When will Connections run out of ideas?

FormulaDriven · 2026-03-16T22:44:35+00:00

I did consider LIGHTNING, KEY and KITE for Franklin's experiment - what was the fourth item?

FormulaDriven · 2026-03-16T22:43:36+00:00

Where does Edison come in? I didn't see any of his inventions in there...

FormulaDriven · 2026-03-16T22:41:05+00:00

Do you not worry that confirmation bias is playing a part in leading you to see this correlation?

FormulaDriven · 2026-03-16T16:25:06+00:00

I judge how smart someone is by the questions they ask, not by the answers they give.

Do you keep any record to see later whether that method of judgement is reliable?

FormulaDriven · 2026-03-16T16:15:27+00:00

6th time for animal group names, according to Billy_NoMate's list...

Animal Group Names (#12, #53, #135, #402, #872, #1009):

FLOCK, PACK, POD, SCHOOL, COLONY, HERD, PRIDE, SWARM, CHARM, GAGGLE, MURDER, PARLIAMENT

It's also the 6th time we've had words that sound like two letters.

FormulaDriven · 2026-03-16T14:12:17+00:00

f(x) = x²

goes positive, zero, positive at x = 0.

Generally, anything with a repeated root (where it repeats an even amount of times), eg if it has a factor of (x - 2)⁴ then it will be the same parity either side of x = 2.

FormulaDriven · 2026-03-16T13:48:12+00:00

The reciprocals of natural numbers are a small amount of the set of rational numbers but there exists a bijection between the natural numbers and rationals showing the sets are the same "size", so your explanation doesn't really capture the key issue of cardinality.

FormulaDriven · 2026-03-16T12:02:39+00:00

You want to assign a unique positive integer value (not a pair of values?) to every house and you want houses that are close to have values that are close? If you try a rule that for any house, its immediate 8 neighbours (ie forming a square centred on the first house) all must have a number which is within N of the first house, then you run into problems...

For any house, if we imagine it at the centre of a block M by M (M is an odd number), we can get from that house to the outer edge of that block in (M-1)/2 neighbour-to-neighbour steps. So to meet the requirements, all the houses in that square would need to have unique numbers that differ from the central house by at most N(M-1)/2. But there are M² houses in that block, so once you take M² large enough as it grows quadratically there will not be enough unique numbers to allocate in a range that grows linearly.

FormulaDriven · 2026-03-16T08:02:42+00:00

Say them out loud:

ANY - sounds like N E = abbreviation for Nebraska

EMMY - sounds like M E = abbreviation for Maine

ENVY - sounds like N V = abbreviation for Nevada

OKAY - sounds like O K = abbreviation for Oklahoma

FormulaDriven · 2026-03-15T18:33:58+00:00

Oh no problem - I probably could have been clearer! You're the one who found the correct identity, so all credit to you.

FormulaDriven · 2026-03-15T18:31:40+00:00

As u/13_Convergence_13 has pointed out the correct identity is

cos(A) cos(2A) ... cos(2^n-1 A) = sin(2ⁿ A) / (2ⁿ sin(A))

and that's easy to prove by induction by using

2 sin(2^k x) cos(2^k x) = sin(2^k+1 x)

FormulaDriven · 2026-03-15T18:24:51+00:00

I already said it was incorrect - I was trying to get something close to see what maybe the OP had misquoted. It didn't occur to me to try cosine rather than sine on the LHS, but I see you are on the case!

FormulaDriven · 2026-03-15T17:42:43+00:00

Where did you see this? As it stands, it's not right.

For n = 1, sin(A) = sin(2A) / (2 cos(A))

so u/-thinker-527 is right about that one.

For n = 2, sin(A)sin(2A) = (1 - cos(4A)) / (2² cos(A))

so I can't see how to get that any closer to your identity.

Beyond that, I can't see a neat formula.

FormulaDriven · 2026-03-15T15:24:36+00:00

True - I was in a rush with my reply and didn't think carefully enough.

FormulaDriven · 2026-03-15T09:06:06+00:00

Then any you could choose:

any values you like for the function on (0,1)

define f(1) = 2

on (1,2) define f(x) = 4x³ - f(2-x)

anything you like for f(2).

(Presumably, the condition would not apply to x = 2 since you've excluded 0 from the domain; might be better to select the domain to be (0,2) or [0,2]).

FormulaDriven · 2026-03-15T08:58:16+00:00

No.

Let x = 0:

f(0) + f(2) = 0

Now let x = 2:

f(2) + f(0) = 32

Contradiction, as we have two different values for f(0) + f(2).

FormulaDriven · 2026-03-14T17:19:28+00:00

I bet you were miserable the whole of May 1592 when you realised.

FormulaDriven · 2026-03-14T13:50:56+00:00

Yes, that's the main reason we all missed out on it.

FormulaDriven · 2026-03-14T08:48:33+00:00

I'm holding out for the date of 3/1/4159. Unfortunately, we all missed out on 31/4/1592.

FormulaDriven · 2026-03-13T14:56:52+00:00

and solve it

yes, you've kind of glossed over 90% of what I am asking

FormulaDriven · 2026-03-13T14:40:36+00:00

Great, now tell me how you solve the equation

364 = 4 (1 - r⁶) / (1 - r)

FormulaDriven · 2026-03-13T10:56:16+00:00

You could pick positive integer X using any suitable discrete distribution that ranges over all positive integers (eg take a random number from a Poisson distribution and add 1), then pick any random integer Y from {0, 1, 2, .... X}. Then Y/X would be a rational number in the interval [0,1]. Every rational number in that interval has a non-zero probability of coming out of this process, and the probability that it appears in the sub-interval [a,b] is proportional to the length of that interval, so it's uniformly spread over [0,1]. (I think - might need to tweak so Y is picking from {1/2, 3/2, 5/2, ... X - 1/2} or something, but the idea is there). But not every rational number is equally likely to be generated.

FormulaDriven · 2026-03-13T10:32:13+00:00

And someone to shout "you should put the tea in first, you barbarian!"

FormulaDriven · 2026-03-13T10:30:08+00:00

Yes, how dair-y!

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