0

## Quick Feats

Published on Thursday, February 28, 2013 in , ,

In many of my posts, I tend to teach feats in great detail.

For a change of pace, today I'll be teaching 3 quick feats that can be learned fairly quickly.

LIGHT OR DARK CHESSBOARD SQUARES? Most chess games today are described in algebraic notation today. Given any square, A1 through H8, and without looking at a chessboard, can you tell me whether the given square is light or dark?

This is surprisingly easy once you get the approach down. First, instead of thinking of the squares as black and white, think of them as light and dark. This is important, as light has an odd number of letters, and dark has an even number of letters. This will be important later.

When given a coordinate in algebraic notation, ask yourself whether the letter is in the word CAGE. If the letter isn't in the word CAGE, you can ignore the letter completely, and just focus on the number part. If the letter is in the word CAGE, simply add 1 to the number, and then ignore the letter.

At this point, you'll have either an odd or even number. If the number is odd, then the given square is light (which has an odd number of letters). If the number is even, then the given square is dark (which has an even number of letters).

For example, is C7 light or dark? Since C is part of the word CAGE, we add 1 to the 7, giving us 8. 8 is an even number, so the square is dark. What about F3? F isn't in the word CAGE, so just ignore it. That leaves us with 3, which is an odd number, so the square is light.

You can have Wolfram|Alpha generate random practice squares for you, and use this diagram to check your answers.

SPEED OF A PASSING CAR? Here's a quick trick for mentally determining the speed of a passing car.

When you see in your rearview mirror that a car in another lane is about to pass you, keep driving at a steady speed, and quickly identify a landmark (lightpost, crosswalk, parking lot entrance, etc.) not too far ahead of you. When the car is even with you, begin counting at a steady rate, starting at 1. When the other car passes the landmark, remember the counting number (let's call this number a), and continue counting. When your car passes the same landmark, stop your count, and remember the final number in your count (we'll call this number b).

Once you have the numbers a and b, all you have to do is divide b by a, and then multiply that times your steady speed to get the speed of the other car.

For example, while driving at a steady speed of 35 mph yesterday, I saw a car coming up fast, and chose a lamppost up ahead as a landmark. When this car passed, I started counting one one-thousand, two one-thousand..., and noted that the other car reached the lamppost on a count of 5, while my car reached it at a count of 8. 8 ÷ 5 = 1.6, and 1.6 × 35 mph (my steady speed) = 56 mph (his speed), so I can estimate the other car was traveling about 56 miles per hour.

I originally ran across this bit of mental math on a now-defunct forum, and have found it to be very useful.

PRECISE SQUARE ROOTS OF NON-PERFECT SQUARES: This one is only quick to learn if you've already practiced and mastered my Estimating Square Roots feat. With just a minor alteration, you can actually give a more precise answer, using what are known as continued fractions.

The process starts the same as in the original feat, right up to the point where you determine the dividend by doubling the integer part of the fraction and then add 1. In this approach, you're going to stop the process before adding 1. You're then going to repeat that same fraction in the manner of a continued fraction.

To better explain this, let's try finding the square root of 261 using both approaches. In the estimation version, you would work out in your heard that the square root is approximately 16533. In this new version, you'll start writing a similar fraction, but don't add the 1. At first, all you would be writing in this new version is 16532 (note the 32 instead of the 33). Take the fractional part, 532, and add that same fraction to the bottom of the current fraction, and keep doing this over and over until you run out of room and/or your writing becomes too small. The result should look like this:

$16\frac{5}{32+\frac{5}{32+\frac{5}{32+\frac{5}{32+\frac{5}{32+\frac{5}{32+\frac{5}{32+...}}}}}}}$

At this point, you can explain that this continued fraction, carried out to infinity, gives the exact square root of 261. Continued fractions are usually expressed with a 1 as each numerator, so the above approach is actually a generalized continued fraction.

With help from Gauss' notation for generalized continued fractions, it could also be entered into Wolfram|Alpha in this manner. Note that, when squared, this returns the exact given number!

This continued fraction approach is only the beginning! Over at Ted's Math World, Ted Muller explains how to create a wide variety of continued fractions for the same given number. If you prefer standard fractions, he also shows you how to create increasingly accurate fractions, as well.

That's enough quick feats for now. I hope you find them useful!

2

## Your Mind In Prime Condition

Published on Sunday, February 24, 2013 in , , ,

When the largest known prime number was discovered recently, computers were used to verify that it was a prime.

While humans won't be working out whether numbers with millions of digits are prime in their head anytime soon, it is possible to quickly and easily test whether smaller numbers, such as those 1,000 and under, are prime. Learning how to do this in your head will also give you a better appreciation for the process used to find larger prime numbers.

BASICS: To be able to determine whether any give number from 2 to 1,000 is prime, you'll need to able to estimate the square root of anumber from 1 to 1,000, as taught here. When checking for primes, you can ignore the fractional part, and just focus on the whole number.

So, if you were a given a number such as 253, all you would really need to know for this feat is that the square root is above 15. This number will serve as the upper limit of the numbers you need to check. The lower limit, of course, is 2.

With our 253 example, does that mean we'll need to check whether 253 is divisible by every number from 2 to 15? No. In fact, we'll only need to check the prime numbers from 2 to 15 in this case: 2, 3, 5, 7, 11, and 13. Any of the remaining numbers above 2 can be made by multiplying 2 or more of those prime numbers, so checking the others is redundant.

So, your first step is to determine the square root, and then realize you'll need to check divisibility for all the prime numbers from 2 to the square root.

Another thing to remember that will speed up this feat is that we're not going to go through the full process to divide a given number by each of these prime numbers, but rather use simple tests to determine whether the given number is divisible by each number. For example, we don't need the answer to 253 ÷ 7, we just need a simple way to say whether 253 is evenly divisible by 7 or not.

We're not looking for an exact mathematical answer each time, just a series of yes-or-no answers up to a limit. If, at any time, we find that the given number is evenly divisible in one of our tests, we can stop there and definitively say that the number is NOT prime. If the given number isn't evenly divisible by any of our test numbers, we can definitively say that it IS prime.

CHECKING AT A GLANCE: The first 3 test can be done almost at a glance.

Is the rightmost digit a 0, 2, 4, 6, or 8? If so, it's not prime, unless the number is 2.

Is the rightmost digit a 5 or a 0? If so, it's not prime, unless the number is 5.

Just with a quick peek at the last digit, you can already tell whether it's divisible by 2 (the first test) or 5 (the second test). Testing for divisibility by 3 is also easy, as explained in the video below:

At this point, you might expect a test for divisibility by 7. I'm going to skip over the test for 7 now, but I will come back to it shortly. For now, we're going to test for divisibility by 11.

DIVISIBILITY TEST FOR 11: The divisibility test for 11 is surprisingly simple. You start by breaking the given number into two smaller numbers, one number consisting of only the rightmost digit, and the other number consisting of all the remaining numbers. Using our 253 example, we'd split into the numbers 25 and 3.

The next step is to treat those two numbers as a subtraction problem, and ask whether the result is evenly divisible by 11. If so, the original number is evenly divisible by 11.

Going back to 253, we work out that 25 - 3 = 22, which is evenly divisible by 11. Therefore, we know that 253 is also evenly divisible by 11, and therefore not prime.

DIVISIBILITY TEST FOR NUMBERS ENDING IN 1: With 1 minor alteration, this break-up-and-subtract approach can be expanded to cover almost any number ending in 1.

Let's take the next number ending in 1, 21, and show you how to test for its divisibility. For 21, take the lone leftmost digit, and multiply it by 2, then subtract. If the result is evenly divisible by 21, then the original number is evenly divisible by 21.

Even though we;ve already proved 253 isn't prime, let's try this test out on it anyway, to show you how it works. We break 253 into 25 and 3, then multiply 2 × 3 to get 6, then perform 25 - 6 to get 19. 19 isn't evenly divisibly by 21, so we know that 253 isn't evenly divisible by 21.

For 31, you'd muliply the lone digit by 3 before subtracting, for 41 you'd multiply the lone digit by 4 before subtracting, and so on. Effectively, removing the 1 shows you what number you'd multiply by for that number's divisibility test. Is a number divisible by 91? Multiply the lone number by 9 and subtract! Is a number evenly divisible by 121? Multiply the lone number by 12 and subtract! And so on.

This is an amazingly simple and useful pattern, and it can even be extended to numbers ending in odd digits without much difficulty.

DIVISIBILITY TEST FOR NUMBERS ENDING IN ODD DIGITS: You've probably already noticed that we're only testing for primes divisors, so we shoudln't need to test for divisibility for 21 itself, as 21 isn't a prime number. Since 21 is equal to 3 times 7, this same test can also be used to test for divisibility for 7. The steps are the same, except you ask whether the result is evenly divisibly by 7 instead of 21!

We've already determined that 253 isn't prime, so as a new example we'll test whether 347 is prime.

347 is bigger than 182 (324), but smaller than 192 (361), so we only need to test divisibility by the primes from 2 to 18.

Is 347 evenly divisible by 2? No, it's an odd number. Is 347 evenly divisible by 5? No, it doesn't end in a 5 or a 0. Is 347 evenly divisible by 3? 3 + 4 + 7 = 14, and 14 isn't evenly divisible by 3, so neither is 347.

Is 347 divisible by 11? 34 - 7 = 27, so 347 isn't evenly divisible by 11. Is 347 evenly divisible by 7? 34 - 2 × 7 = 34 - 14 = 20, and 20 isn't evenly divisible by 7, so 347 isn't evenly divisible by 7.

With just a few quick tests, we've already eliminated divisibility by 2, 3, 5, 7, and 11. We still need to test for divisibility by 13 and 17, so how do we do that?

Just as we used the test for 21 as a divisibility test for 7, we can use a similar approach to quickly find an appropriate divisibility test for other numbers ending in odd digits (except for 5, as we already have a simpler divisibility test for it).

Which multiply of 17 ends in 1? We have 17, 34, 51 - there we go! We can use the 51 test as a divisibility test for 17! Since it's 51, we have to break up the numbers as we did before, multiply the lone digit by 5, then do the subtraction, and ask if the result is evenly divisible by 17.

347 split becomes 34 and 7, just as before, but now we multiply that lone digit by 5 and subtract: 34 - 5 × 7 = 34 - 35 = -1. -1 isn't evenly divisibly by 17, so we've just established that 347 isn't evenly divisible by 17.

For 13, we have to go a bit farther: 13, 26, 39, 52, 65, 78, 91 - there it is! That means we have to multiply the lone digit by 9 before subtracting, and then ask whether the result is evenly divisible by 13.

For 347, this test becomes 34 - 9 × 7 = 34 - 63 = -29. That result, -29, isn't evenly divisible by 13, so 347 isn't evenly divisible by 13, either.

At this point, we've established that 347 isn't evenly divisibly by 2, 3, 5, 7, 11, 13, or 17, which are all the prime numbers in our established range of 2 to 18. This means we can say that 347 is indeed a prime number. Double checking with Wolfram|Alpha, we verify that 347 is a prime number.

With numbers up to 1,000, you'll also need divisibility tests for 19, 23, 29, and 31. We've already talked about 31, so that test is simple enough. 19 becomes 171, so you multiply the lone digit by 17 before subtracting. With 23, you use 161 (23 × 7), so the lone digit is multplied by 16, and with 29, you use the test for 261 (29 × 9), and therefore multiply the lone digit by 26 before subtracting. The BEATCALC website has a shortcut for multiplying by 26 which can be helpful.

FULL PROCESS: As a worst-case scenario, let's test the number 977. The square root is more than 31, but less than 32, so we'll need to test all the prime numbers from 2 to 31.

You should be able to tell at a glance that 977 is not evenly divisible by 2, 3, or 5.

Next, we move up to the break-up, multiply, and subtract tests. 97 - 7 = 90, so it's not evenly divisible by 11. We can also try 97 - 2 × 7 = 97 - 14 = 83 to establish it's not evenly divisibly by 7, and 97 - 3 × 7 = 97 - 21 = 76 to establish it's not evenly divisbly by 31.

97 - 5 × 7 = 97 - 35 = 62, which establishes it's not evenly divisible by 17. 97 - 9 × 7 = 97 - 63 = 34, and 34 isn't evenly divisibly by 13.

With a quick mental run-through, we've established 977 isn't evenly divisible by 2, 5, 3, 11, 7, 31, 17, and 13, in that order. It's an unusual order, until you get used to walking through that order.

The only remaining tests are the ones for 19, 23, and 29, which are the challenging ones. Remember, we use 16 to test for divisibility by 23, 17 to test for divisibility by 19, and 26 to test for divisibility by 29. This also means you should be comfortable mmultiplying any 1-digit number by 16, 17, and 26.

97 - 16 * 7 = 97 - 112 = -15, which isn't evenly divisibly by 23. 97 - 17 * 7 = 97 - 119 = -22, which isn't evenly divisible by 19. 97 - 26 * 7 = 97 - 182 = -85, which isn't evenly divisible by 29.

We've just run through every prime from 2 to 31, and found that 977 isn't evenly divisble by any of them. This means that 977 is prime, and Wolfram|Alpha verifies this for us.

SHORTCUTS: If you're worried about dealing with negative numbers, as often happens in the larger divisibility tests, you can simply do the multiplication needed, and then subtract the larger number from the smaller. For example, when testing 977 to see if it's evenly divisible by 29, you could think of the numbers as 97 and 26 * 7, working out the latter as 182, then performing 182 - 97 = 85. The sign doesn't matter in the divisibility tests, so you can always use this strategy.

If you're not comfortable working with the tests for 19, 23, and 29, you can avoid having to check for those by asking for a number from 1 to 350, instead of 1 to 1,000. That way, 17 is the largest divisibility test you'll need to perform.

NerdParadise.com has more on divisibility tests and testing for primes, if you wish to explore these patterns further.

Even if you don't perform this feat, I hope you read through it enough to understand each part of the process. That way, you'll have a better appreciation for the work required of computers when testing for extremely large primes.

0

## Bringing Pi Digits to Life

Published on Thursday, February 21, 2013 in , , ,

It's one thing to be able to memorize 400 digits of Pi. It's another thing altogether to be able to make the feat memorable and interesting for you audience.

Sure, 400 is an impressive number of Pi digits to memorize, but what do 400 digits really mean? What kind of detail are we talking about?

Numberphile, who has done numerous Pi videos already, has recently released a new video about Pi and the size of the universe, that starts to give you an idea of the sense of Pi's scale:

It almost seems strange that so few digits should be able to take us from the width of a hydrogen atom all the way up to the diameter of the universe. Each place in Pi (as in the tenths, the hundredths, the thousandths, and so on) represents a place that's smaller than the previous one by a factor of 10. There's a classic 1977 film called Powers of 10 that does a wonderful job of dramatizing just how few power of 10 are needed to cover the entire scale of the universe:

This film may look familiar, either because you may have seen it in school, or you've seen one of its parodies, such as in Men In Black, Contact, or The Simpsons.

A good way of keep thing in mind is Tim Rowett's poem, Space:

Seven steps each ten million to one
Describe the whole space dimension
The Atom, Cell’s girth
Our bodies, the Earth
Sun’s System, our Galaxy – done!

– Tim Rowett, Three Limericks – On Space, Time and Speed
BetterExplained.com's article on how the digits of Pi were determined in ancient times gives you a sense of scale in a different manner, and is done so with their usual startling clarity.

As I've mentioned before, memorization and understanding together give you a more complete picture than either could ever do separately.

2

## Yet Again Still More Quick Snippets

Published on Sunday, February 17, 2013 in , , , , , , ,

This month, we're going to delve into math and memory techniques you may have thought were too dificult to develop. With sufficient practice, however, they become powerful additions to your mental toolkit!

• One of the main reasons people want to improve their memory is so they can recall names and faces. This appears difficult to many people, because of the social pressure involved, and the apparent difficulty of connecting a name with the face. As USA Memory Champion Nelson Dellis will show you, it's not as difficult as you may think:

• Another memory skill that comes across as impressive is memorizing the order of a shuffled deck of cards, especially when you can do it in under 60 seconds. Over at the Four-Hour Work Week blog, they have a wonderfully vivid tutorial on memorizing the order of a shuffled deck. They use the easy-to-understand analogy of a software purchase. They start your new brain software off will a trial version they call “Bicycleshop Lite,” where you get the basic process down of memorizing shuffled cards. Once you've done that, you're ready for “Bicycleshop Pro,” which improves your speed. Need some incentive to learn this feat? They're offering \$10,000 to the first person who masters it from their tutorial!

• For those who have mastered squaring 2-digit numbers, you might have wondered about taking numbers to higher powers in your head. To do that, you'll need to develop a few other skills. First, you should know the binary equivalents of the numbers 2 through 10 from memory, as well as getting comfortable squaring 3-digit numbers (Video tutorial: Part 1, Part 2, Part 3). Being able to multiply 3-digit numbers by 1-digit numbers is also helpful.

Once you develop those skills, the following video will teach how to bring them together to take any small number to any small power in your head:

• Multiplying numbers by themslves repeatedly is one thing, but how about multiplying any 2 numbers together in your head, up to, say, 7 digits? YouTube user Joesph Alexander has a series of tutorials on how to develop your mental multiplication skills to this level. He starts by teaching how to handle 2- to 4-digit numbers (presentation, explanation), then moves you up to 5-digit numbers (presentation, explanation).

When you're comfortable with doing those type of problems in your head, you're ready to move up to 7-digit numbers (presentation - shown below, explanation):

Try picking just one of these skills to develop, and you just may amaze yourself at how far you can go!

0

## Love Poems For Valentine's Day

Published on Thursday, February 14, 2013 in , , , , ,

Happy Valentine's Day!

In honor of Valentine's Day, I thought I'd share a few favorite love poems. This might seem like a strange turn to many regular Grey Matters readers, but I've actually discussed poetry quite a bit on this blog.

Let's start with one of the most classic love poems of all time, “How Do I Love Thee? (Sonnet 43)” by...William Shakespeare, right? Surprise! The author is actually Elizabeth Barrett Browning.

Poetry can be an excellent way to improve your mind, in terms of both memorization and understanding. Memorizing a poem, which you can learn to do here, helps you carry it with you, and offers more time to think and ponder the meanings involved. Having a poem in your mind is a big advantage over leaving it on a page in a book. Shmoop.com's poetry section has wonderful guides to classic poetry. Their analysis of “How Do I Love Thee? (Sonnet 43)” is a great example, as it gives you background, summaries, and plenty of other information to help develop a better understanding.

This is love poetry as it is generally imagined - written in a loving, longing tone, full of rich, romantic imagery. That isn't a hard and fast rule, however. A more unusual tone can often help a love poem stand out, such as the darker tone of Edgar Allan Poe's “Annabel Lee”:

Shmoop.com's analysis of “Annabel Lee” starts with a great analogy to music. Once you've been pulled in by his greatest hits, like “The Raven”, it's fun to discover the intensity of this tale of love lost.

Love poetry can also be funny. If you read the text of “Root of Three”, it would sound more like a math equation than a love poem, but watch how the delivery makes a startling difference (NSFW due to language):

I've included this poem not just because it's a good example of tone, but because I know Grey Matters readers often like at least a little mathematics in the post. Fans of this type of love poem might also be tickled by “Wolfram|Alpha, Be My Valentine”.

May your Valentine's Day be filled with love, humor, and thoughtfulness!

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## New Largest Known Prime Number Discovered!

Published on Sunday, February 10, 2013 in , ,

The biggest news in math this week is, of course, the new discovery of the largest known prime number.

Whenever math makes the news like this, many people's eyes glaze over and they say things like, "So what?" Well, let's take a look at what this discovery can actually mean.

There's few better places to review the basics of prime numbers than betterexplained.com. It's a short article, so take the time to go over it all.

Marcus du Sautoy, whom you probably know from his Story of Maths documentary, vividly describes primes as being the atoms of mathematics in the following interview:

The prime number that everyone is talking about this week is a special type of prime known as a Mersenne prime (note the connection to perfect numbers), and is only the 48th one ever discovered. To wrap this post up, here's the numberphile video on the 48th Mersenne prime itself:

0

## Leapfrog Division

Published on Sunday, February 03, 2013 in , , ,

Today, I'm going to get you playing with numbers in a strange yet fun way. I'll save the destination for a surprise, so trust me and play along, and I think you'll like where we wind up.

Even in blogs and other resources that cover math shortcuts thoroughly, division doesn't get a lot of attention, simply because few shortcuts exist. Let's see what we can discover about division, shall we?

STARTING POINTS: Remember when you were first learning about division? First, they taught you about problems that came out evenly, such as 8 ÷ 2 = 4.

Next, they probably introduced you to problems that didn't come out evenly, and taught you about “remainders” They would give you a problem such as 13 ÷ 2, and you were expected to give the answer as “6 remainder 1.” That's the kind of division we're going to work with today. Always think about your answers as “x remainder y” for the purposes of this post.

Even if the reminder is 0, for this post, you'll need to think of the answer as including “x remainder 0” for the practices you'll be taught in this post.

“LEAPFROG” DIVISION: Let's return to the earlier example of 13 ÷ 2 = 6 remainder 1. Write down the quotient (the first number in the answer), the 6 in that example, and keep the remainder (1 in this example) in mind.

Here's the bizarre type of math to which I referred earlier. Imagine the remainder, 1, leapfrogs from its place on the left of the quotient (the 6) and lands to the quotient's immediate left. This would now look like the number 16, wouldn't it?

Let's try dividing that number by 2. What is 16 ÷ 2? It's 8. Don't forget, though, I want you to include the remainder in all answers, even if the remainder is 0. Better stated, 16 ÷ 2 = 8 reminder 0.

Once again, write down the quotient (the 8 in this last problem), and then imagine the remainder (0) leapfrogging over to the left of the quotient.

If we keep repeating this process, we wind up with a list of numbers like this:

• 13 ÷ 2 = 6 (remainder 1)
• 16 ÷ 2 = 8 (remainder 0)
• 08 ÷ 2 = 4 (remainder 0)
• 04 ÷ 2 = 2 (remainder 0)
• 02 ÷ 2 = 1 (remainder 0)
• 01 ÷ 2 = 0 (remainder 1)
• 10 ÷ 2 = 5 (remainder 0)
• 05 ÷ 2 = 2 (remainder 1)

...and so on. It's not hard to imagine that you could take this leapfrogging process as far as you like. For many numbers, but not all, the sequence will eventually repeat at some point.

Remember, you've only been writing down the quotients. Let's take a look at the sequence of quotients from the above series of problems: 68421052. Actually, we could have stopped earlier or later in that process, so it's probably more effective to think of that sequence as a decimal number: 0.68421052.

Let's put that decimal number into Wolfram|Alpha and see what comes up. Probably the most interesting result returned by Wolfram|Alpha is the first one in the Possible closed forms pod, wher it reads 1319 = 0.6842105263157. We started with a 13, so it's very interesting that a fraction with 13 would show up. Is this a coincidence?

Let's pick up from the last division we did above, and see where it takes us:
• 05 ÷ 2 = 2 (remainder 1)
• 12 ÷ 2 = 6 (remainder 0)
• 06 ÷ 2 = 3 (remainder 0)
• 03 ÷ 2 = 1 (remainder 1)
• 11 ÷ 2 = 5 (remainder 1)
• 15 ÷ 2 = 7 (remainder 1)
So, the whole sequence of quotients we've gotten from dividing 13 by 2 gives us 6824105263157 and 1319 = 0.6842105263157. After 13 decimal places, it's looking a lot less like coincidence.

DECIMAL PRECISION: This is actually a reliable division shortcut most identified with performing lightning calculator Alexander Craig Aitken (1895-1967), who used it when dividing by numbers ending in 9.

When given a problem such as 1319 to convert to decimal, when the divisor/denominator (the bottom number) is 9, simply round up the divisor/denominator to the nearest multiple of 10, and ignore the 0. For 1319, you'd mentally change this to 1320, then to 132, and proceed via the leapfrog division approach above.

As it turns out, this doesn't just work for 19, but for ANY number ending in a 9! If you're dividing by 29, you'd round up to 30, and focus on division by 3 repeatedly. When dividing by 39, you round up to 40, and do your leapfrog division by 4, and so on with higher numbers.

Naturally, if the dividend/numerator (the top number) is larger than the divisor/denominator, you'll want to reduce the number to a mixed fraction, first. For example, 6729 should be reduced to 2929 first, and then worked out as taugh above. Let's try that:
• 67 ÷ 29 = 2 + (9 ÷ 29) = 2.????? (do leapfrog division with 9 ÷ 3)
• 9 ÷ 3 = 3 (remainder 0)
• 03 ÷ 3 = 1 (remainder 0)
• 01 ÷ 3 = 0 (remainder 1)
• 10 ÷ 3 = 3 (remainder 1)
• 13 ÷ 3 = 4 (remainder 1)
• 14 ÷ 3 = 4 (remainder 2)
• 24 ÷ 3 = 8 (remainder 0)
• 08 ÷ 3 = 2 (remainder 2)
• 22 ÷ 3 = 7 (remainder 1)
So, if we did this correctly, 6729 should equal roughly 2.310344827 Wolfram|Alpha tells us that, yes, this is roughly correct! If you're rusty on working with mixed fractions, Math Dude has some wonderful refresher courses in the form of free podcasts: What Are Mixed Fractions?, How to Turn Mixed Fractions Into Improper Fractions, How to Add and Subtract Mixed Fractions, and How to Multiply and Divide Mixed Fractions.

Thanks to this approach, if you're comfortable dividing by, say, 8, then you can now feel just as comfortable dividing by 79. In other words, if you're comfortable dividing by a number n in your head, you now know how to easily divide 10n - 1, as well. If you can divide by 12, think how impressed people will be when you divide by 119!

TIPS: This trick can also help with numbers ending in 1, 3, and 7, as well. for numbers ending in 9, multiply both the dividend/numerator and divisor/denominator by 9. For numbers ending in 3, multiply both numbers by 3, and for numbers ending in 7, multiply both numbers by 7.

For example, if you need to work out 813, simply change the fraction to 2439, and proceed from there. What's 617? It's the same as 42119, and if you can mentally divide by 12, you already know how to do this!

Few numbers tend to be evenly divisible by numbers ending in 9, so the decimal equivalents tend to go on forever (usually repeating at some point). Due to this, you can take the leapfrog division process out to as many or as few places as you deem appropriate. If you're presenting this against a calculator, you'll generally want to go beyond the number of places that can be displayed by the calculator.

Take a look at the Mental Division tutorial in the Mental Gym, especially where the numbers ending in 9 are concerned, and you can learn further shortcuts for specific numbers ending in 9, such as 9 and 99.

Practice this, and you'll have an impressive new ability to demonstrate for your family, friends, and any teachers you may know!