Given that:

   1              1
------------- - -----
√(22-10-2022)    √(22)

= 0.00....00ABCD...(N zeros).
Accordingly, exactly N zeros immediately follow after the decimal point, and:
ABCD are base 10 digits, whether same or different, with A being nonzero.

Determine the value of N.

*** As an extra challenge, solve this puzzle without using computer program or excel solver.

The first term is such a small deviation from the second, it’s an excellent candidate for approximation by Taylor expansion. 

1 / √(a + d) ~= 1√a - 1/2 * d * 1/√(a**3) plus a term of order d**2

Where in this case, a = 22 and d = 10**-2022

Since we’ll then subtract 1/√a, the remainder is just  - d / 2√(a**3), and the next term in the expansion, since it’s proportional to d*2, can’t possibly impact the next 2000 digits, let alone the next four.

We just need an accurate approximation of 1 / 2√(a**3)  = 1/ 2a√a = 1/44*√22

Now we may as well use the same trick to estimate root 22. √22 ~= √(25 - 3) ~= 5 - 3 /10 ~= 4.7  with an error of at most 3^2 / 6*5**3 = 12/1000 or just under 1%

4.7 * 44 = 206.8, and we need the reciprocal of that, which clearly lies between 1/200 (0.005 = 5 * 10**-3) and 1/250 (0.004 = 4 * 10**-3).  Combined with the factor of 10**-2022 from d, and the value must lie between 4 * 10**-2025 and 5 * 10 **-2025.

As long as we’re doing Taylor expansions, we may as well use them to approximate 1/206.8. That’s 1/(200 + 6.8) ~= 1/200 - 6.8 / 200**2 or 0.005 - 68 / 400,000. 68/4 = 17 so a little subtraction gives 0.00483 = 4.83 * 10**-3

Now, the number of zeros to the right of the decimal is one less than the absolute value of the exponent in scientific notation, so this value has 2024 zeros immediately to the right of the decimal, followed by 4 and then 8, but maybe not 3 since our overall error is at best 1%. And indeed, the 3 is really a 4, but this still agrees with Charlie’s exact response to within 0.4%

Posted by Paul on 2022-03-09 17:40:36