The Mystery of 6174

6174. It is a number well known to many. Some say it is mysterious, but I would not go that far. It is, however, a very interesting number. Indeed, the number 6174 is also known as the Kaprekar constant, named after the Indian mathematician Dattaraya Ramchandra Kaprekar who studied the mystery behind 6174.

So what is all the hoopla about 6174? Well, first, if you arrange the digits such that you have the highest number (7641) and also the lowest number (1467), and then determine the difference between the two, you arrive at 6174 (7641 – 1467 = 6174).

Well you say, I suppose this is somewhat interesting. But now suppose you take the number 2355. Do what you did before with 6174 – rearrange the numbers to obtain the highest and lowest numbers, and then subtract the lowest from the highest:

2355 -> 5532, 2355
5532 – 2355 = 3177

Now continue the process with the result:

3177 -> 7731, 1377
7731 – 1377 = 6354

And again:

6354 -> 6543, 3456
6543 – 3456 = 3087

Continuing:

3087 -> 8730, 0378
8730 – 378 = 8352

8352 -> 8532, 2358
8532 – 2358 = 6174

As you can see, this process leads to a convergence to 6174. And as you saw before, performing this routine on 6174 results in an endless loop. Cool!

In fact, this same result will occur for any four digit number you choose as long as the number isn’t composed of the same digit (eg., 4444 would not work). You can even use leading zero’s for your four digit number (eg., 0007). Now that is interesting!

Let’s try another example, one with a lot of zeros – 3000:

3000 -> 3000, 0003
3000 – 3 = 2997

2997 -> 9972, 2799
9972 – 2799 = 7173

7173 -> 7731, 1377
7731 – 1377 = 6354

6354 -> 6543, 3456
6543 – 3456 = 3087

087 -> 8730, 0378
8730 – 378 = 8352

8352 -> 8532, 2358
8532 – 2358 = 6174

It is this process that was discovered by our Indian friend Kaprekar, and is known as Kaprekar’s routine. There are other similar numbers that are obtained for other n-digit numbers. For example, for three digit numbers, this process leads to a convergence to 495. Using the routine on two digit numbers results in an infinite loop:

9 -> 81 -> 63 -> 27 -> 45 -> 9

As an interesting experiment, you are invited to try to determine what happens if you perform this routine on higher digit numbers. You may be surprised to see what your results are!

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Happy Pi Day 2012!

I am a little early with this post, but there is good reason.  For Pi Day 2012, I made a game that you can play online, and right here!  I thought really hard about coming up with a name for it, and after mulling through many candidate names, I came up with, “The Pi Game“.  Yea, yea – the creative juices were flowing!

Anyway, you do need Flash installed in order to play it, so if you do, you are in luck!  If not, then I am sorry – but you can download it here.   If you would rather download a windows version of this game, please download it from our download page.

The object of the game is to get as many digits of Pi displayed as you can by hitting the Pi ball with the paddle.  In order to get the next digit, you also have to have the correct digit selected.  Use the keyboard to select the next digit; you can also use the keypad.

If for some reason you do not know the first 80 digits of Pi, hit the “P” key to have it displayed.

High scores are recorded and stored on your local computer.

The game is working well, but this is the first release – thus the reason I posted a few days before the actual Pi Day 2012.  If you find anything buggy with it, please let me know!

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Spirolaterals – A Perfect Match for Doodlers!

Have you ever had to wait in an office for a long time for an appointment without a book?  Sure, they may have magazines at the dentist’s office or what have you, but chances are that they are months old.  What do you do to entertain yourself while you wait?  One solution is to draw spirolaterals – a great way to draw interesting math art with a few simple rules – plus a notepad, pen, and patience!

Spirolaterals are spiral structures generated by setting up simple rules for drawing lines, and performing reiterated iterations of these rules.  Using complex algorithms, the number of spirolaterals that can be generated may indeed be infinite.  These structures were first investigated in 1968 by Hal Abelson and his colleagues.  Spirolaterals are open or closed, and can range from simple squares and rectangles to complex curves.

A very simple spiral is a square.  The rules for drawing a square are:

Draw a line of length one, rotate your paper by 90 degrees and repeat.  Repeat procedure until a closed design is generated.

I have outlined these steps using the excellent spirolateral program at http://math.fau.edu/MLogan/Pattern_Exploration/Spirolaterals/SL.html:

Iteration 1

Iteration 2

Iteration 3

Iteration 4

So now what do you get if you use the above algorithm, but change the angle to 60 degrees?  You get a geometric figure which has
an angle of 60 degrees between all of its line segments:

That’s right – a hexagon!

If you vary the line length as you progress in your iterations, much more interesting shapes can be made.  Of course, you can do all of this roughly with pen and ink, but here I will generate the designs with the program mentioned above.

Using variable lengths 1 and 2 (here it is set to letter codes MZ, which codes for 13 and 26, which is the same as 1 and 2, but the program draws the design bigger this way – LOL!), and an angle of 120 degrees, we get this design:

This one was drawn using 1 5 9 lengths, and 35 degrees.  It reminds me of the designs i used to make with the Spirograph toy when I was a kid:

Length 1 (actually, it is set to Z, or 26) and 30 degrees:

1 2 3 4 5 6 7 and 35 degrees:

Give it a try, either with pen and paper or by using the program.  It really fun and interesting, and you get a sense of how patterns should develop before you actually draw them.

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Quickly Calculating the Product of Two 2-Digit Numbers

A resource about math tricks would not be complete without mention of the techniques of Vedic mathematics.  Vedic math was introduced by Bharati Krishna Tirthaji Maharaja in the first half of the 1900s, and are a collection of sutras which allow the user to quickly solve mathematical problems very efficiently.  Tirthaji claimed that these sutras were found while studying ancient Hindu writings, but confirmation of his explanation has never been made.

Vedic math can be used such that calculations can be performed mentally or very quickly using single-line notation.  This article will demonstrate the Vedic math technique of quickly calculating the product of two 2-digit numbers.

In this example, 32 will be multiplied by 43:

32
x43

First take the product of the right-most digits and multiply them.  Then, write the result under and to the right of the multiplication set:

32
  |
43
6

Next, take the product of each diagonal digits and add them together.  Write the sum to the left of the first result.  I will show this step in two parts:

32
 X
43
(3×3)+(4×2)   6

Which is the same as:
32
 X
43
17  6

Lastly, take the product of the left-most digits, and write the result to the left of the first two results:

32
 |
43
12  17  6

Almost done – now if you have any remainder in the tens columns, be sure to add it to the ones columns to the result to the left.  In our example, the “17” has a remainder of “1” in the tens columns, so add 1 to 12:

32
43
13  7  6

Now just “squeeze” the results together, and you will have your answer:

32
43
1376

And, indeed, if you use a calculator, you will find that 32 x 43 = 1376.

And now for a few more examples:

12×16:

12
16
1   2+6   12

12
16
1  8  12

192 answer

16×36:

16
36
3   24   36

16
36
3   24   36

16
36
3  27  6

16
36
5  7  6

576 answer

25×97:

25
97
18  45+14  35

25
97
18  59  35

25
97
18  62  5

25
97
24  2  5

2425 answer
Practice some more of your own on paper to get the technique down, and then try to do some in your head.  With a little practice, you will be able to do two-digit calculations quickly in your head!

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