Reply to: Chef's Quick Brain Teaser, revisited

[OwlHoot]

Brolly old son, you're making a big meal out of this. I think most if not all of us grasped the idea, namely that you were asking for the predecessor number in Chef's sequence based on the recurrence x_(r+1) = x_r * (x_r + 1).

[BrollyBonce]

1. Chef asked what was next in the series 1, 2, 6, 42, 1806, ? The answer appeared later in that thread.

Is it really the case that none of you use the "first try and solve the number on the left to get the formula - the numbers are usually smaller so you can do it quicker and in your head - then use that to determine the number on the right" technique?

That would have had you immediately thinking "What number, when multiplied by [ itself plus one ] gives the result one?"

That should have had bells ringing and got you thinking of the Golden Ratio: the number that is its own reciprocal plus one, of course.

2. conned_tractor asked where the series 1, 2, 1.5, 1.666, 1.6, ... converges. There was some controversy about the question, but the answer was a special number known as the Golden Ratio.

With the Golden Ratio implied in the last question, I had presumed this had actually triggered the second question.

Although there are always an infinite number of polynomials for any number sequence, the famous Fibonacci sequence was obviously the relevant factor here (Occam's Razor).

Since the sequence tends toward the Golden Ratio, it seemed obvious that had triggered conned_tractor's question.

Those questions are actually related and I thought EO or sasguru or NickFitz or someone would spot the relationship and comment on it. They didn't, so I gave a clue in the form of a question:

3. "In the context of the first question in this thread, what is the significance of the number 0.6180339887498949......"

... it is the number before the first number...

That should have prompted someone to say, "Hey, not only is that number rhubarb regarding question 1, it is also rhubarb regarding question 2!"

4. When you have answered question 3 (and not before), why is that number significant to question 2 and, therefore, how are questions 1 and 2 related?

... and it is the muliplicand that requires the Golden Ratio multiplier.

Come on you clever people. What is the answer to questions 3 and 4?

It really was that simple.

I concur. I'll take that gun, if I may.

[BrilloPad]

Is there a "hanging oneself" smilie available?


I'll settle for this

HTH

[expat]

<expat's>
which is the same as
<Bagbuss's earlier>

I agree a million percent.




x(n+1) = xn * (xn + 1)

so xn**2 + xn - x(n+1) = 0

when x(n+1) = 1,
xn = [ -1 +/- sqrt(1 + 4) ] /2
positive root = [ -1 + sqrt(5) ] /2 = 0.618 etc

I wasn't going to bother going through with it: the maths is only school-level but the notation is painfully ponderous in plain text.

[Bagpuss]

They are algebraically identical. However Churchill's shows the rule that is probably in mind to generate the series.

x(n+1) = xn * (xn + 1)

which is the same as

it should be X(n)=X(n-1)*[X(n-1) +1]

[TimberWolf]

If x1 = x0 * (x0 + 1)

then x0 * (x0 + 1) = x1

Where x1 = 1 calculate x0 and put it in the series.

[1] x1 = (1 + x0)x0, so x0 = ( -1+-sqrt(1-4x1) ) / 2
[2] x1 = (1 + x0)/x0, so x0 = 1/(x1-1)

[EternalOptimist]

They are algebraically identical. However Churchill's shows the rule that is probably in mind to generate the series.

x(n+1) = xn * (xn + 1)

Ah , but now observe


n = x * (x+1)
n = (x*x) + x

therefore

x * (x+1) = (x*x) +x

therefore

x = sqrt((x*(x+1)-x)


I refer to this as 'The Golden Rivet' (Which, colloquially, is an eight incher)

[expat]

All I can see in question 1 is

x = x * (x + 1);

I thought it was

x = (x * x) + x

They are algebraically identical. However Churchill's shows the rule that is probably in mind to generate the series.

x(n+1) = xn * (xn + 1)

[SallyAnne]

Is there a "hanging oneself" smilie available?


I'll settle for this

[ASB]

Another HUGE hint.

What number came BEFORE in the series 1, 2, 6, 42, 1806, ...

I.e., what comes before the '1'?

If you take the obvious solution for the series then yes - and that would then define the series as one of the possible solutions. See OwlHoots link for other solutions that are valid with that sequence.

[Bagpuss]

Told you I was simple

and apparently unable to read ()'s corectly.....

BODMAS mate BODMAS

Where's CUK's self proclaimed maths whiz?

He reads maths books as a hobby so maybe he should be setting the questions

[DaveB]

0*1 = ?

Told you I was simple

and apparently unable to read ()'s corectly.....

[Bagpuss]

Well to my simple mind, if the formula above was correct then if X1=1 X0 = 0

0*1 = ?

[oracleslave]

I thought he already told us that.

Welcome to CUK does GCSE Maths

Where's CUK's self proclaimed maths whiz?

[DaveB]

Another HUGE hint.

What number came BEFORE in the series 1, 2, 6, 42, 1806, ...

I.e., what comes before the '1'?

Well to my simple mind, if the formula above was correct then if X1=1 X0 = 0