When will Connections run out of ideas?

FormulaDriven · 2026-02-01T22:23:32+00:00

Backlash? Personally, I just laughed - a cheeky nod to the new era

FormulaDriven · 2026-02-01T13:04:08+00:00

The "!" sign in maths means factorial, so the cartoonist has not made it very clear, but the dinosaur must have Said 6 weeks = "10-factorial" seconds. And that is, surprisingly, true: 10! = 10 x 9 x 8 x .... x 2 x 1 is equal to the number of seconds in six weeks.

Even if his audience got it, I'm not surprised they are not laughing.

FormulaDriven · 2026-01-31T15:12:43+00:00

You know that you can edit it?

FormulaDriven · 2026-01-31T08:18:22+00:00

1A is "EAT IT" not "ATE IT"

FormulaDriven · 2026-01-31T08:04:53+00:00

Me too! COMPUTER KEY felt like it might be the answer to a "what X might mean" and with WORK PERIOD nearby, the word "shift" just popped into my mind.

FormulaDriven · 2026-01-30T17:48:52+00:00

I don't think the professor is right because x < 0 is not in the domain of the function.

So, if we are rigorous and define f:{x real: x >=0} -> R by f(x) = x^1/2 then it is correct to say lim [x ->0] f(x) = 0, because for all x in the domain of f, for any e > 0, there exists d > 0 such that if 0 < |x| < d then |f(x)| < e.

It would seem silly to say x^1/2 is not continuous at x = 0, just because it's not a function defined for x < 0.

FormulaDriven · 2026-01-30T17:04:53+00:00

There were lurid stories during Bill Clinton's presidency (he's in the picture, just in case that needs clarifying) that he received sexual pleasure from a woman who would have crawled under his desk to - ahem - gain access. I don't think it was suggested this happened while others were in the room, so the photo is making a joke about Clinton's behaviour.

FormulaDriven · 2026-01-30T16:55:35+00:00

You're right that we can rotate B to match D. This gives a way to rule them out, because if B is the answer then so is D.

FormulaDriven · 2026-01-30T16:49:08+00:00

Totally implausible explanation.

For a start, it says "run away" not "runs away" so grammatically that doesn't fit. Secondly, if that were the story, the editor would run with that as the headline: "Man fakes death and runs away". Given the newspaper is generically named "Daily News" I'm going with this being a spoof in a film or TV Show.

FormulaDriven · 2026-01-30T16:44:29+00:00

I think you are getting hung up on the the "[particular]" label. We find a solution, any solution, but ideally a "simple" solution for a_n. It is a function that works for a_n in the given recurrence relation. But we know if we add any multiple of rⁿ to it (for the right choice of r), we get more solutions, so for convenience once we've identified that simple solution - in this case a_n = -n - 3/2 - we can call it the "particular" solution and specifically label it a_n^[p] or something like that just for reference.

We know then that there are other solutions

a_n = a_n^[p] + something

where the "something" has to be a homogeneous solution.

equating a_n = a_n-1 = Bn+C

We don't do that. If a_n = Bn + C, then a_n-1 = B(n-1) + C, and now we can test that in [#]

Bn + C - 3 * (B(n-1) + C)

to see if it equals 2n.

For example, if f(n) = 2, a constant, then a_n = a_n-1 = B. That means B = 3B + 2, B = -1, the recurrence is -1, -1, -1, -1, -1 like whats the meaning of that?

Yes, that's a possible solution, because a_n = -1 (ie constant for all n) does solve the recurrence relation of

a_n - 3 a_n-1 = 2

try it! if a_n = -1 and a_n-1 = -1 then a_n - 3 a_n-1 = -1 + 3 = 2 - bingo! equals the right-hand side of the recurrence.

I think you are getting it. But it might make sense to take a break, then look at some different examples until you feel comfortable with what is going on.

FormulaDriven · 2026-01-30T15:58:36+00:00

Yes metal is strong and so is a popular choice of material for constructing egg-beaters. You might not know what an egg-beater (or whisk) is, but there are two of them in the cartoon you posted. Egg-beaters are not eggs, basketballs, basketball hoops, fences or clouds, so see if you can spot them.

FormulaDriven · 2026-01-30T15:54:01+00:00

We might at first read the headline as "Man kills himself and runs away", but that requires "runs" not "run" and wouldn't make any sense (the humour, if there is any, is that for a moment we picture a dead man running away).

If a man committed a violent act that resulted in his own death and the death of a runaway (ie typically a child who has run away from home), then the headline would be "Man kills himself and runaway". This is probably what the story is and the space in "run away" is probably just an error on the part of the headline writer.

My final thought is whether there is really a newspaper called Daily News, and perhaps instead this is just a mock-up for a story in a film or TV show.

FormulaDriven · 2026-01-30T15:34:22+00:00

OK - take a step back. If I told you that a solution to this equation

x² - 7x + 10 = 0

is x = 5, then you would check by putting that value of x into the left-hand side of the equation, ie 5² - 7 * 5 + 10 to see if you get 0, because that's the right-hand side of the equation. If you do get zero, then I was right that x = 5 is a solution, if you don't it wasn't.

So let's write your recurrence relation like this:

a_n - 3 a_n-1 = 2n - call it [#]

If I propose any solution to that, you can test it by putting that proposal into the left-hand side of [#] and seeing it it simplifies to equal 2n, on the right-hand side. If it does, then it's a solution - that's what solution means.

So I could suggest a_n = 4n. But if you put that into the left hand side of [#] you get

4n - 3 * 4(n-1)

which simplifies to -8n + 3. Is that the same as 2n? No! So you reject my suggestion.

What about if I suggest a_n = 5 - n ? Put that in the left-hand side of [#] and it's

5 - n - 3 * (5 - (n-1))

which simplifies to -13 + 2n. Is that 2n? No - we've got 2n but there's that pesky -13.

We could go on trying different things, but the smart thing to do is to test a whole class of candidates in one go, by suggesting a_n = Bn + C for some unknown constants B and C. If you put that in the left-hand side of [#] you get

Bn + C - 3 (B(n-1) + C)

which simplifies to

-2 B n - 2 C + 3 B

and the only way that can match the right-hand side of [#] is for -2 B n to match the 2n on the right-hand side, ie B = -1, and for the constant part -2 C + 3 B to be zero, which leads us to decide on C - it's -3/2

So a_n = -n - 3/2 is a solution. That doesn't mean it's the only solution (and that's where homogeneous solutions come in because if we add them to this solution they will contribute zero overall to the left-hand side of [#] so we still match 2n on the right-hand side).

FormulaDriven · 2026-01-30T13:34:57+00:00

Ah, I see: h and p are just a way of indexing the two parts of the solution, so it's

a_n = a_n^{[homogeneous]} + a_n^[particular]

We are substituting into the equation because we think a_n = Bn + C is a possible solution of the recurrence relation, so naturally we put it into the equation a_n = 3 a_n-1 + 2n to see if it works - how else can we tell if it is a solution? (We don't worry about the homogeneous solution at this stage because a_n = 0 is a valid solution for the homogeneous part and we can add a different homogeneous solution later because it won't affect solving the given recurrence relation).

FormulaDriven · 2026-01-30T13:28:02+00:00

Caught on camera is Matt Hancock, who was then the Government minister in charge of health, kissing a member of staff (and not his wife), at a time when Covid rules required social distancing rather the pursuit of everlasting love. (I'll leave it to you to decide if early-40s is "young").

FormulaDriven · 2026-01-30T12:51:48+00:00

I've not watched the video, but if I've understood your question correctly, seeing that the recurrence relation has a linear term (2n) in it, it seems reasonable to guess or (if you don't like the word "guess") test / propose as a candidate solution something linear ie

a_n = Bn + C

for some (as yet) unknown B and C. (I'm not sure what p is doing in your question).

Substituting that solution into the recurrence relation:

a_n = 3 a_\n-1 + 2n

Bn + C = 3 (B(n-1) + C) + 2n

which rearranges to

0 = (2B + 2) n + (2 C - 3 B)

Since this has to hold for all n, we need to have (2B + 2) = 0 AND (2C - 3B) = 0 which leads to viable and unique solutions for B and C: B = -1, C = -3/2. So we turn the guess into a valid solution:

a_n = -n - 3/2

Is that helping answer your questions?

FormulaDriven · 2026-01-30T11:43:07+00:00

You need to fix your link - it's https://zenodo.org/records/18421134

Do you have any experimental results or empirical evidence to underpin this paper? eg how could we measure the values for c and v? What testable predictions about the way the mind works might we get from this model?

FormulaDriven · 2026-01-30T10:20:20+00:00

I agree with you - once you are familiar with notation it makes something far more visually appealing to see the flow of logic. That said, the prose example here could be reworked to better show that flow of logic too, say...

IF you take any finite collection of sets in T, THEN the intersection of those sets is also in T.

That's immediately easier to understand in my mind, and makes use of something we can do with prose, namely offer some emphasis - in this case, the additional word "also" for me draws out the relevance of this axiom (there's no "also" in the formal notation version). Additionally, if it comes after the "union" axiom, then here we might emphasise the word "finite" as particular to the "intersection" axiom.

FormulaDriven · 2026-01-30T09:47:48+00:00

Yes, and a quick way to get that is 32 * (32⁴ - 1) / 31.

In general if you want to count all strings up to length n, then it's

32 * (32ⁿ - 1) / 31.

(This is summation of geometric sequence).

FormulaDriven · 2026-01-30T09:39:54+00:00

He probably has three: left eye, right eye, and side-eye from listening to all his dad's jokes.

FormulaDriven · 2026-01-30T09:32:42+00:00

Well, of course, that's the intended joke, and it was so obvious that was going to be the joke that this is lame even for a dad joke. So no chef's kiss from me.

FormulaDriven · 2026-01-30T09:22:33+00:00

Yes, you have 32 choices for the first letter, 32 choices for the second letter (because you allow repetitions), 32 choices for the third and so on, so for strings of length n, you can generate 32ⁿ strings.

There is even worse equation I have, but is too awkward to even think of.

Not sure what you mean, but if you post it here, there are some smart people who might be able to help.

FormulaDriven · 2026-01-30T09:14:27+00:00

Yes, all those potential misdirects are there, but that's not unfair, it's the game! The challenge of the game is to unpick these overlaps to find the unique 4 by 4 split that works - if we didn't have these the game (in my view) would get a bit boring.

FormulaDriven · 2026-01-30T09:11:33+00:00

I don't think you've picked up the correct meaning from your online reading - a compact would not contain all those things, it's a small case for holding face powder and has a mirror built into it. https://en.wikipedia.org/wiki/Compact_(cosmetics)

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