[medvezhenok]

I understand the argument, but I think you're missing the nuance somewhat. There are a series of things that are learnable mental tricks; I have read Moonwalking with Einstein and am well aware about rhyme techniques, memory palace techniques, etc. I memorized ~250 digits of pi in the 6th grade, so I'm also aware of techniques for that. I wouldn't consider either of those a domain of savants.

(sidenote - I would be impressed with the people that could memorize millions of digits of Pi, given that the world record is either 70,000 digits or ~110,000 digits last time I checked (depends on the source), and it takes ~6 hours just to recite that many digits)

I'm talking specifically like things like Hypercalculia: https://en.wikipedia.org/wiki/Hypercalculia , which are documented feats that cannot be explained by "tricks". Usually people with savant syndrome also have co-occurring autism and other neurological conditions like synesthesia.

Here is a an ABC profile of one of these savants: https://abcnews.go.com/2020/autistic-savant-daniel-tammet-so...

I don't think you could learn a "trick" to compute 27^7 in a few seconds.

[wizzwizz4]

Funny you should give 27^7 as an example, because I actually did get good at powers of 2 and 3. 27^7 is 3^21 is 3×81^5, which is easy to calculate in your head if you're good at multiples and powers of 8: it's just binomial expansion with the next row after 1 4 6 4 1, i.e. 1 5 10 10 5 1. (I used to be able to directly recall 1 5 10 10 5 1 and 1 6 15 20 15 6 1, but this is literally the first time I'm doing non-trivial mental arithmetic in a decade.) Multiplying a power of 2 by 5 is the same as halving and multiplying by 10, which reduces the problem to a simple addition of digits of small powers of 2 (^0 = 1, ^2 = 4, ^6 = 64, ^9 = 512, ^11 = 2048, ^15 = 32768), then a multiplication by 3 – both of which are easy to perform in a streaming fashion, if you have a suitable delay line. (I use the auditory processing delay line ("echoic memory"), which would probably work better if I spoke a language like Mandarin, where all digits have the same syllable length – but I got by. Some find the mental abacus more reliable, but I have no training in this approach.)

I only memorised my powers of 2 up to 2^16, powers of 3 up to 3^5, and powers of 5 up to 5^5, because the part that made it fun was only memorising things I'd calculated myself, in my head, and this was only an occasional game. If my goal had been "develop the skill of quick arithmetic", I would've memorised the first 12 powers of every prime below 12, and my times tables up to 100×100 – but I resented times tables, so never really memorised them until I (briefly) got really into division.