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Good morning. Today I am going to test your mental arithmetic.

To do so I shall ask you to give me the sum of a series of ones. I shall say

One and one and one and one and one and one, etc.

Of course you won't know ahead of time when I will stop.

I shall speak rhythmically in time with a metronome and lead you in with my conductor's baton so you know when to start.

As you know, I have already tested you and I know that you cannot carry more than one syllable in your head at a time at the speed I will go.

I know that you are a concert pianist and so have remarkable finger control and can think independently of your finger movements. I know that your only language is English and that you have an excellent education in all aspects of that language. I also know you are quite knowledgeable about computer theory.

You will not have time to devise a new and wonderful system so you must use your current knowledge and skills to the best effect.


Question

Assuming that the tester runs a series of tests and keeps increasing the number of ones each time, how far can the pianist get in this test just using one-syllable mental arithmetic, plus her current knowledge plus her fingers?

She is allowed to do a few minutes of calculation when the tester has stopped speaking.

Thumbs are counted as fingers. No other parts of the body are allowed.

Assume an unlimited amount of mental and physical stamina. This person can play whole concerts of long pieces after all! (and probably practices for 40 hrs per day - [See note])

Options

  1. 1 - 10
  2. 11 to 100
  3. 101 to 1,000
  4. 1,001 to 10,000
  5. 10,001 to 100,000
  6. 100,001 to 1,000,000
  7. More than that.

Can you be even more precise?


syllable

  1. a unit of pronunciation having one vowel sound, with or without surrounding consonants, forming the whole or a part of a word; for example, there are two syllables in water and three in inferno. Powered by Oxford Dictionaries

Note: Practising 40 hours per day is an in-joke for many musicians. Consult Google.

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  • $\begingroup$ What constitutes a syllable, exactly? What sorts of representation using fingers are allowed? This seems very subjective, and the answer will depend on the precise rules you use. $\endgroup$
    – Deusovi
    Jul 10 '20 at 21:31
  • $\begingroup$ Why is this off-topic? $\endgroup$
    – Chipster
    Jul 10 '20 at 21:52
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    $\begingroup$ That doesn't say what precisely counts as a syllable. "gvprtskvni" is a single syllable in the Georgian language. Does that count? Meanwhile, Japanese only allows a consonant and a vowel, and an optional ending N sound. Even sticking to English doesn't help: is "psst" a single syllable? It's a Scrabble-legal word. What about "pssk", then? Phonotactically there's no reason the former would count but the latter wouldn't. Can you nasalize vowels to distinguish them? Can you use syllables that are legal according to English phonology but happen to not appear? $\endgroup$
    – Deusovi
    Jul 10 '20 at 22:19
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    $\begingroup$ I'm not talking about what languages they know, just what counts as a syllable. My point was that the concept of "syllable" is actually very vague, and cannot be precisely defined. You need a way to specify exactly how many syllables exist. Ditto for the hand positions -- "obviously" you can distinguish between two or three hand positions, and "obviously" you can't measure at the atomic level. But then where is the dividing line? You need to precisely draw this line, because the answer directly depends on that division. If you don't, the question is subjective. $\endgroup$
    – Deusovi
    Jul 10 '20 at 22:31
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    $\begingroup$ "I don't know," said Alice. "I lost count." $\endgroup$
    – RobPratt
    Jul 11 '20 at 17:58
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As a concert pianist, the testee will be very comfortable with subconsciously counting in $\large{^9_8}$.

We can hold the current beat we're on in our head (replacing "seven" for "sev"), and then add on a finger every time we reach a new bar.

Additionally, if we count on our fingers in binary (as opposed to the unary system we use day to day), our maximum bar number is $2047$. We need a simple way to count in binary (because on the go addition isn't really an option).

When we need to add $1$, to any digit:

  • We switch the finger from down to up, or up to down.
  • If the finger was up, we do the same with the next finger.

Furthermore, since every concert pianist is (at least) comfortable with subdivisions of $4$ at very high speeds, we can add those to our count: every time a "one" is said, we go forwards a subdivision.

This brings our maximum to $4\cdot 9\cdot 2047$, which is $73692$.


Note: This answer assumes you're counting very fast, which I assume would be the case if you got to the point where the professional pianist could only hold one syllable in their head.
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    $\begingroup$ It's worth noting that a talented pianist could most likely reach much higher than this ($^9_8$ and subdivisions of $4$ are very small in terms of professional ability), but this presents a simple solution that allows us to count very high and very fast. $\endgroup$
    – Helen
    Jul 11 '20 at 17:47
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    $\begingroup$ My daughter does in fact count in binary when she needs to count on her fingers. Including e.g. bars of rest in music. $\endgroup$
    – Gareth McCaughan
    Jul 11 '20 at 18:33
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    $\begingroup$ Wow that's pretty impressive! I thought it was a bit of a stretch to use binary finger counting but clearly it's possible $\endgroup$
    – Helen
    Jul 11 '20 at 18:35
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As a musician myself, I will count the instances of "one" in batches of 4, and move one finger down for each batch of 4. When done I know how many fingers I have down, and I know how many extra ones since the last one.

On the offchance that there are more than 40 ones, I then start again, but this time turning my feet in a certain way (I have insufficient control of my toes) and hope I can keep track of the number of "hands" of 5 fingers' worth of 4's before the sadist giving me the test finally stops.

Or am I missing something obvious?

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  • $\begingroup$ Feet aren't allowed I'm afraid - only the things I mentioned. Can you put a number to the maximum sum you can reach using your method? Could you reach 1,000 ones for example? (Assuming you have the mental stamina and don't die of boredom!) $\endgroup$ Jul 10 '20 at 21:38
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    $\begingroup$ I'd havedied of boredom before even reaching the end of the first hand. $\endgroup$ Jul 10 '20 at 21:48

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