• Maybe your mother likes music? In the video below, Vi Hart plays with mathematical symmetry groups, and gives them a surprising musical twist:
• Perhaps you're giving your mom flowers or plants for Mother's Day? Vi Hart has you covered there, as well. She posted a video series she dubbed Doodling in Math: Spirals, Fibonacci, and Being a Plant. Part 1 is here, followed by Part 2 and Part 3.
In the process, I've run across a few new and fun Knight's Tour videos I thought you might enjoy.
The first clip is from a late '90s British game show called The Moment of Truth. Contestants are given one week to practice some impressive feat, and then perform it before a live audience, often under time pressure, in order to win exotic and expensive prizes.
One of the best things about the video below is that it's a wonderful example of how to create suspense. Between the time clock, the live audience, the player's immediately family, and the possibility of winning prizes, this Knight's Tour has plenty of tension. The anticipation created can be felt strongly.
The next Knight's Tour video is more informational. The basics of the Knight's Tour are explained, and then a example solution is shown. This Knight's Tour video has the rather unusual feature of being shot in 3D! Instead of embedding it, I'll link to the video, so you can use the 3D menu to choose your preferred method of viewing the effect. If you prefer, it can also be viewed as a standard 2D video instead.
The final video in the set is a school lecture about the approaches and history of the Knight's Tour. Just listening to the terminology they use, I can't help but wonder whether these two ran across my Knight's Tour lessons in their research.
While there's not as much tension as the first video, there is plenty to learn from it. As a bonus, you have to love how they chose to end the lecture.
That post only discussed how to win the game on a single board. What about winning on more than one board?
Whether or not you've practiced the technique from the previous Notakto post, try the game for yourself. You can play it either online here or download the iPad app here. If you're not familiar with the game, here are the rules, excerpted from the iPad app homepage:
Notakto is a two-player game that is similar to Tic-Tac-Toe, except that both players make X's, and whoever completes three-in-a-row LOSES the game.
For a challenge, play a multiboard game of Notakto. On your move, make an X on just ONE of the boards. The computer may respond on the board you played on, or on some other board. A board that already has a three-in-a-row configuration is considered out of play. To win, force the computer to complete the last three-in-a-row configuration on the last available board still in play.
Even when you catch on how to win at 1-board Notakto, figuring how to win with 2 or more boards can still be tough.
To make the strategy for multi-board Notakto clearer and easy to follow, I've added 2 new posts to the Mental Gym. The first Notakto post teaches you the basics, and how to win 1- and 2-board games specifically.
The second Notakto post reviews what you've learned, and then expands upon that to show you how to properly apply what you've learned to games using 3 or more boards. This part also includes how to present it as a game to play, and adds some other tips and resources to explore.
The tutorials are set up so that you learn a concept, practice it, and then move on to the next concept. Ideally, this helps make each part clearer, and easier to learn and remember.
If you have any suggestions or questions about these tutorials, or even if you want to share any fun stories relating to the game, post them in the comments!
This is the final post in the age guessing series.
As with the previous post, this will mix the approximate judging of someone's age with math, but this version hides the math better. That's because it focuses on calendar dates!
In this approach, you have a spectator give you their birthday, without the year. You then have the spectator look up the day of the week on which they were born in a perpetual calendar, and tell you that day of the week (again, without telling you the year). You then look them over, and announce their exact age!
For example, let's say your spectator tells you they were born on June 16th. After looking up the day of the week they were born, they tell you it was a Friday. After looking them over briefly, you announce (correctly) that the person was born in 1978!
Basic concepts
This is the most advanced age-guessing routine, and you'll need to master two other skills before learning this feat.
In the classic calendar formula, you add m (month key) + d (date key) + y (year key) = a (answer key for day of the week). In this feat, we're focusing on age, so we need to rework this formula to provide the year key as an answer. Subtracting d and m from both sides, we get a - d - m = y. In other words, we start with the key for the day of the week, subtract the key for the date, then subtract the key for the month, and we'll wind up with the key number for the year they were born.
Remember, though, that a, d, and m can all be numbers from 0 to 6, so you might wind up with problems such as 4-6-5. To avoid dealing with negative numbers, it's best to start by adding 14 to a, the day of the week key, right at the beginning. This will keep the day of the week key large enough to prevent negative numbers as answers. Since 14 is a multiple of 7, it won't change anything if you need to subtract multiples of 7 later on (I told you that would be important!). Modifying the formula to take this into consideration, and make the mental math easier, we have: (a + 14) - d - m = y.
Step by step
1) Begin right as you select the person whose age you will determine. Use your age-approximation skills to determine the person's approximate age, and save this for later as your preliminary guess.
Example: Let's say our spectator looks to be in his mid-30s, so you make a preliminary guess of 35. You don't mention this guess out loud. If you're performing this in 2012, you work out that being 35 means that he would have been born in 1977. For now, just keep the year 1977 in the back of your mind.
2) Ask them to give you their birthday, but without the year. As you explain about looking up the day of the week on which they were born, you'll need to recall the month key, and reduce the date they give by subtracting the multiple of 7 lesser than or equal to the date you were given.
3) Have them look up the day of the week in which they were born, and announce that day of the week. This can be done using a perpetual calendar you bring, or by using an app or website on their mobile device. Mentally convert the date they give you into its key number (according to the day of week key chart here).
Example: They look up June 16th in a perpetual calendar, and announce that they were born on a Friday. The key number for Friday is 5.
4) Now that you've got all the numbers you need, plug them into the formula: (a + 14) - d - m = y.
Example: Since a (answer key for day of week)=5, d (date key)=2, and m (month key)=3, we work out (5+14)-2-3=19-2-3=17-3=14.
Example: Since our spectator definitely appears to have been born in the 1900s (1977 was our preliminary guess), we work out 14-1=13.
6) At this point, if your mental running total is greater than 6, subtract the nearest multiple of 7 equal to or less than the total to get your final year key.
Example: Our running total is 13, and the nearest multiple of 7 equal to or less than the total is 7, so we subtract 13-7=6. 6 is the final year key in this example.
7) Recall your preliminary mental guess from step 1. Using your memorized list of year keys, ask if the corresponding year in the 2000s (the year 100 years later) has the same year key as the one you calculated.
Example: Our preliminary guess was 1977, so we think about 2077 (remember, we subtracted 1 to adjust the year to the 2000s, which we've already memorized), and recall that it has a year key of 5. The year key we're looking for is 6, so '77 is obviously not the correct year.
8) Try changing forwards or backwards by one year, and find the closest year with the correct key. While you're mentally searching for a year, you can pretend to be studying the person closely for signs of their age. This not only gives you more time for your mental search, but can potentially be very entertaining, as well.
Example: Since 1977 was wrong, yet very close, we move forward a year and try 1978. Recall that 2078 has a year key of 6, which is the year key we're looking for!
9) Once you've found the closest year with correct key, work out the age that would make them and announce that as your guess! Assuming you're correct, bow to thunderous applause!
Example: We worked out that 1978 (well, actually 2078, but in 2012, we can be sure they weren't born in 2078) has the correct key. Being born in 1978 means they'll turn 34 in 2012, so we make a guess of 34 out loud!
Even if you get their age wrong (hopefully by guessing too young, as people will always forgive that), you can still save the trick by pointing out that the age you guessed would've put their birthday on the correct day! This is still quite impressive, and implies a seemingly impossible knowledge of dates.
Unlike the original day of the week for any date feat, the emphasis here isn't on speed. As I mentioned in step 8, you can do your mental calculations while walking around the person and pretending to be examining them for signs of their age, which you've already secretly done before the trick even started.
As always, don't forget that age can be a touchy subject, and treated with caution. Explain at the beginning that you want someone who is willing to not only state their actual age, but have their age announced out loud before an audience, as well.
I hope you've enjoyed this series on age guessing. If you have any questions about any of the posts, please let me know in the comments, and I'll do my best to answer them.
In this post, you'll learn an approach that seems to be a math trick, yet seems impossible to explain in that way.
You starting by asking a spectator to put any 5-, 6-, or 7-digit number in their calculator. You then tell them to multiply that number by 9. The last steps are to add their age to that number, and then show you the resulting number.
You examine the number for a few seconds, and instantly announce their age!
Why this is deceptive
Let's say you perform this, and the resulting number on the calculator is 1,248,695, and you announce that the person's age is 26, which is confirmed by them.
Mathematically, all they have is the formula 9x + y = 1,248,695. With two variables, that equation (known to mathematicians as a Diophantine equation) has an infinite number of solutions. How is it possible that you could narrow down the possibilities so quickly, and in your head?
How it works
When you're shown the total, you first add the digits of the answer up in your head. Next, using the age-guessing skills you learned in the previous post, ask yourself if the person could be that age.
If the person seems older than that, add 9 to the number you got and ask if that seems to be a more reasonable age. If that doesn't seem right, move up or down in 9-year increments, and keep doing that until you find an age that seems right.
In our example, you'd see the answer 1,248,695, so you add 1+2+4+8+6+9+5=35. Ask yourself if the person could reasonably be 35. Let's say they look younger than that, so you subtract 9. 35 - 9 = 26, so you consider 26, which we'll say seems more reasonable, so you guess that number out loud.
When you sum up the digits, the result is known as a digit sum. The digit sum of 99 is 18 because 9+9=18. In the video above, notice they keep repeating the process of taking the digit sum until they get a 1-digit number. If you do this, the 1-digit number you get is called the digital root. The digital root of 99 is 9 because 9+9=18, and 1+8=9. The point of the above video, of course, is that any number multiplied by 9 will have a digital root of 9.
What happens when you add a number to a multiple of 9? Let's take 5 as an example. 9+5=14, and the digital root of 14 is 5 (1+4=5). 18+5=23, and the digital root of 23 is 5 (2+3=5), and so on. Let's 18+14, which is a multiple of 9 plus a number with the digital root of 5. 18+14=32, and 32's digital root is 5! Also notice that the answers remain spaced by multiples of 9: 5, 14, 23, 32, and so on. In short, any time you add a number to a multiple of 9, the answer will always have the same digital root as the number you added, and you'll always be a multiple of 9 away from another number with the same digital root.
Applying this to the trick, when you multiply by 9 and add the age, the digit sum (1+2+4+8+6+9+5=35 in our above example) will not necessarily be their age, but will have the same digital root as their age, and be some multiple of 9 away from the correct age (even if that multiple is 0).
Try this out for yourself. Get a calculator, enter any 5-, 6-, or 7-digit number, multiply that by 9, then add your age. Take the result, and enter it into the widget below, then click Submit. A window will pop up showing all the possible ages (listed as the variable a) between 0 and 100 you could be, based on the number you entered.
Sneakier ways of getting to a multiple of 9
If someone is familiar with the effects of multiplying by 9, they might suspect what you're doing. There are other less obvious ways of getting to a multiple of 9:
• From a sidebar in in Karl J. Smith's Nature of Mathematics (available at Amazon.com): Mix up the serial number on a dollar bill. You now have two numbers, the original serial number and the mixed-up one. Subtract the smaller from the larger. Assuming you didn't create two identical numbers, the result will have a digital root of 9, because you're subtracting 2 numbers with identical digital roots (More about this principle here).
• Also from the same sidebar in in Karl J. Smith's Nature of Mathematics: Using a calculator keyboard or push-button phone, choose any 3-digit column, row, or diagonal, and arrange these digits in any order. Multiply this number by another [3-digit] row, column, or diagonal. As it happens, most numeric keypads are arranged in such a way that any row, column, or diagonal of the numbers 1-9 will make a multiple of 3. Multiplying two multiples of 3 together will always result in a multiple of 9.
• You could also adapt Scam School's first Pi Day Magic Trick (YouTube link). Have them multiply 1-digit numbers together as shown in the video, until you get to a number somewhere between 1 million and 1 billion. Instead of having them remove a digit as in the original routine, however, have them add their age instead. As you see in the video, though, it is possible to get a number like 8,100,000,000. Adding their age to that would be obvious (assuming the guy in the video is 22, he'd get 8,100,000,022). To prevent this, tell them to avoid pressing the 5 and 0 keys, as this will just result in a lot of zeros at the end (or just one in the case of multiplying by 0).
These aren't the only secret ways to get to a multiple of 9, but are varied and interesting enough to get you started.
If you'd like an age-guessing routine that has the precision of math, but without the appearance of math (or even use of a calculator), I think you'll enjoy the next post, which will be the final installment in our series on how to guess people's ages.
Just knowing these tips isn't of much good without practice. Thankfully, there are several sites where you can practice guessing the age of random people:
Even though carnival age-guessers aren't having you put any numbers in a calculator, they're still able to use some very subtle math tricks. For example, instead of advertising that they'll hit your exact age, you'll usually see a margin of error such as, “I'll guess your age within 3 years!” That sounds quite close to most people.
Even more central to an age-guesser's actual purpose is the simple economics of the situation. Let's assume that the cost to have the carny make a guess is $3, and the cost per stuffed animal to the carnival is $.25 (since they buy them in bulk). If we assume the guess is wrong every time, perhaps to keep every customer flattered, they're making an 1100% profit on each prize!
As the guesser becomes more skillful, the profit margin goes up! If we assume the age-guesser can correctly guess the ages of 4 out of 5 people (an 80% success rate), then that's 5 people times $3/person or $15 they're taking in. Only 1 wrong guess out of those 5 means that they're giving up $.25 for every $15 they take in, a staggering 5900% profit margin!
So, when it comes down to it, age-guessing as a skill is all about the margin of error and the profit margin. And that's assuming they don't employ standard scams like writing two ages and then covering up the one that's farther away, using magician's techniques to write down a close answer after you state your age, or simply pickpocketing your wallet and looking at your ID.
Guessing ages is a skill, but only ever an approximate one at best. The mathematical approaches, as we've seen, offer precision. Perhaps the best approach is to develop the skill of age-guessing, and use math in a way that doesn't detract from the skill.
That's the approach we'll start developing in the next post in this series.
Back in 2008, I wrote a post about guessing ages. Unfortunately, it was several approaches compacted into one long post and lacked clarity, as a few readers have noted.
I've decided it's time to update the post. I'll break age-guessing up across several posts in an effort to improve the clarity, as well.
In this post, I'll start with the methods for finding someone's age using purely mathematical methods.
In the original Age Guessing post, I also linked to this age plus a secret number approach, which explains it's own algebra, and these two algebraic approaches, one of which breaks up the age into two different numbers, and the other that makes use of the year the person was born.
These types of tricks can be very impressive for an audience unfamiliar with the basic concept of algebra, and can also be a great way to introduce new students to algebra. Anyone beyond that stage, even if they can't work it out at the moment, will recognize that there's some simple pattern that will get you the answer. Since this is the case, perhaps there's a mathematical approach that is more deceptive.
A deceptive approach that's long been a favorite of magicians is one known as the Age Cards. You can find an interactive version of it at this link. Look for your age in each group. If you see your age in a given group, click the checkbox for that group. Once you've checked all six groups for your age, and clicked where appropriate, click on the CALC button. The computer will tell you your age!
It works simply by adding up the smallest number (the one on the upper left corner) on each card on which the age was seen. If your age was 27, you would only click the boxes of Group One (smallest number is 1), Group Two (2), Group Four (8), and Group Five (16). Adding 16 + 8 + 2 + 1 gives 27, so the chosen age is 27.
That's how the trick is done, but why does it work that way?
The method here is better hidden than the algebraic methods because instead of using our usual base 10 numbering system, which uses the digits 0 through 9, the Age Cards trick is based on the base 2 numbering system, better known as binary, which only uses the digits 0 and 1. Working with a different number base can seem scary and confusing, but BetterExplained points out that you work with different number bases more than you might think.
How does this all relate to the Age Cards? Note that there were six Age Cards used. Each card acts like one of the places in the binary number. Note that the smallest number on each card corresponds to one of the binary places, as well: 32, 16, 8, 4, 2, and 1.
To find out where a given number goes, we use it's binary code. As mentioned, 27 converts to 11011. We're working with 6 cards, though, so just like our regular base 10 numbering system, we can add zeroes to the left side without changing the value. Doing this, 11011 becomes 011011.
The rightmost spot in binary is the 1s spot, and if there's a 1 there, as there is in our 27 example, we put that number on the 1 card. There's a 1 in the twos place, so we also put 27 on the 2 card. There's a 0 in the 4s place, so we don't put 27 on the 4s card. The 1 in the 8s place and the 16s place indicate that the 8 and 16 cards do have 27 put on them. Finally, the leftmost 0 in the 32s place tells us not to put 27 on the 32 card.
In the video above, 38 only appeared on the 32 card, the 4 card and the 2 card because 38 in binary is 100110, which only has 1s in the 32s place, the 4s place, and the 2s place. Get the idea?
The Age Cards is well-known among magicians, so even this routine could benefit from a better disguise. Fortunately, Werner Miller has come up with some very creative work on the Age Cards!
First, there's his ingenious Age Cube, which is presented as a giveaway with five magic squares on it. You ask someone who is 31 or younger (because we're only working with 4 binary places) on which magic squares they see their age, and thanks to your secret addition of the numbers in the upper left corner of each magic square, you can magically divine their age!
His other approach comes as a webapp that works in any modern browser, and also as a Windows executable file. It's called Age Square, and builds impressive from the Age Cube. It only uses 4 binary places, but thanks to a secret better described in the original Age Square post, it still manages to cover ages from 30 to 85! Instead of giving the age directly as an answer, the app generates a new magic square, with their age as the total.
Divining someone's age purely using math can be interesting, but what about getting someone's age with some help from their appearance? That will be the topic of the next post in this series.
I teach quite a few fun mental challenges over in the Mental Gym.
While I teach methods in as simple and straightforward a manner as possible, there isn't always just one approach. In this post, I'll take a look at new approaches to feats in the Mental Gym.
Besides making the squaring of two digit numbers easier, this video also illustrates a good point about algebra. Algebra lets you see patterns of which you may not have been previously aware, and help you see a shorter, and possibly better approach.
These lessons are especially handy for students taking trigonometry. Here's a handy approach to memorizing the unit circle, especially useful for tests, that works solely by taking advantage of several simple patterns:
We'll wind up this post by focusing on two of the puzzles.
For much of the world, today is Mother's Day!
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