32 votes
Accepted

The computer can't do anything I couldn't do with pen and paper

An elementary approach:
  • 4,537
32 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 ...
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19 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,
17 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.
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12 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:
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11 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:
9 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
  • 23.9k
9 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} &...
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7 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.
7 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^...
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5 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
4 votes

Can you Avoid the Spear-Wielding Gladiator?

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3 votes

The computer can't do anything I couldn't do with pen and paper

Another 80 based solution.
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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?

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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2 votes

The computer can't do anything I couldn't do with pen and paper

I learned in surely you're joking mister Feynman, that one of the most powerful tools for mental arithmetic is to know some logarithms by heart. In particular with the logarithms of 2, 3, 5, 7 and 10, ...
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