Squared twice?

  x -> x^2^2 = x^4

Doubled and squared?

  x -> (2x)^2 = 4x^2

Difficulty factor of 2 is squared?

  x -> (2^2)x = 4x

Or was that squared twice?

  x -> (2^2^2)x = 16x

All of these functions agree at x = 0, so can we conclude that it's gonna be way easy??

Lesson: don't handwave your math when you're talking to lay journalists, because nerds are gonna read it and be horrified.

>> So, four times as difficult?

Or, by "twice squared" they mean to say "It has been squared ... twice". That would be 16x, which is probably more in line with the total deltaV requirements.

Well, if the original difficulty is 1, and we square it twice, that is 1^2^2, which ... is still 1.

Did you just linearize squaring?

"not just twice ...; it's twice squared"

So "twice"(2) and "twice squared" (x^2^2) = 2^2^2 = 16

Where are you getting the third two to put in for x? Your number would be phrased as "twice squared twice."

Read "not just twice" as "not only twice".

Then you can read "it's twice squared" as (the first use of twice) + ("twice squared")

The 'it' reference can change. The first 'it' is referencing the difficulty. The second 'it' is referencing the first 'twice', creating the option that 'it' is further "twice squared". Not the most elegant interpretation but a possible interpretation.

Learning to avoiding this vague language is why math majors should still take the occasional poetry class.

English is a strange language. "Twice squared" could be interpreted both as "Twice as hard, squared" or "Twice as hard, squared two times."

Think of the sentence, "A man, twice fooled, shall not be fooled again."

If x is the difficulty, the phrase either means “twice the difficulty, squared”: (2x)^2 or 4x^2.

Or “the difficulty squared and squared again”: (x^2)^2 or x^4.

The GP is asking why x == 2 in the second example rather than just “the difficulty”. English interpretation doesn’t explain that.

Or (twice as difficult)^2 which is 4x for difficulty 1, but in general 4(prior difficulty squared)x as difficult.

it was a test .. you're hired

> compile error: figure of speech is not a number

I see the urge to avid the tired cliche of "exponential", but being creative has risks.

Or maybe (2n)2 as difficult?

Fun fact I realised only recently: 2 ∘ 2 = 4 for any operator ∘ in the Ackermann sequence other than the zeroth (successor) operator.

2+2 = 2×2 = 2² = … = 4

At least, unless I messed up my proof. My highest qualification is two A-levels in maths and half of brilliant.org…

https://kitsunesoftware.wordpress.com/2020/07/27/hᵤ2-2-4-for...

Tangential to the topic of this post, but how have you found brilliant? I’ve been considering it but don’t know if it’s worth adding another subscription to my monthly spend when I could presumably get a lot of the same knowledge from YouTube (albeit in a less structured way). Any thoughts?

Plus:

Brilliant’s computer science courses cover slightly more than my actual software engineering degree from Aberystwyth university, and their maths courses cover more than I did at A-level.

The various introductions to matrices (in different contexts each time), are better than the introduction I had in my A-levels (with a good teacher!), and I would say they are of comparable goodness to 3blue1brown. Only, all of it is that good.

If you get stuck because a topic assumes knowledge you don’t yet have, they have a course to teach you what you need to learn.

I can’t comment on the levels of Brilliant’s physics, chemistry, or biology as I have only GCSEs in those subjects — but I can say they’re well-described and engaging.

Negatives:

Limited practice materials, e.g. I still suck at probability because I’m not groking the broader implications of their simpler lessons.

This is neat. I was taking a Trigonometry course online. Was great at first but I kept running it references to “as we learned in Algebra or Geometry” Well dang it, I aced trigonometry in college, but apparently I have completely forgotten some basic building blocks. So I need to start back at basic algebra and work my way up to find the missing gaps.

Thanks for replying! Sounds like I should try it out. My maths could use a refresher and I could use an engaging way to learn it. Much appreciated.

I figured this out when I discovered hyperoperators and spent a good day or so telling everyone "two and two are always four!"

Pretty sure it's just a metaphor, and not literal rocket surgery.

Even in the most obnoxiously nerdy interpretation, it would be (2*n)^2, not 2^2.

Mmm, easier than I thought so.

twice = (* 2) squared = (^ 2)

twice . squared = (* 2) . (^ 2)

(difficult * 2)^2