Math Is Fun Forum

There are 28 people who are going to share a four course meal. But not all in the same place.  Some will go first to house 1, while others are in house 2 etc. There are 4 houses. Then they shuffle who goes where for the next course.

So the challenge is to come up with a rotation plan so that, over the 4 courses, everyone meets everyone else for at least one of the courses.

eg if abcdefgh meet in house 1 for the first course, you might try aijklmno in house 2 for the second course.

One trivial solution is for everyone to start in house 1 for course 1, then all move on to house 2 for the next and so on.

I'm assuming that 28 in one venue is too crowded, so I'm trying to spread people out more but still meet the criteria.

As 28/4 = 7, the first thing I tried is 7 in each house. But that must fail as 'a' only meets 4 x 6 = 24 of the friends.  So at least 8 in some venues is needed.

So far I haven't come up with a plan using 8 people at each venue.  I think that there may be some underlying maths that either helps to a solution or proves it cannot be done but I don't know what it is. Maybe you could set up a program that runs through the possibilities???

Bob

It's here: https://www.mathisfunforum.com/viewtopic.php?id=32566

Bob

When I meet a new problem for which I don't have a method, one approach that sometimes helps me is to simplify the numbers to an easier case first and see if I can establish some rules.

So far I've filled lots of sheets of paper with arrangements but not yet got anywhere. 

Bob

What you describe sounds like a Galilean frame of reference and I'd think it would  work.  You could add equations that accurately give the position in orbit for the planets and moons and so we'd have a way of taking account of all the movements.  I'm not sure if Einstein's theory messes with that nor if the uncertainty principle throws a spanner in, but something like that got astronauts to the Moon and back, and also all those exploration rockets that have gone out to the outer planets.

Is this the moment to remind you that the Sun is part of a spiral arm of the Galaxy which is both spinning and moving through space?  It's a similar dilemma to the one that says the Earth is sitting on the back of a giant turtle.  And what holds the turtle? Another larger one. And what holds that?  "Oh, you don't get me like that", said the lady,"It's turtles all the way down". *

Bob

*https://apilgriminnarnia.com/2014/03/26/turtles/

Thanks for the promotion.  Do I get a smart uniform?

Thanks phrontister, well spotted. If I see a post that is pure advertising I remove it and the new member/guest.  This one wasn't selling anythng (I think) so I've edited it out but left Echo_23 as a member for now. It can get tricky when the 'offender' has put in actual mathematical content as I don't want to fragment the thread.

Any ideas on the dinner party post?

https://www.mathisfunforum.com/viewtopic.php?id=32566

Bob

There's not much chance of observing uniform motion without external forces here on Earth chiefly because of friction and to a lesser extent air resistance.

In space it's happening all the time.  The space craft that were shot off to escape the solar system are just continuing in the direction they were going when the fuel ran out.

And any object in orbit just keeps going.  There is an external force; gravity; and that's what causes the orbit but .for example the Earth's tangential speed doesn't change significantly .

You could think about examples where the friction is very low such as in ice skating.

Bob

You could call it a throttle as that's how it works.

Once you've reached cruising speed there is still a force trying to slow you up  made from wind resistance and friction so you still need to input some gas to create a force to overcome these.

Once there are no externally impressed force's Newton says we keep going at constant velocity.

Bob

hi Marjanovic

I have now had a more careful look at your thesis.  I haven't understood passed figure 2.  Please explain whether 3 underlined is the space between 2 and 3, or between 3 and 4.  Figure 2 does not make this clear at all.

Bob

I would call

trig identities.

Def: identity --- a formula that is true for all values of the variable(s).

https://www.mathwords.com/i/identity.htm

I started with your proof, which works if you allow tan(90) = ∞ and tan(90).tan(0) = 1 .

I just juggled the algebra to avoid messing with ∞.

Bob

In any area of maths you have to be careful not to assume things that you haven't been given.  So I would advise caution.

Just that. The graph is all you have and all you need.

Yes. You'd have to be told something about the rate of acceleration and for how long.  If a is constant, then the V-T graph is a straight line and the D-T graph is a quadratic. This comes from calculus but you can also get it from s = ut + ½at^2

If a isn't constant that's a one term course at first year Uni.

Bob

Constant velocity definitely means no acceleration.  The given information does not state that u=0 and I think that's why you're confused.  If you take u = v = 5

then v = u + at     => a = (v-u)/t = (5-5)/3 = 0

Bob

hi kittenlover206

There is a way to answer this without checking every single integer.

Except for 1, all numbers have at least two divisors, 1 and the number itself.

If a number has another then mostly it will have another. If f is a divisor of N then N/f is another divisor.  eg. 48 has these divisor pairs {1, 48, 2, 24, 3, 16, 4, 12, 6, 8}

So we can divide the integers into sets like this.

{1} has a single divisor.

{primes} have 2 divisors. eg divisors of 7 = {1,7}

{many non primes} have an even number of divisors: 1, the number, one or more  divisors  f, and another N/f for each f. (Such as 48 see above)

So what integers have an odd number of divisors? From these which have only one extra apart from 1 and the number?

Bob