north and end up in the same spot. Is this the only place on the planet where you can follow those exact directions and end up in the same spot? If not, where else can you do this?

Any point one mile directly north of the latitude line with a circumference of 1 mile closer to the south pole than the north pole

 

[qaz]

might do more later

I suspect there are some people on here who know how to solve these:
1) Radium has a half-life of 1600 years. Suppose there are initially 10g of radium. Find out how much radium is left after 100 years.
10g * .5^(100/1600) = 9.576g
2) A kilogram of a toxic material has been spilt into a tank containing 1000 litres of water and is thoroughly mixed. There is only one way to get rid of it; allowing the mixture to drain out of the bottom while refilling the tank with fresh water at the same time. The mixture runs out and is replaced at a rate of 20 litres per second. Find how much time has passed when only 0.1 kilograms of the toxin remain.
dM/dt = -(20/1000)M,
ti = 0, Mi = 1
tf = ?, Mf = 0.1
move around terms -> -50dM/M = dt
integrate both sides -> -40ln(M) = t + C
ln(1) = 0, so C = 0 -> -40ln(M) = t
-40ln(0.1) = tf -> tf = 92.1 seconds
3) You are floating on an airbed on a swimming pool checking Facebook on your phone. Because you are a clumsy idiot, you accidentally drop your phone into the water and it sinks to the bottom of the pool. Does the water level go up, down or stay the same (and can you explain why)?
4) Let's say the distance from London to Liverpool is 200 miles. At 10:00, a train sets off from London to Liverpool travelling at 50 mph. At the exact same instant, a fly sets off from Liverpool going to London along the train tracks. When the fly hits the windscreen of the train, instead of getting crushed as we would expect, it simply travels at the same speed it started with back to Liverpool before starting the journey to London again. If the fly travels at 100 mph, how far does it travel by the time the train reaches Liverpool? (this has two methods; one is easier calculationwise but requires some lateral thinking, the other one is brute force- I would be interested to see anyones method if they solve it).
at their first meeting, since the fly is traveling twice as fast, it has traveled 133.33 miles and the train 66.67 miles. When the fly makes it all the way back, the train has traveled another 66.67 miles, leaving 1/3 of the original distance left. Therefore with a distance d the fly travels 2(2/3)d or (4/3)d. Since each d is 1/3 the previous d, the equation for the total distance traveled by the fly is sum(200*(1/3)^i*(4/3)), i = 0 to infinity, or (800/3)sum((1/3)^i), i = 0 to infinity. The formula for the sum of a geometric series such as this is just a/(1-r) or (800/3)/(2/3) = 400 miles.
5) Same context as above but this time we know the fly travels 600 miles by the time the train reaches Liverpool. Find its speed.
6) A farmer shares a small circular island with a sheep that is 10 m in diameter. Ten farmer wants to make sure the sheep can only graze on exactly half of the island but the only way he can restrict the sheep is by tying a rope to it and attaching the other end to a stake. He decides he does not want the inconvenience of halving to walk around the sheep's territory so decides to put the stake on the edge of the island. How long must the rope be?
7) At the north pole, you can walk a mile south, a mile east and a mile north and end up in the same spot. Is this the only place on the planet where you can follow those exact directions and end up in the same spot? If not, where else can you do this?
8) If you fire a cannonball up a hill, what is the angle you can fire at relative to flat ground that means the cannonball travels furthest up the hill if the hill is at an incline of x rad (drag is negligible)?
9) What angle should you fire a cannonball at so that the distance travelled in the air (not where it lands) is the greatest when on flat ground?
10) Show that any number which is a multiple of four or an odd number can be written as the difference of two squares. Also show that no number of the form 2n+2 can be written as the difference of two squares.
11) Finally, probably the hardest problem: show any even number can be written as the sum of 2 primes and any odd number can be written as the sum of 3 primes.
are they unique primes?

 

[shadowslice e]

Let n be odd, so n = 2m+1. We have (m+1)² - m² = 2m+1.
Let n be a multiple of 4, so n = 4m. We have (m+1)² - (m-1)² = 4m.
For 2n+2 there is the simple counter example of n = 1.

EDIT: But we can prove that no number of the form 4n+2 can be written as the difference of two squares. Just calculate x² - (x - 2y)² and x² - (x - 2y - 1)² to find out that a difference of two squares is either odd or a multiple of 4 as all cases are covered.

Oops I meant to type 4n+2 so gj for working that out.

 

[TDM]

11) Finally, probably the hardest problem: show any even number can be written as the sum of 2 primes and any odd number can be written as the sum of 3 primes.

lol

I've seen 6 before but didn't try it or see any solution, so I'll have a go at that later. Most of these I've done/seen before, but 8 and 10 look interesting. I'll try those too.

E: for 8, do we start at the bottom of the hill, or a certain distance away?

 

[shadowslice e]

lol

What's so lol about it? I was perfectly serious when I said it was the hardest problem.

I've seen 6 before but didn't try it or see any solution, so I'll have a go at that later. Most of these I've done/seen before, but 8 and 10 look interesting. I'll try those too.

E: for 8, do we start at the bottom of the hill, or a certain distance away?

Well, this question was posed as at the foot of the hill though you can try x metres away if you want an extra challenge (I may try that at some point)

 

[TDM]

Well, this question was posed as at the foot of the hill though you can try x metres away if you want an extra challenge (I may try that at some point)

I'll try the original version first (and tomorrow - I can't do maths at 11pm!)

Hopefully I haven't forgotten any mechanics, I've not done it for a year.

 

[shadowslice e]

I'll try the original version first (and tomorrow - I can't do maths at 11pm!)

Hopefully I haven't forgotten any mechanics, I've not done it for a year.

Cool. Btw, just a hint; when you do it, after you've done all the projectile motion and Cartesian equations, it's actually much more to do with trig and calculus (mostly differentiation) than the other mechanics principles.

But then I suppose half of the more advance mechanics at AS & A2 boils down to Co-ordinate geometry, trig and stuff like that (at least from the STEP questions I've done).

...

For 4, while your answer is completely correct, there is a way to do it without any sum to infinity I'll leave it to you to see if you can find it.

Also for 11, the primes do not need to be unique so 7+7=14 would be perfectly permisable.

 

[Daniel Lin]

k i'm trying to figure out number 9...
please don't solve it before me

 

[TDM]

Cool. Btw, just a hint; when you do it, after you've done all the projectile motion and Cartesian equations, it's actually much more to do with trig and calculus (mostly differentiation) than the other mechanics principles.

But then I suppose half of the more advance mechanics at AS & A2 boils down to Co-ordinate geometry, trig and stuff like that (at least from the STEP questions I've done).

Yeah, finding the best angle for furthest distance along a flat plane did that, and I imagine it will be similar enough to that. I've just noticed I misread Q9 (I thought it was furthest distance) - that again looks similar, with parametric arc lengths. The formula for that is an integral so differentiating that should be easy

I didn't try any Mechanics STEP questions this year, not even while preparing for STEP. I just did stats ones - I've heard the mechanics ones were usually easy but I hadn't practised mechanics at all so I didn't want to risk it. But I did badly on STEP anyway so it didn't even matter.

 

[shadowslice e]

Yeah, finding the best angle for furthest distance along a flat plane did that, and I imagine it will be similar enough to that. I've just noticed I misread Q9 (I thought it was furthest distance) - that again looks similar, with parametric arc lengths. The formula for that is an integral so differentiating that should be easy

I didn't try any Mechanics STEP questions this year, not even while preparing for STEP. I just did stats ones - I've heard the mechanics ones were usually easy but I hadn't practised mechanics at all so I didn't want to risk it. But I did badly on STEP anyway so it didn't even matter.

Well, I've still got another year so fingers crossed but I feel like I will probably get crushed by it too but I can hope... :/

I tend to do only the pure maths ones though I occasionally do some mechanics if I have no other options. I will always leave stats as a last resort though because I got put off it by S1 and D1 but I suppose the STEP will be more interesting so I guess I should break that bad habit...

Maybe...

Or I could just get better at the other stuff and hope I don't need to...

Oh well, I'll worry about that later

 

[TDM]

(mods: if this is getting too off-topic then maybe move the posts about shadowslice's problems to a new thread?)

Well, I've still got another year so fingers crossed but I feel like I will probably get crushed by it too but I can hope... :/

I tend to do only the pure maths ones though I occasionally do some mechanics if I have no other options. I will always leave stats as a last resort though because I got put off it by S1 and D1 but I suppose the STEP will be more interesting so I guess I should break that bad habit...

Maybe...

Or I could just get better at the other stuff and hope I don't need to...

Oh well, I'll worry about that later

Good luck next year! If you prepare enough then you won't feel like you'll get crushed before the exam. Of course, in the exam you will get crushed, but you won't be expecting it so that's a good thing. Right?

I found the stats questions easier than most of the pure ones, but yeah, I think S1 does put a lot of people off Stats modules get easier as you go along though. There's almost nothing to learn at all in (Edexcel) S4. I think you'll probably be able to answer 4-6 pure questions as you get better, like you said, and not even need any applied ones unless you want to answer them.