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hi envy1987

Although they want you to avoid the truth table, I see no harm in separately trying it, just to see if it is a tautology. (You don't have to submit that as part of your answer.) It looks to me like it isn't, so you need to find an example. Have you been given any examples for other non tautologies so I can get an idea what is expected here?

Bob

I think this scheme has been devised by fat-heads. They haven't even worked out if you have to repay if you put weight back on.

A while back the education minister floated the idea that students would have to remain in school indefinitely if they failed to reach a certain standard in their exams. The idea was quietly dropped when people explained to him why this was unworkable.

Papers like the Mirror and Mail love making headlines like this because it sells more papers and hence gets them more advertising revenue.

Two famous ones from the past were the plane (with picture) found on the surface of the Moon

http://elitedaily.com/news/world/britis … ight-moon/

and one that has since become the standard by which all other such stories are measured : "Freddy Starr ate my hamster!"

http://en.wikipedia.org/wiki/File:Freddiehamster.jpg

One of the problems of having a free press.

Bob

Simplest solution:

I thought of this early on but rejected it as not working. Now I can see it does work. Shame I didn't have more confidence back at post 50 or so.

If there are no restrictions then the number of orders is 6! = 720.

Now look at the guardian pairs like this

G1 G2 C1 G3 C2 C3 and

C1 G2 G1 G3 C2 C3

In this pair of solutions four of the positions are repeated and I have just swapped G1-C1

Now introduce the guardian first restriction. For every pair like the above, one solution is allowed and the other isn't. So divide by 2.

This reduces the number of solutions to 720/2 = 360.

Further reduce by 2 and 2 to removed the 'not-allowed' C2-G2 pairs and the C3-G3 pairs.

Final number of solutions = 360 /2 /2 = 90.

Bob

hi Georgeyyyyyyyyyyyyy

I'm having a lot of difficulty making sense of the question.

she remembers that in 1961 she needed $3 more for each £1

More than what?

she is now receiving £40 more for each $100 exchanged

Again, more than what?

Here's my attempt to make an equation.

If the exchange rate is $d for £1 then $100 will become £(100/d)

In 1961 it took $3 more to get a £ so the exchange rate would be $(d+3) to make one £1. So $100 got £ 100/(d+3) in 1961.

So if she is now receiving £40 more that means 100/d = 100/(d+3) + 40

=> 100(d+3) = 100d + 40d(d+3)

This will simplify to the required equation.

Hope that helps.

Bob

hi bobbym,

Thanks. I just missed them when trying to find all the possibilities. That adds 3 x 2 x 1 x 3! = 36 more and so I also now get 90.

Bob

G1 G2 G3 C1 C2 C3

G1 G3 G2 C1 C3 C2

G2 G1 G3 C2 C1 C3

G2 G3 G1 C2 C3 C1

G3 G1 G2 C3 C1 C2

G3 G2 G1 C3 C2 C1

G1 G2 G3 C1 C2 C3

G1 G3 G2 C1 C3 C2

G2 G1 G3 C2 C1 C3

G2 G3 G1 C2 C3 C1

G3 G1 G2 C3 C1 C2

G3 G2 G1 C3 C2 C1

G1 G2 G3 C1 C2 C3

G1 G3 G2 C1 C3 C2

G2 G1 G3 C2 C1 C3

G2 G3 G1 C2 C3 C1

G3 G1 G2 C3 C1 C2

G3 G2 G1 C3 C2 C1

G1 G3 G2 C1 C3 C2

G2 G1 G3 C2 C1 C3

G2 G3 G1 C2 C3 C1

G3 G1 G2 C3 C1 C2

G3 G2 G1 C3 C2 C1

G1 G3 G2 C1 C3 C2

G2 G1 G3 C2 C1 C3

G2 G3 G1 C2 C3 C1

G3 G1 G2 C3 C1 C2

G3 G2 G1 C3 C2 C1

G1 G3 G2 C1 C3 C2

G2 G1 G3 C2 C1 C3

G2 G3 G1 C2 C3 C1

G3 G1 G2 C3 C1 C2

G3 G2 G1 C3 C2 C1

I did do it by hand. If the 'correct' answer is 90, then please give one element that is missing from my set.

Bob

Ok. Here's my solution set:

LATER EDIT: Due to carelessness, I missed a whole load of solutions. There should have been 36 more making the answer 90. See post 72 for the missing ones. My thanks to bobbym for helping me with this.

G1 C1 G2 C2 G3 C3

G1 C1 G2 G3 C2 C3

G1 C1 G2 G3 C3 C2

G1 C1 G3 C3 G2 C2

G1 C1 G3 G2 C2 C3

G1 C1 G3 G2 C3 C2

G1 G2 C1 C2 G3 C3

G1 G2 C1 G3 C1 C3

G1 G2 C1 G3 C3 C1

G1 G2 C2 C1 G3 C3

G1 G2 C2 G3 C1 C3

G1 G2 C2 G3 C3 C1

G1 G3 C1 C3 G2 C2

G1 G3 C3 G2 C1 C2

G1 G3 C3 G2 C2 C3

G1 G3 C1 C3 G2 C2

G1 G3 C3 G2 C1 C2

G1 G3 C3 G2 C2 C3

G2 C2 G1 C1 G3 C3

G2 C2 G1 G3 C1 C3

G2 C2 G1 G3 C3 C1

G2 C2 G3 C3 G1 C1

G2 C2 G3 G1 C1 C3

G2 C2 G3 G1 C3 C1

G2 G1 C2 C1 G3 C3

G2 G1 C2 G3 C1 C3

G2 G1 C2 G3 C3 C1

G2 G1 C1 C2 G3 C3

G2 G1 C1 G3 C2 C3

G2 G1 C1 G3 C3 C2

G2 G3 C2 C3 G1 C1

G2 G3 C2 G1 C1 C3

G2 G3 C2 G1 C3 C1

G2 G3 C3 C2 G1 C1

G2 G3 C3 G1 C1 C2

G2 G3 C3 G1 C2 C1

G3 C3 G1 C1 G2 C2

G3 C3 G1 G2 C1 C2

G3 C3 G1 G2 C2 C1

G3 C3 G2 C2 G1 C1

G3 C3 G2 G1 C1 C2

G3 C3 G2 G1 C2 C1

G3 G1 C3 C1 G2 C2

G3 G1 C3 G2 C1 C2

G3 G1 C3 G2 C2 C1

G3 G1 C1 C3 G2 C2

G3 G1 C1 G2 C2 C3

G3 G1 C1 G2 C3 C2

G3 G2 C3 C2 G1 C1

G3 G2 C3 G1 C1 C2

G3 G2 C3 G1 C2 C1

G3 G2 C2 C3 G1 C1

G3 G2 C2 G1 C1 C3

G3 G2 C2 G1 C3 C1

Bob

I'm doing that now. But I've only got 48. Have to take a break and then have another look.

Bob

Yes, But it just goes to show my devotion to MIF.

Bob

ps. As it's only 54, I'm tempted to write out all the solutions. Then you'll have something to agree / disagree with.

hi

I'm getting 54.

Like this:

(On paper I keep muddling Gs and Cs because of my writing so I'm using A for the guardians and C for the children.)

First interview must be a guardian, let's say Ax. 3 ways.

Hereafter I'll list the interview sequence:

Second interview

AxCx (3 x 1)

AxAy (3 x 2).

Third interview:

AxCxAy (3 x 1 x 2)

AxAyCx (3 x 2 x 1)

AxAyCy (3 x 2 x 1)

Fourth interview:

AxCxAyCy (3 x 1 x 2 x 1)

AxCxAyAz (3 x 1 x 2 x 1)

AxAyCxCy (3 x 2 x 1 x 1)

AxAyCxAz (3 x 2 x 1 x 1)

AxAyCyCx (3 x 2 x 1 x 1)

AxAyCyAz (3 x 2 x 1 x 1)

Fifth and sixth interview:

AxCxAyCyAzCz is forced (3 x 1 x 2 x 1 x 1 x 1 = 6)

AxCxAyAzCyCz (3 x 1 x 2 x 1 x 1 x 1 = 6)

AxCxAyAzCzCy (3 x 1 x 2 x 1 x 1 x 1= 6)

AxAyCxCyAzCz is forced (3 x 2 x 1 x 1 x 1 x 1 = 6)

AxAyCxAzCyCz (3 x 2 x 1 x 1 x 1 x 1 = 6)

AxAyCxAzCzCy (3 x 2 x 1 x 1 x 1 x 1 = 6)

AxAyCyCxAzCz is forced (3 x 2 x 1 x 1 x 1 x 1 =6)

AxAyCyAzCxCz (3 x 2 x 1 x 1 x 1 x 1 = 6)

AxAyCyAzCzCx (3 x 2 x 1 x 1 x 1 x 1 = 6)

Total = 9 x 6 = 54

Bob

hi Agnishom,

Thank you for clearing up a puzzle for me. When I try to start a 'chat' session with Amazon there is always one person ahead of me in the queue and their problem seems to take ages to resolve. Now I know why.

You could only use EM to get an estimate on delivery by repeating the order many times and collecting data. To avoid the high cost of the experiment here are some bounds for you, obtained by theory rather than EM:

Further information on this can be found at

http://tardis.wikia.com/wiki/End_of_the_universe

Hope that helps.

Bob

hi Arna

I also am confused by what you have written. Maybe you are using the wrong symbols?

= equal

Two numbers are equal if they are the same.

eg.

: is used to show a ratio.

eg. If there are three times as many children in a room as adults then children : adults = 3 : 1

you wrote:

>y=z>y:z

This line is incomplete. You have said that y equals z implies the ratio of y to z is ?????

you wrote:

x=y

x=zSubstituting x, y, and z

2=4

2=3

If the numbers are equal then you cannot have different numbers for x y and z.

I'll go back to the original problem.

curved area : total area = 1:2

implies curved area : curved area + top and bottom = 1 : 2

implies curved area : top and bottom = 1 : 1

This means they are equal so

Bob

You are welcome.

If you ever find out how this question was expected to be done, please let me know.

Bob

Eeeekkk! Don't do that. Without your answer, I wouldn't have thought I could do the problem.

Teamwork!

Bob

hi phro,

Funnily enough it was your answer of 45 that made me think properly about this one. It's got loads of irrationals from the roots, but somehow they must all cancel out and leave a simple number. So I started to think seriously about how that might happen.

If you times each term by a fraction thus

then the denominators simplify to just 2. Thus the roots are moved to the numerators and with a minus sign now instead of a plus. It's a bit like rationalising a surd.

Now put all over the common denominator of 2, and you get a load of cancelling like this

The only terms that don't cancel are -root(100) at the start and + root(10000) at the end. As both have integer roots that does it.

Bob

hi thedarktiger

If pq + qr + rp = 20 (correct so far) you need to divide by three for the average.

Bob

1) Use the difference of two squares thus:

Do that for every term, cancel all but the first and last, and it comes out (as 45, thanks Phro )

4)Let that expression = N

Then

Using the quadratic formula you want b^2 - 4ac to be a perfect square.

So you just need to find the biggest t (under 10000) that gives such a perfect square.

Bob

hi thedarktiger

Wow! I see your algebra has come on a lot.

1) and 4) are beyond my powers but others will hopefully jump in on those and give you (and me) an idea what to do.

2) I don't have a super number cruncher for this, so I'd have to slog through elimination. Do-able but tedious.

3) Arhh! One I can contribute to. If you expand (p+q+r)^2 you'll find you can re-arrange this so that pq+qr+rp is the subject and the other side of your equation can be substituted with numbers. So it's easy from there.

Bob

Might have an idea for 2). I'll be back if it works.

LATER EDIT: Yes it does.

Add all five equations and simplify to get the value of a+b+c+d+e

Now subtract the first from the second, substitute in for a+b+c+d+e and you'll get the value of e.

Do similar tricks with subtractions of other pairs of equations to get other letters.

hi Niharika,

When I was taught this it was called Newton's experimental law. If two balls collide with velocities u1 and u2 before the impact and v1 and v2 afterwards, then the coefficient of restitution is given by

and the law of conservation of momentum is

In real life 0 < e < 1. But the kinetic theory of gases assumes that for molecules e = 1.

For the impact of the ball with the ground, e must be 1 for otherwise the ball will not climb back to its original height. As we are not told the value of e for the impact with the block, I am assuming it, also, must be 1. There's no way to do the problem without knowing e, because it affects the velocity that the ball bounces off the block.

By taking e = 1 we get

Substituting the given velocities (and taking right movement as positive):

So the ball bounces back with a horizontal component of velocity of 3v/2.

Alternatively, what happens if e = 0.5, say.

So the ball bounces back with a horizontal component of velocity of 7v/8.

So you can see the value of e is critical.

That's the best I can do with the information given in the problem.

Bob

hi Dawn0388

Welcome to the forum.

The x-y system of coordinates was invented by the mathematician Descartes.

http://mathworld.wolfram.com/CartesianCoordinates.html

But you don't have to use x and y at all. For example to draw a graph of velocity changes with time, you might have 't' across and 'v' up.

You'll find more about coordinates at

http://www.mathsisfun.com/data/cartesia … nates.html

Bob

hi Al-Allo

Yes, those answers look good to me.

Bob

hi Niharika,

Any information about 'coefficient of restitution' ? ie. How 'elastic' are the collisions ?

I could assume it's the same for the block and the floor and hope it will cancel out I suppose.

More thinking: If e = 1 for perfectly elastic collisions, then the ball would come back to its starting position without A moving at all. As A is moving, this suggests that the energy lost during the collisions is exactly the same as the energy imparted to the ball by the block. Hhhmmmm.

LATER EDIT:

Ok. I've thought about it. Let's just think about the vertical motion first.

The initial velocity is zero and the ball accelerates under gravity. When the ball hits the wall the vertical velocity is unaffected because it is parallel to the plain of the block and perpendicular to the motion of the block. So we can just treat the vertical motion without worrying about the block at all.

So use

to get the time to the bottom.

Now the ball will only rise up to the start point if no energy is lost on impact with the ground, so the collision must be perfectly elastic (e=1). This means the time for the ball to rise up is the same as the time to drop, so you can get the total time by doubling.

Now for the horizontal motion. At impact with the block, the velocity is still V and the block is moving at -V/4. We aren't told the mass of the block so no momentum equation is possible. But we can use the elasticity equation. We'll have to assume e=1 again as we aren't told anything else. So

You'll need to be careful with negatives here, but you should be able to get the velocity of the ball after the impact (assume block is still -v/4).

So now you can get the total time for the two horizontal motions and hence get an equation for x in terms of everything else.

Bob

hi bobbym,

It looks like you've got the same woman dancing with two men?

This is what I did (before I researched it).

Get all 30 into the ballroom. Expel everyone else. (except the band of course).

Line the men up against the wall. (So they cannot escape if they don't like their partner )

Man one can have any of 15 partners. Once chosen, man two can choose from 14, man three from 13 etc etc and man fifteen gets the last woman.

Let's simplify.

Say three men Al, Bob and Colin and three women, Di, Elaine and Fiona.

Al-Di; Bob-Elaine; Colin-Fiona

Al-Di; Bob-Fiona; Colin-Elaine

Al-Elaine; Bob-Di; Colin-Fiona

Al-Elaine; Bob-Fiona; Colin-Di

Al-Fiona; Bob-Di; Colin-Elaine

Al-Fiona; Bob-Elaine; Colin-Di

That's 3 x 2 x 1 = 6 possibilities.

Bob

Sorry, that took a while. I had to find a four variable solver on-line. I eventually found

http://www.zweigmedia.com/RealWorld/simplex.html

which gave b = 0, w = 350, m = 0 and e = 150.

How does that seem?

Bob

hi Fiona,

Yes, those look good to me. So give me a moment to get an solution.

Bob