27
votes
26
votes
The computer can't do anything I couldn't do with pen and paper
Not sure if this counts as "pencil-and-paper" but certainly counts as "no computer..."
$$
3^{100} = 10^{100 \log 3}
$$
We can compute this logarithm using a slide rule. The D and L ...
- 16.2k
14
votes
The computer can't do anything I couldn't do with pen and paper
Let's solve this problem using ... music!
Because of this,
But also,
- 423
14
votes
The computer can't do anything I couldn't do with pen and paper
I took an extra challenge to solve this in my head without writing.
From here, we can use that
Now
which is quite close to the true value.
- 23.8k
11
votes
The computer can't do anything I couldn't do with pen and paper
This seems like a perfect application for
Looking at
Now for the inverse operation:
Bringing it all together,
Error analysis:
- 849
9
votes
The computer can't do anything I couldn't do with pen and paper
Simple approach... I don't find a simple way to quantify the error:
- 792
8
votes
The computer can't do anything I couldn't do with pen and paper
Second answer, using exponentiation by squaring and one (!) digit of precision:
$$
\begin{align}
3^{1} &= 3 \\
3^{2} &= 3^{2} = 9 \\
3^{4} &= 9^{2} = 81 \approx 8 \times 10^{1} \\
3^{8} &...
- 16.2k
6
votes
The computer can't do anything I couldn't do with pen and paper
Pretty similar to other solutions:
$3^{10} = 9^5$
$9^n = 9, 81, 729, 6561, 59048$. (Can be done on paper, or even in the head)
$3^{10} = 59048 \approx 6 \times 10^4$
$3^{100} \approx 6^{10} \times 10^...
- 22.7k
6
votes
The computer can't do anything I couldn't do with pen and paper
Note first of all that
Well,
Now
At this point
The errors here are at most about
so comfortably within the requested factor of 2.
- 112k
4
votes
3
votes
The computer can't do anything I couldn't do with pen and paper
I already posted one approach. Several others are possible. Here's one that gives a pretty accurate result. It's similar in spirit to my other answer.
Now
So
We can already
but
- 112k
3
votes
The computer can't do anything I couldn't do with pen and paper
Short and rough:
$3^5=243\approx 250 = 10^3/2^2$
So
$3^{100}\approx (10^3/2^2)^{20} = 10^{60}/2^{40}$
But $2^{10}=1024\approx 10^3$
So $2^{40}\approx 10^{12}$
Hence $3^{100}\approx 10^{48}$
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3
votes
How many digits does $7^{100}$ have?
Once again, I solved this in my head, without writing or looking anything up.
We use the approximation
and also rewrite and estimate
to get
which is quite close and gives the right digit count of
- 23.8k
2
votes
1
vote
How many digits does $7^{100}$ have?
A simple upper bound:
And a semi-simple lower bound:
Or, borrowing an answer from the previous problem: If you have
then you can determine that
which produces an estimate of
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