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## Fibonacci Addition: A 2nd Look

Published on Sunday, October 28, 2012 in , , ,

Back in July of 2011, Scam School presented a classic lightning addition feat involving the Fibonacci numbers.

I recently took a closer look at the Fibonacci addition feat, and ran across some interesting findings.

First, we'll start a quick review. Here's the original video:

Why does this work? If you replace the first starting number with a, and the second starting number with b, here's how the math works out:

``` 1.   a
2.         b
3.   a +   b
4.   a +  2b
5.  2a +  3b
6.  3a +  5b
7.  5a +  8b
8.  8a + 13b
9. 13a + 21b
10. 21a + 34b
--------------
55a + 88b
```
Notice the 7th number from the top is 5a + 8b. If we multiply this by 11, as explained in the video, we get 55a + 88b, which is the total we need!

Also, as mentioned previously, you can learn and practice multiplying by 11 in your head over at The Math World's Math Tricks: Multiply by 11 page. That's it for the review.

### Practicing with Wolfram|Alpha

I got the idea to use Wolfram|Alpha to practice this routine, and was hoping they had some ready-made command where I could type in something like “FibonacciSeq[{6, 2, 10}]”, and it would generate a generalized Fibonacci sequence starting with the numbers 6 and 2, and run for 10 terms. Unfortunately, there is no such command.

There is a Fibonacci command, entered as Fibonacci[n], which returns the nth Fibonacci number from the standard series {0, 1, 1, 2, 3, 5, 8, ...}. Counting 0 as the 0th Fibonacci number, Fibonacci[4] would return 3, since 3 is the 4th number in that set. Wolfram|Alpha also accepts a shortened form of this command, using a capital F with a subscripted number, such as F_4 (effectively F4).

Looking at the sequence for variable b in the above addition, we see that we could use this quite easily, since the series goes {0b, 1b, 1b, 2b, 3b, 5b, 8b, ...}. What about the variable a, though? Somehow, we have to work it to get the series {1a, 0a, 1a, 1a, 2a, 3a, 5a, ...} at the same time.

Fortunately, it turns our there are negative orders of Fibonacci numbers. They don't often get discussed, but are important for our purposes. It turns out that F-1 is 1, which gives us the start of the sequence we need for the variable a!

The sequence we need, then, should start with F-1×a + F0×b, which is 1a + 0b = 1a = a. The next step would be F0×a + F1×b, which works out to 0a + 1b = 1b = b. This will continue up to F8×a + F9×b, which is 21a + 34b, our 10th term.

If we add Wolfram|Alpha's table command to display the entire sequence, a variable x for the length of the sequence we desire (1 through 10 for this trick), and two starting variables set to random numbers from 1 through 20, we find the way to generate a random generalized Fibonacci sequence is this: a=RandomInteger[{1,20}], b=RandomInteger[{1,20}], Table[F_(x-2) a+F_(x-1) b, {x, 1, 10}].

I just tried this out, and got the following sequence: {7, 11, 18, 29, 47, 76, 123, 199, 322, 521}. Using the principle behind the trick, I worked out that this sequence should total 1,353. As an added bonus, you can verify your mental math simply by clicking on the generated sequence! Clicking on that particular sequence, you'll see the sum in the Total pod (among numerous other ways of looking at that set).

The Successive ratios pod is also interesting, as you can see the ratios get closer and closer to approximating φ (Phi).

### Other Patterns

What happens if we make the sequence longer or shorter? Let's try 8 terms on the left, and 12 terms on the right, including their totals:
``` 1.   a                 1.   a
2.         b           2.         b
3.   a +   b           3.   a +   b
4.   a +  2b           4.   a +  2b
5.  2a +  3b           5.  2a +  3b
6.  3a +  5b           6.  3a +  5b
7.  5a +  8b           7.  5a +  8b
8.  8a + 13b           8.  8a + 13b
--------------          9. 13a + 21b
21a + 33b          10. 21a + 34b
11. 34a + 55b
12. 55a + 89b
--------------
144a +232b
```
Take a look at the 12th term on the right, 55a + 89b, and compare it to the earlier total, 55a + 88b. Those two are very close, differing only by the amount b.

Similarly, if we look at the total of 8 numbers on the left, 21a + 33b, we notice that it's very close to the 10th term in the sequence, 21a + 34b. Once again, it differs only from the term by the amount b. Not surprisingly, if we continued the algebraic list to the 14th term, you'd see that it was 144a + 233b. Do you see the pattern?

This turns out to be a consistent pattern, and there's a very simple way to take advantage of it. Start as you would before, but instead of telling them to stop at the 10th number, tell them to keep on going to, say 15 terms. Let's say they generate the sequence {1, 15, 16, 31, 47, 78, 125, 203, 328, 531, 859, 1390, 2249, 3639, 5888}, but written vertically. Tell them to draw a line between any two numbers on the list (the line must be drawn somewhere above the last two numbers on the list), and then to add up all the numbers above the line on the calculator. You use only your head, and you still beat them!

Keeping with the previous example, let's say they draw the line in the sequence below 1390, like this:
```   1
15
16
31
47
78
125
203
328
531
859
1390
----
2249
3639
5888
```
All you have to do is look at the 2nd number after the total line, 3,639 in this case, and subtract the 2nd number on the list, 15 in this example. 3,639-15=3,624, so you give 3,624 as your instant answer. Sure enough, Wolfram|Alpha verifies as correct!

There's another way to do this with a list of any length, but the calculation is more difficult. If we call the last two numbers on a list r and s respectively, then the number following s will be r + s, and two numbers after s will be s + r + s = r + 2s. Subtracting b means the total will be r + 2s - b.

Applying this pattern to our 1390 example above, you'd double 1,390 to get 2,780, add 859 to get 3,639, and subtract 15 to get 3,624. This approach has the advantage of not needing to know any numbers below the total line, but the disadvantage of being harder to calculate mentally, depending on the chosen numbers.

I hope you enjoyed this closer look at the Fibonacci addition trick, and that it gets the gears turning for other possible approaches.

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## Calendar Craft

Published on Thursday, October 25, 2012 in , , , , , ,

For Day One, my calendar calculation routine, I recently released a custom-designed calendar clipboard which helps make the feat visible to a small crowd.

In today's post, I'll show you how to put together your own calendar prop inexpensively, and even some other directions you can take the basic idea.

To start, you'll need a magnetic dry-erase board with a blank calendar pre-printed on it. I used an 11 by 14 dry erase calendar from Expo, which comes with two magnets. The two important features in the board are that it be magnetic, as well as small enough to use and carry for a performance, while still being visible for your audience. You'll also need to make sure your chosen board has 5 weeks to mark.

You'll also need a dry erase marker (usually included with dry erase calendars), a dry erase eraser, and a permanent marker, such as a Sharpie. Optionally, you may want a ruler for making your marks consistent. If you choose a ruler, I suggest a cork-backed ruler to minimize damage to the board.

What you're going to do is use the permanent marker to write the dates in the squares from 1 through 28, similar to the way the calendar clipboard is laid out. Most dry erase calendars have a small space in a corner for the date, such as the corner notches seen on this dry erase calendar, but for better audience visibility, you'll want to use as much of each date's square to write the date.

If you prefer, you can use the ruler and a dry erase marker to create even guidelines for your dates first. Personally, I didn't do this. In performance, I need to write and/or erase 29, 30, and 31 on the board, and those will usually be written freehand, so they tend to stand out if the other dates aren't written freehand, as well.

What happens if you make a mistake when writing with a permanent marker on your dry erase calendar? Don't worry, there are numerous ways to remove permanent marker off of your dry erase board. The simplest and most surprising of them uses only a dry erase marker and eraser to remove permanent marker.

Once you've written the dates from 1, in the upper-leftmost square, through 28, in the rightmost square of the 4th week, using large numbers as discussed above, you can put the permanent marker away.

Not surprisingly, most dry erase calendars have the days of the week permanently marked at the top. Yet, you need to be able to change the days of the week in routines like Day One. To solve this problem, I simply use magnets printed with days of the week. This is why I emphasized the importance of a magnetic dry erase calendar earlier. Even on larger boards, these days of the week magnets cover the pre-printed days, and highlight the days printed on the magnets.

Before each performance, you'll use a dry erase marker to write 29, 30, and 31 on the first 3 squares of the 5th week on the calendar, and have the days of the week magnets arranged in the remaining 4 days of the 5th week in the calendar.

When you're given the month and year, write them in the space for the month at the top, and use the Day One technique to determine where to place each day of the week magnet. After placing the magnets, erase any of the last 3 dates as needed (For example: If you're given a February in a leap year, erase the 30 and 31, leaving the 29) and your calendar should be arranged correctly!

After each performance, erase all the dry erase markings, and put the magnets back down in the 4 empty squares of the final week. If you're about to do another performance, write the 29, 30, and 31 back in. If you're not, you can simply put the board and magnets away until you're ready to perform again.

The basic idea of using permanent markers to create a custom design (and knowing how to remove it in case of mistakes), should start the gears turning for other ideas. Starting from a blank dry erase board, you could create things like a grid for a magic square or a chessboard for the Knight's Tour.

The cork-backed ruler I mentioned earlier is an essential when designing grid-based layouts, of course. For the chessboard, I recommend creating the board in a blue that's dark enough to be distinguished easily from the white squares, yet still light enough to contrast with dry erase numbers written in black.

Don't forget that using a magnetic dry erase board can also be a great way to display magnetic playing cards, either in full-size or in miniature.

For one last idea to inspire you, how about a Sudoku grid? You could use it to display your apparent Sudoku genius as taught in Werner Miller's Swindle Sudoku routine!

That should be enough to be enough to inspire you and get you thinking about different ways to customize and present the mental feats you've learned here on Grey Matters.

For my next puzzle, I'll try figure out why custom-printed dry erase Sudoku boards are so much more readily available than custom-printed dry erase chess boards.

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## 98 Years of Martin Gardner

Published on Sunday, October 21, 2012 in , , , , , , , , ,

On October 21, 1914, Martin Gardner was born. After having an enormous and positive effect on the fields of magic, mathematics, philosophy, literature, science, and many other fields, he died on May 22, 2010.

On the 98th anniversary of his birth, I thought it would be fun to look back at Martin Gardner's influence.

Back in 1956, Martin Gardner submitted an article on flexagons to Scientific American. They not only ran the article, but offered Gardner his own monthly Mathematical Games column. The first column ran in January of 1957, and he kept it going regularly for the next 25 years! Martin continued to write the occasional column until the May 1986 issue, retiring both himself and the column.

In honor of the flexagon column that started it all, and Martin Gardner's birthday, Vi Hart has dubbed October 2012 the month of the flexagon. Below is the first video she made to honor this month:

Vi Hart also made a second video on their surprising mathematical qualities, and even a third on flexagon safety.

All of Martin Gardner's columns were assembled into a series of 15 books. The books themselves have since been assembled into the Martin Gardner's Mathematical Games CD-ROM. This series of PDF eBooks is searchable and portable, making the joy of surprise and discovery even more accessible.

Even though it's Martin Gardner's birthday, you can get a few free goodies to get an idea of his influence. The January 2012 issue of the College Mathematics Journal was entirely dedicated to his work, and remains free to download as either the full issue or individual articles. The Gathering 4 Gardner Foundation has several Martin Gardner posters available for free download, and even a full free PDF copy of a tribute book called The Mathematician and the Pied Puzzler.

Probably one of the most famous Gathering 4 Gardner giveaways, thanks largely to this YouTube video, would be the paper dragons whose heads seem to turn and follow you. The PDFs that will let you build your own green dragon, red dragon, and/or blue dragon are still available as free downloads.

Many of the puzzles he created and/or popularized over the years can be found in Puzzles.com's Martin Gardner section. It's surprising how many now-classic puzzles can be traced back to him. He even gets a mention in Bill and Ted's Excellent Adventure:

[Bill and Ted are working on their history report]
Bill: Okay, Ted, George Washington. One: the father of our country.
Ted: Two: born on President's Day.
Bill: Three: the dollar-bill guy.
Ted: Bill, you ever made a mushroom out of his head?
That mushroom reference is to a puzzle originally published in Martin Gardner's April 1968 column on dollar bill puzzle, and later reprinted in both Mathematical Circus, as well as Perplexing Puzzles and Tantalizing Teasers (solution can be found here).

Even today, people are help keeping Martin Gardner's memory alive. Besides the Gathering 4 Gardner Foundation above, you can also follow @WWMGT on Twitter. WWMGT stands for “What would Martin Gardner tweet?”, and the author does an excellent job of living up to that title.

I'll wind this with a full 1996 episode of David Suzuki's The Nature of Things which focused on Martin Gardner. It gives a great idea of not only his life, but his legacy, too:

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## Power of 2

Published on Thursday, October 18, 2012 in , , , ,

The recent Age Cards posts showed us one amazing application of the number 2 (base 2, in that case).

Today, we'll investigate some more amazing and surprising uses of the number 2, including some that can make you look like a genius!

For much of what you're going to learn here, you'll want to be comfortable with multiplying by 2 in your head, as well as the ability to easily divide by 2 mentally.

Incredibly, even if you can only multiply and divide by 2, you can still multiply any 2 numbers together! This is made possible by an unusual approach known as “Russian peasant multiplication” (sometimes also called “Egyptian multiplication”). It's explained in the following video:

Yes, this works with any 2 numbers, but depending on the numbers chosen, you may wind up adding more or fewer digits. If you're interested as to why this works, Dr. Math has some excellent explanations on this page, as well as this page.

Using a similar process, including the habit of dropping any decimals after dividing by 2, you can also quickly and easily convert any decimal number to a binary number. How? You take any given number, ask yourself whether it's odd or even. If it's odd, you're going to write down a 1, and if it's even, you'll write down a 0. You then divide by 2, and repeat this process. When you've completed this step all the way down to 1, you're done.

To make this clearer, let's try a small example, such as 19. Working this out step-by-step, we get this:

19 is odd: 1
(19/2 = 9.5, round down to get 9)
9 is odd: 1
(9/2 = 4.5, round down to get 4)
4 is even: 0
(4/2 = 2)
2 is even: 0
(2/1 = 1)
1 is odd: 1

Working from top to bottom, we write these numbers down from right to left (opposite of the way you would normally read) to get 10011. Double checking with Wolfram|Alpha, we see that 10011 is indeed the binary for 19.

As you can see, this is actually a bit simpler than the Russian peasant multiplication, but comes across as more impressive.

Let's try this with a higher number, such as 76, and simulate how you'd think and write it when showing this to someone else (the steps of dividing by 2 and rounding down are implied from one step to the next), noting that each step's result is written to the left of the previous result:

Think: 76 is even...
Write: 0

Think: 38 is even...
Write: 00

Think: 19 is odd...
Write: 100

Think: 9 is odd...
Write: 1100

Think: 4 is even...
Write: 01100

Think: 2 is even...
Write: 001100

Think: 1 is odd...
Write: 1001100

Yep, 1001100 is binary for 76! From the point of view of anybody watching, you seem to just work out the binary equivalent of the given number in your head without breaking a sweat!

Once you're comfortable converting decimal to binary, it's not hard to learn to work the other way.

To take a binary number and convert it, you're going to start from the leftmost 1. The leftmost binary digit will always be a 1 because, just like in decimal, you can put as many zeroes as you want to the left side without changing the number.

You start at the leftmost 1 with a mental total of 1. Every time you move to the right, regardless of the digit, you're always going to multiply by 2. If the digit you're at is a 1, then you're going to add a 1. Next, you move to the right again and multiply by 2, adding 1 if the digit is 1, and so on until you get to the rightmost digit.

Let's how this system is used to translate the binary number 11001. We start at the leftmost 1, with a mental total of 1:

11001
Mental Total: 1

(Move a digit to the right...)
11001
Mental Total: 1 * 2 = 2, 2 + 1 = 3 (the 1 is added because this digit is a 1)

(Move a digit to the right...)
11001
Mental Total: 3 * 2 = 6

(Move a digit to the right...)
11001
Mental Total: 6 * 2 = 12

(Move a digit to the right...)
11001
Mental Total: 12 * 2 = 24, 24 + 1 = 25 (the 1 is added because this digit is a 1)

Our final mental total is 25, so if we did this right, 11001 in binary should be 25 in decimal. A quick double-check with Wolfram|Alpha proves this is correct!

You can even work through this verbally quite easily, in this manner: “1...2.3...6...12...24.25!”

These binary and decimal conversions are derived from what's known in mathematics as Horner's method. This was originally developed as a way of evaluating polynomials, such as 3x2+2x+7 (Here's a quick tutorial on how to do that). However, since any number in any number base can be treated as a polynomial, such as 327 = (3 * 102) + (2 * 101) + (7 * 100), Horner's method also turns out to be useful for converting numbers from one base to another. You can learn the generalized method for converting any number in any base to decimal here.

Now you have 3 new amazing, and even amusing, abilities - Russian peasant multiplication, decimal-to-binary conversion, and binary-to-decimal conversion. Practice them, and you'll find them fun and maybe even useful!

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## More Quick Snippets

Published on Sunday, October 14, 2012 in , , , ,

October's snippets are ready for your perusal!

Today, we'll use math videos to take you into surprising and bizarre aspects of mathematics. Don't worry, you're safe on this side of the play button.

• Let's start with a classic tale, which teaches the surprising nature of exponential growth:

How many grains of wheat are on a given square? On the first square, we have 1 grain, or 20. On the second square, we have 2 grains, or 21, and on the third square, we have 4 grains, or 22. It's not hard to see that on the nth square, we'll have 2n-1 grains of wheat on the chessboard.

That's only the amount of grains on a given square, however. What about the total number of grains? With the first square, we have 1 grain of wheat. On the first two squares, we have 3 grains of wheat. On the first 3 squares, we have 7 (1+2+4) grains of wheat, and so on. Notice that 21=2, 22=4, and 23=8, each just 1 more than the total number of grains on each of the corresponding squares. In other words, the total grains of wheat on the first n squares will be (2n)-1.

With a little help from Wolfram|Alpha, we can follow the story more closely. After the 17th square, the inventor would have 131,071 grains of wheat. After the 26th square, there would already be 67,108,863 grains of wheat. What about the 42nd square? More than 4 trillion grains of wheat (4,398,046,511,103 to be exact) had been given to the inventor by the surprised king!

The video implies that the process had to be ended there, but were it continued to the 64th square, the total number of wheat grains would come to 18,446,744,073,709,551,615 (read as 18 quintillion, 446 quadrillion, 744 trillion, 73 billion, 709 million, 551 thousand and 615)!

Just for fun, if we tell Wolfram|Alpha we're actually talking about grains of wheat, we can discover stranger things about so much wheat, like the total nutrition information and physical properties, that the king could never have known.

• Back in May of last year, I wrote about John Conway's game of life. Since it was first described in 1970, there have been many ingenious variations, such as Life For Two. When you search for conway's game of life on Google, Google even runs a sample game of it!

The classic version is built around integers (whole numbers) and grid squares, giving the game its classic blocky look. There's a new version called Smoothlife that's instead built around floating point numbers and distance measurements, giving the game of life a much more organic look and feel:

You can learn more about this project at the SmoothLife link above, as well as watching the videos on the SmoothLife playlist. There are even sample runs of SmoothLife in 3D later in the playlist, so take the time to explore!

• I imagine all this talk about life, death, and grains of wheat is probably making you hungry. Even if the above topics didn't, James Grime's and Numberphile's trip to McDonald's to discuss McNugget numbers will probably do it:

As mentioned, the largest number of McNuggets (or other objects) that cannot be made with given sets is known as the Frobenius number of the set. More colloquially, the problem is known as the coin problem. It was Henri Picciotto who first drew attention to McNugget numbers back in the 1980s. He still discusses McNugget numbers on his website.

Wolfram|Alpha has a built in function for finding the Frobenius number of any given set. Here's the classic McNugget problem set, as well as the modified McNugget problem set, including the boxes of 4.

If you enjoy the McNugget numbers problem, you might also the semi-related world of Golomb rulers. Golomb rulers are measuring rulers optimized so as to use the fewest markings possible that still allow you to measure any whole distance from 1 unit up to a given limit (say, 12 units).

That's all for now, so have fun exploring!

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## Happy 10-11-12!

Published on Thursday, October 11, 2012 in , , , ,

Here in the US, where we use a month-day-year order, today is 10-11-12 for short.

I thought I'd take advantage of this unusual date sequence to take a look at the amazing power of some other mathematical sequences.

If I handed you 13 cards of different values, the simplest sequence would be to put them in Ace through King order. What if I challenged you to put them in a sequence such that when you spell the word Ace, you get the the Ace, then spell the word two, and you get a two, and so on? That's the basic trick taught in the following video:

How is this done? Here's the explanation video.

How did anyone come up with this sequence in the first place? That's actually very simple. Take out 13 cards of all the same suit, and put them face-down in Ace through King order, with the Ace as the top card, and the King as the bottom card (if the cards were turned face-up, you'd be looking at the King).

Next, spell Ace by putting one card for each letter on the bottom, then taking the next card, just as in the above video. Since you've ducked 3 cards and removed the 4th, this will naturally be a 4 of whatever suit you removed. This means that the Ace should be the 4th card in the sequence. Continuing with spelling t-w-o, you'll turn up an 8 next, meaning that the 2 should be placed as the 8th card.

Doing this all the way up through spelling k-i-n-g, the order you'll get will be 4-8-A(1)-7-K(13)-6-3-2-Q(12)-J(11)-9-5-10. Note that in the explanation video, note that the Ace is the 4th card, the 2 is the 8th card, the 3 is the first card, and so on. If you ever forget the sequence as taught in the video, now you know how to work this out.

Let's take this sequence to the next level. In the book Scarne on Card Tricks, there's a version of this trick called “The Uninvited Joker”, which can be learned online (part 1, part 2). One performing tip I'd add from experience: If you switch the roles so that the spectator successfully spells the card, and you (the performer) keeps getting the joker, the audience is put more at ease, and is more likely to be entertained.

Another simple, yet amazing, sequence took the internet by storm earlier this year was the fact that 1/998001 = 0.000001002003004...996997999000001.... In other words, the decimal representation is a sequence of almost every number from 000 up to 999 before repeating. Naturally, James Grime couldn't resist making a Numberphile video about it:

As you can see, even simple sequences can have amazing and surprising results. As recently as the previous post on the Latin ESP Square, you can see evidence of just that. There are numerous amazing sequence to explore here on Grey Matters, including the Fibonacci sequence, de Bruijn sequences, the Gilbreath Principle, the Kruskal Count (part 1, part 2), and even Penney's sequences.

How do we wind up this discussion of sequences? The only possible way is with a sequence of sequences! The always amazing Pascal's triangle, as seen in the presentation below, only scratches the surface of its amazing qualities:

In my 12 Days of Christmas post, I talk about many of the other aspects of Pascal's triangle, including the fact that you can get a Sierpinski triangle by differentiating odd and even numbers, as demonstrated here on Wolfram Alpha. Try it modulo 3, modulo 4, and so on (try clicking the More Rows button) to see even more impressive patterns.

Have fun exploring these amazing sequences!

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## Werner Miller's Sub Rosa 1 and 2

Published on Sunday, October 07, 2012 in , , , , , ,

In the past, Werner Miller has generously shared his creative brand of mathematical magic with Grey Matters readers numerous times.

He's just released two new books, Sub Rosa 1 and Sub Rosa 2, and has allowed Grey Matters readers to take a free peek at an effect from the book.

Sub Rosa is Latin for “in the strictest confidence.” In Werner Miller's case, it also means a multi-volume series of books containing a treasure trove of deceptive and original mathematical-based magic, none of which requires any difficult sleight-of-hand.

In Sub Rosa 2, there's an effect called Latin ESP square. You shuffle a set of standard ESP testing cards, and then have the spectator cut the cards. You then deal them out without looking at the faces, so that no row, column, or diagonal contains duplicates of any symbol!

The method is described below, and you can download this PDF at this link.

In the description above, you start by giving the deck a special type of shuffle. You can learn that shuffle in just a few minutes via this YouTube tutorial. When using it with just 25 cards, it's very deceptive. Although it looks like a very thorough mixing procedure, the only effect it has is the same as a cut.

Beyond just setting up an amazing pattern, this is also a great way to secretly set up a Latin square for a subsequent trick. Colm Mulcahy, whom I wrote about just last month, has some excellent trick involving Latin square set-ups, including Splitting the Pot and Amazon Arrays (Large Action). You can find much more about Latin squares on the Mathematical Association of America's website.

Thanks again to Werner Miller to allow Grey Matters to share this great routine. For more of this ingenious thinking, check out Sub Rosa 1 and Sub Rosa 2, as well as his other works!

0

## Age Cards: Game Show Theme

Published on Thursday, October 04, 2012 in , , , ,

In the Age cards discussion in my previous post, I tried to spark ideas about creating your own original versions. If you haven't already seen it, I recently added a James Grime video to that post, about other approaches to the Age Cards using sets such as Fibonacci numbers and prime numbers.

Brian O'Neill recently e-mailed me his brilliantly conceived variation of the Age Cards. In my opinion, it makes the routine more deceptive, as well as engaging for the audience.

Brian's idea has several layers to it. First and foremost, he's given it the theme of a popular game show that involves choosing from numerous suitcases.

Thanks to this theme, there's now a reason that not all numbers are visible on the card. As you see below, it's designed as if some of the suitcases are open, with prize amounts showing, and others are closed, with their suitcase numbers displayed. The logic here is clear and consistent with the audience's expectation.

You can have someone think of any suitcase, numbered 1 through 26, and use the standard Age Cards procedure (noting the bottom-leftmost visible number on each card) to divine that number. Already, this is not a bad trick, but Brian has ingeniously added another level to his version.

You can also have someone think of any of the prize amounts shown on the last card, and divine the spectator's prize amount, as well! The method here is a little different.

When you have someone think of a prize amount, add the bottom-leftmost visible suitcase numbers of the cards on which they see their prize amount, just as you would for the suitcase-number version.

Once you have that number, however, the last image listing all the prize amounts becomes your secret key. You'll have a number from 1-24, and that will tell you where on the list to look for their prize amount. The amounts in the left column (\$1 to \$1,000) correspond to the number 1 through 12 (respectively, from top to bottom), and the amounts in the right column (\$5,000 to \$1,000,000) correspond to the numbers from 13 through 24 (again, respectively from top to bottom).

For example, if someone was secretly thinking of \$300,000, they'd note that they see that amount on the 3rd card (lowest number showing: 4), and the 5th card (lowest number showing: 16). In this example, you'd add 16 + 4 to get 20. Since 20 is higher than 12, you'd look on the right side of the prize amount list (the easiest thing to do in this example is to count backwards from 24 starting at the bottom). Sure, enough, the 20th prize amount on the list is \$300,000!

When performing this for two people at the same time, you'll want to prevent the confusion of calculating two separate totals at the same time, so have the first person think of a suitcase number, and go through asking them about the cards on which they see their number, then turn to the 2nd person and have them think of the prize amount. This way, you complete one total before starting another one. You still have to make sure and remember both totals, as well as to whom they apply.

Download each of the images below, and try it out for yourself. Thank you once again, Brian, for your creativity, and your willingness to share!