## Posts tagged ‘computation’

### Mental Math and the MCWC

How long would it take you to find the sum of 2 two-digit numbers?

What about 3 three-digit numbers?

Or 4 four-digit numbers?

Okay, let’s get really crazy… how long would it take you to find **the sum of 10 ten-digit numbers**?

You can decide whether you’ll do the calculation in your head, on a calculator, or with paper and pencil. Your choice.

With a calculator, it took me **91 seconds** to find the sum of 10 ten-digit numbers.

Without a calculator, the winner of the Mental Calculation World Cup 2014 needed only 242 seconds to complete 10 problems in which participants were asked to add 10 ten-digit numbers. On average, that’s just **24 seconds** to do in his head what took me a minute-and-a-half with technology.

**Holy smokes!**

Competitors at the MCWC do a number of mental calculation tasks. The following exercises will give you an idea of the computations that they do.

**Exercise 1.** Sum of 10 ten-digit numbers.

**Exercise 2.** Multiplication of 2 eight-digit numbers.

**Exercise 3.** Square root of a six-digit number.

**Exercise 4.** Day of the week for a calendar date.

**Exercise 5.** Multiplication of 3 three-digit numbers.

These sample exercises are taken from the examples in the MCWC 2014 Official Rules.

If you were able to complete all five of those exercises in, say, less than 3 minutes, then you might be ready for MCWC 2016. (Note that for Exercise 3, you needed to be accurate to within 5 × 10^{-6}.)

But if you’re like me, you’ll probably want to skip the competition and keep your calculator close at hand.

### Exponentially Smarter, Literally

To show my sons what Siri can do, I asked her (it?) the following question:

What is 6 + 4?

Siri told me, “The answer is 10.” But she also provided a bunch of other information pulled from Wolfram Alpha, including the following data:

This data appears to be taken from dissertation research by B. A. Fierman which was furthered by psychologist Mark H. Ashcraft. What it shows is that we get exponentially smarter — or at least faster at calculating — as we get older.

According to Excel, this data can be modeled exponentially by *y* = 8.36 · *e*^{–0.129x}, though this model has obvious limitations. For example, it implies that a one-year-old would be able compute this sum in 7.35 seconds, yet I know no one-year-old who understands addition. Further, it claims that it would take me 0.03 seconds to compute the sum, but I would argue first that I don’t compute the sum, I merely recall it; and second, my reaction time when asked for the sum would be greater than 0.03 seconds.

Playing around with the generic function *y* = *ab*^{x} + *c* using the world’s best graphing calculator from Desmos, I found a model that may approximate the data a little better:

*y* = 57 · 0.65^{x} + 0.9

With this model, it would take a one-year-old 37.95 seconds to compute sum. That’s still not reasonable for any one-year-old that I know, but at least the model says it would take me 0.9 seconds to recall the fact, a far more reasonable estimate than the 0.03 seconds given by the Excel model above.

Interestingly, How To Geek claims that Siri uses Wolfram Alpha for 25% of its searches. Yet if you ask Siri, “What is the meaning of life?” it will respond,

I can’t answer that right now, but give me some very long time to write a play in which nothing happens.

or

Try and be nice to people. Avoid eating fat. Read a good book every now and then, get some walking in, and try to live together in peace and harmony with people of all creeds and nations.

On the other hand, if you ask Wolfram Alpha, “What is the meaning of life?” it will respond,

42.

Proper.

All this talk of exponentials reminds me of a joke.

Q: How do you know that your dentist studied algebra?

A: She tells you that candy will lead to exponential decay.

Perhaps the most famous joke about exponentials is not one of which I’m terribly fond. I share it here only to honor my mission of providing math jokes to the world, not because I think any of you will enjoy it.

Several functions are sitting in a bar, bragging about how fast they go to zero at infinity. Suddenly, one hollers, “Look out! Derivation is coming!” All of the functions immediately cower under the table, but the exponential function sits calmly on the chair.

The derivation comes in, sees the exponential function, and says, “Don’t you fear me?”

“No, I’m

e,” says the exponential confidently.^{x}“That’s all well and good,” replies the derivation, “but who says I differentiate with respect to

x?”