In April, in the first of my age-guessing series of posts, I discussed the magic trick known as the “Age Cards.”

In that post, I focused solely on their use in finding someone's age, but today I'm going to look at the trick from several other perspectives.

As a quick note before we proceed, you should be familiar with the basics of the Age Cards from that original post to follow most of this post.

Binary is important in this trick, because all the numbers are being uniquely identified by whether or not each of the powers of 2 are involved in adding up to that number.

The numbers are distributed on the Age Cards in accordance with their binary equivalent. For example, 53 in binary becomes 110101. The 1s in 110101 appear only in the 1s, 4s, 16s, and 32s places, so 53 will only appear on the cards with 1, 4, 16, and 32 in the upper left corner. This is why you can add the numbers in the upper left corner, 1+4+16+32 in this case, to get the person's number.

What if the range of numbers were smaller? If I limit your choice to the numbers 1 through 7, I would only need 3 cards, since 7 is 111 in binary. However, the trick would seem far less impressive, as it wouldn't take that long to just look through the cards and find a number common to the cards containing the number, and making sure those numbers aren't on the remaining cards.

This seems to suggest that the trick becomes less impressive with smaller sets. However, this is actually only true when you're using cards filled solely with numbers. What if the numbers were represented with something else?

Let's try using months. January will replace 1, February will replace 2, and so on up to December replacing 12. How many cards will we need for the month version? The largest number we'll deal with is 12, which is 1100 in binary, a 4-digit number, so we'll only need 4 cards.

How do we distribute the months? We can test each bit using the AND operation with the first 4 powers of 2 to work out where they go. With Wolfram|Alpha, this is achieved using the BitAnd command.

Testing each of the numbers from 1 to 12 by performing the AND operation against each of the first 4 powers of two, we get the rather odd-looking result of {{1, 0, 0, 0}, {0, 2, 0, 0}, {1, 2, 0, 0}, {0, 0, 4, 0}, {1, 0, 4, 0}, {0, 2, 4, 0}, {1, 2, 4, 0}, {0, 0, 0, 8}, {1, 0, 0, 8}, {0, 2, 0, 8}, {1, 2, 0, 8}, {0, 0, 4, 8}}. What does this mean?

This is the distribution of months we need! Formatted more clearly, the answer would look like this:

January (1) = {1, 0, 0, 0}
February (2) = {0, 2, 0, 0}
March (3) = {1, 2, 0, 0}
April (4) = {0, 0, 4, 0}
May (5) = {1, 0, 4, 0}
June (6) = {0, 2, 4, 0}
July (7) = {1, 2, 4, 0}
August (8) = {0, 0, 0, 8}
September (9) = {1, 0, 0, 8}
October (10) = {0, 2, 0, 8}
November (11) = {1, 2, 0, 8}
December (12) = {0, 0, 4, 8}

In other words, if we consider the various cards as 1, 2, 4, and 8, as we would were we using just numbers, November would appear on cards 1, 2, and 8, while May would only appear on cards 1 and 4. The final lists, then, would look like this (each column represents a singe card):

January        February       April          August
March          March          May            September
May            June           June           October
July           July           July           November
September      October        December       December
November       November

In my previous post, I introduced a new version of Nim called Abacan, and introduced one strategy for winning it.

In this post, I'll teach the rest of the strategy you'll need to win Abacan every time. Before we do that, though, I'll go through a quick review.

In Abacan, you have 5 rows of beads. There are rows of 1, 3, 5, 7, and 9 beads. Players alternate taking turns, and can only move 1, 2, or 3 beads, all in the same row, on their turn. The last player to move a bead is the loser.

The strategy taught last time involves first reducing the number of beads in each row to a number from 0 to 3. In situations where the rows reduced to all 1s and 0s, or all 1s and 0s with a single 2 or 3, you simply move so that the reduced rows become an odd number of 1s. For example, rows of 1, 1, 4, 6, and 8 beads can be reduced to rows of 1, 1, 0, 2, and 0 beads. From there, it's a simple matter to see that the “2” row (reduced from 6) should have 1 bead removed from it, leaving the other player with 1, 1, 4, 5, and 8 beads (which reduces to 1, 1, 0, 1, and 0 beads - the equivalent of 3 piles of 1).

You should always reduce the numbers of beads first, then see if you can apply the strategy above first. This “odd 1s” approach is effectively the endgame strategy of multi-pile Misère Nim, as taught in Secrets of Nim (Part 3). In Abacan, however, it can be used well before the end.

What happens, though, if you reduce the rows of beads and find that there's more than one row containing the equivalent of 2 or 3 beads? In that case, you'll need a second strategy.

Strategy 2: In the Secrets of Nim (Part 2) and Visualizing Multi-Pile Nim posts, I discuss the importance of pairing up similar values using the binary. Thankfully, there won't be any need to delve into binary conversion in the Abacan version.

If you have a set of reduced rows containing multiple rows of 2 and/or 3 beads, you'll use the second strategy. In this strategy, you'll want to leave your opponent with rows that, when reduced, have any 1s, 2s, and 3s paired up with each other. You don't need to worry about pairing up 0s.

This can sound confusing, so let's use the starting arrangement of 1, 3, 5, 7, and 9 beads as an example. The first step is to reduce the arrangement from 1, 3, 5, 7, and 9 beads to its equivalent, which is rows of 1, 3, 1, 3, and 1 beads.

Since we can't use strategy 1 from the previous post, we move on to strategy 2, pairing up all the 1s, 2s, and 3s. With the reduced rows of 1, 3, 1, 3, and 1, it's not difficult to see how we can leave our opponent with a pair of 1s and a pair of 3s.

Taking 1 bead from the first row would make leave your opponent with 3, 5, 7, and 9 beads, which reduces to 3, 1, 3, and 1. You could also remove 1 from the 3rd row, leaving your opponent with rows of 1, 3, 4, 7, and 9, which reduces to rows 1, 3, 0, 3, 1. Another alternative is leaving your opponent with rows of 1, 3, 5, 7, and 8 beads, which reduces to rows of 1, 3, 1, 3, and 0 beads.

Now you know that it's to your advantage to move first, and that your first move should be sliding over 1 bead from the row of either 1, 5, or 9. Since 1, 5, and 9 are all equivalent after performing modulus 4, all 3 of these moves have the same effect.

Full approach: This is almost everything you need to know to win Abacan:

1) Mentally reduce the number of beads in each row by using mod 4.
2) If the reduced rows are all 1s and 0s, or all 1s and 0s with a single 2 or 3, remove enough beads to make the reduced rows consist of an odd number of 1s (strategy 1).
3) If using strategy 1 doesn't apply, use strategy 2. Remove beads in such a way that all any 1s, 2s, and 3s remaining are paired up.

I just added a Nim game called Abacan to the Grey Matters Store. The main piece is a triangular shape with 25 beads in it, distributed as rows of 1, 3, 5, 7, and 9.

Abacan has rules that are slightly different from the previous versions of Nim discussed here on Grey Matters. To make this simpler, I'll review some previous versions of Nim before moving on to Abacan's peculiarities.

Let's start with single-pile Misère Nim (last player to move loses), using 25 objects, and each player is only allowed to remove 1, 2, or 3 objects on their turn. With your previous knowledge and a little practice, or the use of the Single-Pile Nim Strategy Calculator, you should have no problem determining that the other person should go first, and you need to hit the key numbers of 21, 17, 13, 9, 5, and 1.

For multi-pile Misère Nim, with rows of 1, 3, 5, 7, and 9, you'll need the binary understanding of multi-pile Nim taught here (and made easy to visualize here) and the endgame play knowledge of multi-pile Misère Nim taught here. The Multi-Pile Nim Strategy Calculator can help you learn the ideal strategy for this version.

If Abacan went by either of these rules, you'd be all set. However, as you'll see in the video review below, there's a subtle difference. You're only allowed to move 1, 2, or 3 beads, all from the same row, on each turn. The last player to move a bead loses.



Reducing the problem: The first step in Abacan strategy is, not surprisingly, treating each row as if it only had 0, 1, 2, or 3 beads in it, regardless of the actual number of beads in each row. This works out as follows:

Actual number of beads:   1  2  3  4  5  6  7  8  9
Reduced number of beads:  1  2  3  0  1  2  3  0  1

With everyone settling into school and the holidays approaching, this seemed like a good time to start adding some new items to the Grey Matters store!

Looking around, I've found several new items that belong in the store. Some have already been added, and others will be added in the coming days.

Looking to improve your memory? Thomas Harrington's The Magic of Memory was the first of the new additions. It features an expanded version of the classic Major/Peg System of memorizing numbers that is very easy to use.

Timothy Hyde's Magazine Memory, which I reviewed back in 2009, is an addition to the store that is long overdue. This is probably the best resource for learning to perform the magazine memory feat which I've ever come across.

Last year, I reviewed Persi Diaconis' and Ron Graham's Magical Mathematics. I'm thrilled to have added this impressive work both as a hardcover and as an ebook.

I've long featured Danica McKellar's previous books in the Grey Matters store, so her newest book, Girls Get Curves: Geometry Takes Shape was a definite and early must-add.

Not all the new additions are books and ebooks, as you'll see. Fans of my Nim strategy posts will definitely enjoy Abacan. The rules are a mix of single-pile and multi-pile Nim, as you'll see in the review video. You may want to read up on those classic versions of Nim before trying to work out a strategy specifically for Abacan.

As I mentioned, this is only the start of many new items to come. If you enjoyed reading Grey Matters, and would like to help support future posts, consider buying items through the Grey Matters store.

It's time for September's snippets!

In this edition of snippets, we're going to catch up with the latest news related to math and memory feats taught here on Grey Matters.

• Google announced a new feature this week: The “Bacon number” search. You can now find out how close any actor is to having starred with Kevin Bacon in a movie. For example, to find out how close Will Smith is to Kevin Bacon by searching for the term Will Smith Bacon number, and you'll learn that Will Smith starred with Josh Brolin in Men In Black 3, and Josh Brolin appeared with Kevin Bacon in Hollow Man, so Will Smith's Bacon number would be 2.

Back in 2006, I created a memory feat based on the classic 6 Degrees of Kevin Bacon game. In the memory feat, you have people pick 2 names from names written on slips of paper, and then you recall the shortest connection between them. Since you need connections between actors who aren't Kevin Bacon, The Oracle of Bacon is actually a better tool for this feat.

• Also in the news this week, people in the San Francisco Bay Area got treated to Pi in the Sky on Sept. 12. Five skywriting planes filled the skies with the first 1000 digits of Pi. If you've practiced memorizing 400 digits of Pi, you should be able to recognize some of the locations in Pi in videos such as the one in the Pi in the Sky link, as well as this video (below) and this video.



• If you like my text-memorization app Verbatim (tutorial video here), and are looking for some classic poems to memorize, YouTube user 0adam0ant0fan0 has posted a series titled Famous Poems that are well worth your consideration. At this writing, they include The Raven (below), The Road Not Taken, Jabberwocky, From Paradise Lost, and If.



• Mental math fans will enjoy TheClickMyVideos' YouTube channel. TheClickMyVideos has reposted several mental math videos made by others, including many of the long lost totalbreeze videos I mentioned back in Feb. 2011. They could be removed, so view them while you have the chance!

While I often teach how to perform mental feats, especially over in the Mental Gym, it's also important to get an idea of how professionals present them.

In the past, you've seen how performers including Maths Busking and Dr. Arthur Benjamin. Today's post features some other professional performers of mental feats.

First, we have Anton Zellman, who has made a good long living as a trade show performers, using his mind to entertain and educate prospective clients.

Because it's one of my favorite mental challenges, check out his Day For Any Date video. Watch it once just to see the performance, then watch it again to pick up on the finer details. For example, note that he only gives dates in two recent years (1990 and 1991 in the video). I have no doubt he could handle many more years, but this limitation makes the feat current, quicker to do, and easier to verify using only the 2 calendars hanging behind him.

Also, notice that he teaches how to do the effect. He's not performing a magic trick, and he's not trying give the impression that he's superior to you. He's simply showing that the boundaries of the human mind are much bigger than we may think. The multiplication table analogy is a wonderful tool to get the idea across of how something that seems hard can quickly become much easier with practice.

Among Anton Zellman's other videos are ones on remembering names and memorizing lists. There are also other videos on Zellman's own website. Watch these, keeping in mind that he's working in an environment where you often have only seconds to attract and keep the attention of attendees. If you don't engage it, that's a potential lost sale.

Scott Flansburg, also known as the Human Calculator, performs amazing mathematical feats for business meeting, schools, and, fortunately for us, the occasional TV appearance. Here's his appearance on a Discovery Channel program called More Than Human:



A few of the feats you see on there can be learned right here on Grey Matters, including the long division feat and the cube root feat.

Again, the attitude here is important. Just like with Anton Zellman, he's sharing, not showing off. Indeed, the most recent tweet (well, retweet) from Scott Flansburg at this writing says:

RT @haleymillsap: Work for a cause, not for applause. Live life 2 express, not 2 impress. Don't strive 2 make your presence noticed, mak ...

— Scott Flansburg (@HumanCalculator) September 13, 2012

To the math students of Spelman College, he's simply known as Colm Mulcahy. On the internet, however, he's better known as Card Colm.

It's a well-earned nickname, as he's been examining and creating principles of mathematical magic in his MAA column since October 2004!

If you're not familiar with Card Colm's column, you can find an index to all of the columns here. Up through June 2011, the columns are hosted on the Mathematical Association of America's (MAA) own site, and the columns after that are hosted on Colm Mulcahy's blogspot site.

Regular Grey Matters readers will recognize Colm from his G4G10 lecture I posted back in April. For those who need an introduction to his work, however, this video will give you a quick idea of his style:



At this writing, his newest column is titled Gilbreeath Shuuffling (pronounced gil-BREEATH SHOE-fling), and focuses on original routines for the classic Gilbreath Shuffle principle involving guessing the orders of suits.

The column before that, Something Old, Something True, Something Borrowed, Something New focuses on the poker deal taught in the first video in my 2010 Cheating At Poker post.

Besides his own original takes on each routine, one of the best things about Colm's work is that he is very detailed in his resources, so you know where to learn more. For example, there's plenty of Martin Gardner-inspired works among them.

As a matter of fact, between the search engine here on Grey Matters, Colm Mulcahy's columns, and their references, you should be able to grasp the basics of any principle, follow Colm's original variations, and then start creating your own variations! You can follow his latest routines on Twitter as @CardColm.

With the links above and a deck of cards in your hand, you're equipped to amaze yourself and other. Start exploring and have fun!

Our old friend Scam School is back for a visit!

This week's episode features some ingenious match puzzles in which Grey Readers readers may have an advantage.

Let's get right to the episode. Besides finding it below, it's available on the Scam School site, as well as YouTube. Stop the video, and try and figure out the first one:



If you've read through Grey Matters enough, the first puzzle is effectively one of the strategies you learn for playing Notakto (Part 2 here). See if you can use just that knowledge to help in solving it.

The 2nd puzzle uses just the matches (straws, actually, in the video). Try and give it time. As you'll see in the video, some people come agonizingly close to the solution without realizing it. If you like this type of puzzle, check out these other match puzzles at Puzzles.com, including the Three Squares puzzle.

As seen in the video, it is possible for people to figure out these puzzles, so be ready to pay up if you've bet on them not finding the solution. Take a closer look at the game of Notakto (linked above), if you're looking for something a little more surefire. You can also widen your arsenal of match puzzles with the Puzzles.com link, so have fun exploring!

I've written quite a few posts about the game of Nim, but practice is the real key.

In this post, we'll take a look at iOS Nim apps that you can download and use to practice for free on the go! Yes, there are free Nim apps for Android devices, but the variety is a little wider for iOS devices.

Most of the Nim apps you find are based on either single-pile Nim or multi-pile Nim.

For single-pile nim, try out Take Apples and NeonNim: The Subtraction Game. Multi-pile Nim can be practiced with the Stick Showdown and myQuickGame Free apps. If you want to switch between the two, there's the simply-named Nim Game and the aptly-named Appointed Win Game.

Back in June 2010, Scam School featured an episode of Nim without props. You'd wouldn't think you could turn a propless Nim game into an app, but it has been done! The calendar Nim variation is available as The Date Game. The other version is available as Race To 100.

There's also the ingenious tic-tac-toe variation of Nim known as Notakto. Notakto's winning approach isn't as easily picked up as some of the other variations of Nim. However, you can learn via my tutorials over in the Mental Gym, How to Play and Win Notakto and How To Play and Win Notakto: 3+ Boards.

There are a few variations of Nim out there, however, for which there are few (if any) ready-made tutorials. If you're up for the challenge, try a version with unfamiliar rules and strategies, and see how well you can do at working them out yourself. There's TacTix, in which you remove items from any contiguous columns OR rows, or the similar Triangular Nim, which throws diagonals into the mix, as well!

Probably the most challenging version of Nim is Hakenbush, which involves erasing lines, and eliminating any lines which are no longer connected to the base. Since there are endless possible shapes and color combinations to use, working out a general strategy is tougher than it may first appear!

I hope you find these versions of Nim fun and thought-provoking. If you have a favorite free Nim app, for any device, let me know about it in the comments!