This thread is born from a remark
Jaybarti made in a recent discussion:
If you took the complete timeline of life on earth and put it to a 24 hour clock, we probably only came into existence in the last minute on that 24 hour clock.
How do you people deal with complex computations, when you don't have a calculator at hand (or accept the challenge of not using one)? It seems everyone have their own tricks and shortcuts, and I'm curious about yours. I'm not a mental computation specialist, although I'd rate myself above average, and I just love smart math quizzes.
Finding the correct solution to this 'clock' case is actually rather trivial (no real trick involved). Here's how I did it (reasoning broken down to its 'atomic' components - note there aren't any complex computations):
Assuming life is 3.5G years old, and Homo Sapiens emerged 200,000 years ago -
3.5G = 3.5 * 10^9
200,000 = 2 * 10^5
3.5 / 2 = 1.75
9 - 5 = 4 (comparing the powers)
1.75 * 10^4 = 17500
I.e., the life history of Homo Sapiens is 1/17500th of the history of life on Earth.
Now, an hour is 60 * 60 * 24 seconds = 86400 seconds. Actually I didn't remember the number and had to mentally compute it as -
3600 * 24 =
36.. * 2.
36.. * .4
---------
72...
144..
---------
864..
Now to the final result -
17500/86400 = 17.5/86.4 (I find it easier to compare and approximate 2-digit numbers)
17.5 * 5 = 35 + 35 + 17.5 = 70 + 17.5 = 87.5 which is superior to 86.4 (the actual ratio is 1/4.937...)
Thus my reply to Jaybarti:
make that minute less than 5 seconds.
OK, this one was pretty straightforward and dull. Here's a much more interesting example: I once was in a train with two college students doing their math homework. They had only one pocket calculator to share. One of them asked:
"9 to the power of 4?"
"Wait a minute, have to finish with that first..."
My brain instantly clicked:
9^4 = (9^2)^2 = 81^2 = (80 + 1)^2
(a + b)^2 = a^2 + b^2 + 2ab
Thus (80 + 1)^2 = 80^2 + 1^2 + 2*80*1
64..
...1
.160
----
6561
It took about the same time to compute and give the result, than it took them to check using the calculator. These two guys were much better at math than I ever was, but until I explained the whole thing, they thought they were sitting close to a math genius. It was sort of a shock for them to realize I needed only a pretty basic formula, 8*8, 9*9, and a simple addition (without carry digits). Actually I was just as stunned myself.
Any similar trick or story to share? If not, math quizzes that are best solved with outside-the-box thinking?