It's time for October's snippets, and all our favorite mathematical masters are here to challenge your brains!
• I'm always looking for a good mathematical shortcut, in order to make math easier to learn. More generally, I'm always looking for better ways to improve my ability to learn. I was thrilled with BetterExplained.com's newest post, Learn Difficult Concepts with the ADEPT Method.
ADEPT stands for Analogy (Tell me what it's like), Diagram (Help me visualize it), Example (Allow me to experience it), Plain English (Let me describe it in my own words), and Technical Definition (Discuss the formal details). This is a great model for anyone struggling to understand anything challenging. This is one of those posts I really enjoy, and want to share with as many of you as I can.
• If you enjoyed Math Awareness Month: Mathematics, Magic & Mystery back in April, you'll love the 31 Tricks and Treats for October 2014 in honor of the 100th anniversary of Martin Gardner's birth! Similar to Math Awareness Month, there's a new mathematical surprise revealed each day. It's fun to explore the new mathematical goodies, and get your brain juices flowing in a fun way!
• Over at MindYourDecisions.com, they have a little-seen yet fun mental math shortcut in their post YouTube Video – Quickly Multiply Numbers like 83×87, 32×38, and 124×126. As seen below, it's impressive, yet far easier than you might otherwise think:
They've also recently posted three challenging puzzles about sequence equations that you might want to try.
• If that's not enough, Scam School's latest episode (YouTube link) at this writing also involves three equations. If you have a good eye for detail, you may be able to spot the catch in each one before they're revealed:
That's all for this October's snippets, but it's more than enough to keep your brain puzzled through the rest of the month!
It's time for October's snippets, and all our favorite mathematical masters are here to challenge your brains!
About 2 years ago, I posted about Russian/Egyptian multiplication, and included a technique for mentally converting decimal (base 10) into binary (base 2).
Recently, Presh Talwalkar covered this same technique on his Mind Your Decisions blog. I've only just realized that with a little modification, this technique can be used to quickly and mentally convert decimal to any base 2 through 9!
We'll use the Mind Your Decisions binary conversion video as a starting point. It's less than 3 minutes long, so it's a quick study:
In both my original Power of 2 post and the above video, the idea of ignoring the remainder is emphasized. Funnily enough, changing the technique to focus on the remainder makes this basic idea much more usable. If you remember division problems with answers like, 22 ÷ 6 = 3 remainder 4, that's the type of division we'll be using in this post.
The first step is simply to take the given number and divide it by whatever base you're using, so that you have a quotient and a remainder. For a starting example, we'll convert the decimal number 84 into base 5. 84 ÷ 5 = 16 (the quotient), remainder 4.
The second step is to write down the remainder. In our example, we'd simply write down the 4.
Step 3 is to divide the quotient by the base again. This time, we'd work out 16 ÷ 5 = 3 remainder 1.
Step 4 is to write down this remainder to the immediate left of the previous remainder. Writing down the 1 to the immediate left of the 4 gives us 14.
Repeat steps 3 and 4 until you get a quotient of 0, at which point, you've got your answer! Finishing up our example, we'd use our current quotient of 3, divide that by 5, getting an answer of 0 remainder 3, write the 3 down to the left of the previous remainders, giving us 314. Since our quotient is 0, we also know we're done! Checking with Wolfram|Alpha, we see that 84 in base 5 is indeed 314!
TIP #1: Once your quotient is a number less than your base, you can simply write that to the left of the remainders and know you're done. In the above example, once we got down to 3, and we realize this is less than 5, we know this is the final step. Because of this, we can simply write the 3 down and stop.
In short, as long as you're given a decimal number and a base by which you're comfortable dividing that number, you can convert that number to that base in your head with little trouble. Not surprisingly, knowing division shortcuts and divisibility rules can be of great help here.
What about 147 (in base 10) to base 4? As long as you realize that the closest multiple of 4 is 144, and that you can handle 144 ÷ 4 in your head, the rest of the conversion shouldn't be a problem. 147 ÷ 4 = 36, remainder 3. Write down the 3, and then work with 36. 36 ÷ 4 = 9, remainder 0, so write the 0 to the left of the 3 (03), and work with 9. 9 ÷ 4 = 2, remainder 1. Write down the 1 to the left of the previous remainders (103). Tip #1 above tells us that, since 2 is less than 4, we can just write down that 2 to the left of the other numbers (2103) and know we're done. Sure enough, 147 in base 4 is 2103!
TIP #2: If the given number is less than the square of the base to which you're converting, you can do everything in a single step. All you have to do is work out the quotient and the remainder, write the quotient to the left of the reminder, and you're done! For example, what is 59 in base 8? 59 ÷ 8 = 7 remainder 3. Write down the 3 as before. Thanks to tip #1, we know that we can write the quotient down to the left, since 7 is less than 8, so we just write the 7 down next to it!
For base 8, this will work for any number less than 8 × 8, or 64. Similarly, for base 5, this will work for any number less than 25 (5 × 5), and so on. 44 in base 7? 44 ÷ 7 = 6 remainder 2, so we can quickly give the answer as 62!
Being able to convert to base 2 and base 8 in your head can be a great asset when working with computers. Practice this skill and have fun with it. You'll not only have a useful skill, but something with which to amaze and amuse others, as well!
A little over a year ago, I teased Grey Matters readers with a mystery skill. First, they had to learn to easily multiply by 63, then learn how to easily multiply by 72. The skill itself was revealed to be how to roughly convert any whole number of years into seconds!
In this post, you'll learn a similar skill: how to convert weeks into minutes instantly!
Back in the days before computers and calculators, this was a popular feat among entertainers who performed as human calculators. It was quick and direct to perform, yet was highly impressive to audiences.
One week has 7 days, and each day has 24 hours. Every hour, of course, has 60 minutes, so if we multiply out 1 week × 7 days/week × 24 hours/day × 60 minutes/hour, we get 10,080 minutes in a week. The number 10,080, as it happens is very easy to multiply by almost any number of weeks. If you keep the number of weeks at or below 124 (about 2.37 years), the numbers are even easier to work out.
STEP 1: Ask for any number of weeks less than 2 years (104 weeks). As an initial example, we'll say an audience member gave the number 36.
STEP 2: Write down the number they just gave you. In our example, you'd write 36.
STEP 3: Multiply this number by 8 in your head, and write this result to the immediate right of the first number you wrote. This is simpler than it sounds; all you have to do is double the number 3 times. For 36, doubling once gives you 72, doubling a second time gives you 144, and doubling a third times gives you 288. Writing 288 next to the 36 you wrote earlier gives you 36288.
NOTE: In step 3, it's very important to always treat the answer as a 3-digit number. For weeks from 13 to 104, it will be, but for weeks from 2 to 12, it will be a 2-digit number. You can change this into a 3-digit number simply by adding a 0 to the left of it. If you're given 7 weeks in step 1, you write down the 7 as in step 2, then multiply 7 × 8 to get 56, which becomes 056. You would write 056 as your step 3 answer, giving 7056, and then continue with step 4.
STEP 4: Write a zero to the immediate right of the other numbers, add commas where appropriate, and you're done! In our example, we add the zero to the right, giving us 362880. With commas, that result is 362,880. This means that there are 362,880 minutes in 36 weeks!
With a little practice, you'll be astounded as to how quickly you can pick this impressive skill up. You can quiz yourself by having Wolfram|Alpha give you a random number of weeks from 2 to 104, and then use it to verify whether you've worked out the correct answer.
HANDY BONUS TIP: You can make this more impressive for an audience by having someone with a calculator verify this in a long, drawn-out manner. Tell them to put in the number of weeks given, then multiply by 7 for the number of days in a week, then multiply by 24 hours in a day, and then multiply by 60 minutes in a week. Multiplying it out the long way makes this feat seem more difficult, as you're hiding the simple 10,080 conversion factor.
Try it out and amaze your friends!
As you can probably tell from this recent post and this recent post, I've spent quite a bit of time thinking about the Knight's Tour lately.
These thoughts have reminded of a different type of Knight's Tour puzzle. This unusual variation involves moving the knight around a calendar.
It was 4 years ago, during September or October, that I was looking for blog post inspirations and ran across a thread on the XKCD forums, titled Knight's Tour revamped, which suggested playing the Knight's Tour on a calendar.
There was an added challenge, however: With your starting square being considered as move #1, how many dates could you land on that were the same as the move number? For example, if move #1 started on the 1st of the month, both the move number and the date would be 1.
As you can see in the original thread, the original poster used a July 2010 calendar and managed to find a complete Knight's Tour on which the 2nd, 6th, 11th, and 23rd moves landed on the dates of the 2nd, 6th, 11th, and 23rd respectively. Not surprisingly, it was Jaap of Jaap's Puzzle Page who found an 8-match solution.
I filed this in the back of my mind, but never really did anything until I ran across the Solving the Knight’s Tour on and off the Chess Board post which I mentioned last week. I liked the basic idea of being able to input a shape, and have the computer work out the tour, and especially the idea of using it to work out the XKCD forum's calendar challenge.
With a little knowledge of Java and graph theory under my belt, I managed to work out a program to solve it. For my fellow Java programmers, here's the main portion of my program, and here's the KnightsTour class I wrote to support it. Most of the hard work is done by lines 590 to 749. Those and lines 20 to 23 can removed if you're interested more in the general Knight's Tour than the particulars of the calendar challenge.
One of the first things I did, not surprisingly, was to find out how many day-to-move matches I could find in this month's calendar. I also found 8, all of which are highlighted below in red:
Yes, I've gone through every possible calendar, starting on every possible date, and learned quite a few interesting things in the process:
• Due to the fact that the number of days in a week (7) is odd, and the fact that the knight always moves an odd number of spaces (3), this means that a Knight on a calendar will always move from an odd date to an even date, and vice versa (just like what your teacher taught you about adding even and odd numbers). This, in turn, means that it's impossible to get ANY date matches if move #1 begins on an even-numbered date, as all the odd moves will land on even dates, and vice-versa.
• The above fact also means that if you start on an even date in a month with an odd number of days (29 or 31), you won't be able to complete a Knight's Tour.
• Yes, Jaap's 8-match path is the best one possible for July 2010 in particular. It also happens to be the only way to get 8 date-to-move matches in a Knight's Tour of a 31-day month beginning on a Thursday.
• Given any random month and year, you can always find a complete Knight's Tour and at least 6 date-to-move matches. Surprisingly, these minimum matches aren't found in the shortest months, as you may expect. With 30- and 31-day months starting on a Saturday, as well as 31-day months beginning on a Friday, 6 is the highest number of date-to-move matches you'll be able to find.
• There are months with 9 date-to-move matches, but none with more than that. 9 date-to-move matches can be found in a 29-, 30-, or 31-day month starting on a Tuesday or a Wednesday. In a 29-day month starting on a Thursday, or a 31-day month starting on a Monday, you can also find 9 date-to-move matches. You can often find more than 1 way to get to these matches, as well.
As it happens, next month (October 2014) is a 31-day month starting on a Wednesday, and here's one of the 3 possible ways to get 9 date-to-move matches:
I chose this one simply for the elegance of the column containing 16-23-30 and the diagonal containing 12-20-28. I also find it interesting that so many powers of 2 have date-to-move matches (2-4-8-16).
For the more math-inclined geeks, I'll wind this post up with all the maximum number of matches I've found, including the dates on which they start:
28-day months, starting on: Sunday: 7 matches, beginning from the 1st or 23rd Monday: 7 matches, beginning from the 1st or 21st Tuesday: 8 matches, beginning from the 25th Wedneday: 8 matches, beginning from the 1st Thursday: 8 matches, beginning from the 1st or 5th Friday: 8 matches, beginning from the 1st or 15th Saturday: 7 matches, beginning from the 23rd or 25th 29-day months, starting on: Sunday: 7 matches, beginning from the 1st Monday: 7 matches, beginning from the 1st or 27th Tuesday: 9 matches, beginning from the 25th Wedneday: 9 matches, beginning from the 1st or 11th Thursday: 9 matches, beginning from the 1st Friday: 7 matches, beginning from the 1st, 5th, 27th, or 29th Saturday: 7 matches, beginning from the 1st 30-day months, starting on: Sunday: 7 matches, beginning from the 7th or 23rd Monday: 8 matches, beginning from the 27th Tuesday: 9 matches, beginning from the 25th Wedneday: 9 matches, beginning from the 11th Thursday: 8 matches, beginning from the 1st Friday: 7 matches, beginning from the 1st or 7th Saturday: 6 matches, beginning from the 1st or 25th 31-day months, starting on: Sunday: 8 matches, beginning from the 23rd Monday: 9 matches, beginning from the 7th or 31st Tuesday: 9 matches, beginning from the 1st, 23rd, or 25th Wedneday: 9 matches, beginning from the 7th Thursday: 8 matches, beginning from the 5th Friday: 6 matches, beginning from the 1st, 5th, 7th, or 31st Saturday: 6 matches, beginning from the 1st, 23rd, 29th, or 31st
Since I've changed my posting schedule, I seem to have neglected my monthly snippet posts!
Not to worry, however, as we're kicking off September with a good round-up of different takes on some of my favorite mental feats.
• One of the longest-standing tutorials on Grey Matters is the classic Knight's Tour. The traditional version usually happens on an 8 by 8 chessboard. What about other irregular, non-rectangular shapes?
Over at the Wolfram Blog, Jon McLoone explored that question using Mathematica in his post Solving the Knight’s Tour on and off the Chess Board. If you're interested in the programming and the math, there's plenty in this article. Even if you don't care for all the math and programming, the variety of boards with successful Knight's Tours is amazing and amusing. Who knew Pac-Man could play the Knight's Tour so well?
• Over in the Mental Gym, I have a full tutorial on squaring 2-digit numbers in your head. I've often wanted to move on to squaring 3-digit numbers, but never really found a method that suited me. However, I recently ran across a video tutorial from Mind Math called Mental Math Trick to Square 3-digit Numbers for Faster Calculation. It breaks the problem up into 2 steps, working with the hundreds digit followed by the remaining 2 digits as a group. If you're used to squaring 2-digit numbers, this method isn't difficult to learn and adapt:
• Back in March, I wrote a post about calculating powers of e in your head. At the time, I was unaware of Colin Beveridge's post, Secrets of the Mathematical Ninja: Estimating Powers of e, which featured a quicker, yet less accurate estimate.
After seeing my post, Colin took it upon himself to develop an improved method, which he posted as Powers of e Revisited: Secrets of the Mathematical Ninja. When you're done exploring those posts, check out the rest of Colin's Blog!
• Another favorite blog topic of mine is calendars. Beyond the standard day of the week for any date feat, there's plenty of interesting mathematical patterns and shortcuts waiting to be discovered in the calendar. One of the best round-ups I've found on the internet is P.K. Srinivasan's Number Fun with A Calendar (PDF version). Besides the PDF version, there's a zipped .DOC version and even a video demonstration of some of the topics from the book:
That's all for this month. I hope you found these enjoyable and useful!
One of my favorite mental challenges, as many regular Grey Matters readers know, is the Knight's Tour. The challenge is, using only the chess knight's L-shaped move, to land on each of the 64 squares once.
Mentalist Richard Paddon recently released a download resource titled The Knight's Tour: A Scenic Journey. In this post, I'll take a close look at this new take on a classic feat.
We'll start with a quick peek at Richard Paddon himself performing the Knight's Tour, via the teaser ad:
The Knight's Tour: A Scenic Journey comes as a set of 3 files: A 45-page PDF of the same title, a 16-minute MPEG file of Paddon's performance, and the Knight's Tour Windows application used by Paddon, and programmed by Dave Everett.
In the PDF, right away, the author emphasizes the importance of developing drama in the Knight's Tour presentation. The first parts of the actual instruction, however, focus on developing the path through the board. Much of this part of the book may be familiar to readers of the Knight's Tour section of Paul Brook's Chrysalis Of A Polymath. However, Richard Paddon does add some new and helpful notes, such as the section on what he has dubbed delta values, which are familiar to those who have programmed a Knight's Tour, but little discussed in the use of performances.
In the next half of the book, Paddon discusses the presentational details. He starts with the benefits of the Knight's Tour, including its uncommon nature, and its huge potential on an emotional and theatrical scale. The thoughts behind the presentations are well laid out. Even if you disagree with any aspect of the presentation as written, you at least have a good starting point of why particular choices were made.
One of the more interesting choices is ending on a selected square, as seen in the above video. As the board empties, the chosen square becomes a more and more important focus, and becomes a natural point of building tension. The PDF winds up with a detailed description of how to use the program.
There are very few weaknesses in this product overall. One of the one that stands out to me as both a programmer and a blogger of mental feats was the choice of the Comic Sans font for the numbering of the board. If you're taking as much care as this author does to make an impact on the audience, there's probably better ways to label your board than a font designed specifically to have a comic-book appearance. On an equally minor note, the lightning in many of the shots of the performance video could be better. The importance of the video is for a more complete understanding of the presentation, so this isn't a huge drawback.
Overall, this is an excellent value for anyone seriously interested in performing the Knight's Tour. The basics of working through the path may be easily accessed in multiple sources, but the depth of knowledge that is presented, as well as the use of multiple media to demonstrate this make this the most complete lessons about all aspects of the Knight's Tour and its proper performance.
It's available for only $9.95 over at Lybrary.com and is a remarkably great value for that money. If the Knight's Tour interests you, Richard Paddon's The Knight's Tour: A Scenic Journey is a must-read.
People are often confused as to why the dates of Easter moves around so much from year to year. It moves so much much because Easter is the first Sunday after the first full moon after the first day of spring.
If this sounds confusing on its own, consider that the Roman Catholic and Eastern Orthodox churches use different calendars, which can yield different dates as a result!
Thanks to the work of John Conway, though, it is possible to work out the date of both Roman Catholic and Orthodox Easters in your head!
Some basic understanding and practice are all that's really needed to be able to calculate the Easter date in your head for any year from 1900 to 2099. In order to help make everything clearer, I've posted my new Easter Date For A Given Year tutorial over in the Mental Gym. To make it easier to learn, the tutorial is broken up into several steps:
• The introduction explains the rules for Easter calculation in detail, as well as what you need to know to get started.
• The next section explains how to calculate the date of the traditional Roman Catholic Easter. After learning how to work out the date of the Paschal full moon (the first full moon after the first day of spring in a given year), you then learn how to work out the date of Easter for that same year.
• If you want to impress others by performing this feat, there's an entire section of presentation tips that can help make this feat entertaining.
• The method for calculating the date of Orthodox Easter is covered another section, as well. Assuming you can work out Roman Catholic Easter, there are surprisingly few changes involved in working out the Orthodox Easter date.
• Finally, there's another section for those adventuresome souls who want to venture on and work out Easter dates in other centuries. Here you can find out what changes need to be made to the original calculations.
Since practice is important, I've also developed a set of interactive Easter date quizzes. Since you work through each section verbally in a step-by-step manner, the quizzes work the same way. In the first quiz, you simply work out the paschal full moon date for Roman Catholic Easter. In the next quiz, you're asked about the paschal full moon and Easter dates. The Orthodox quizzes are similar, and start with the Roman Catholic dates first, since you need that information as a starting point.
If you put in a little understanding, a little practice, and a little time, you may surprise yourself (and others) with an impressive new skill!
One of the more popular Mental Gym tutorials, probably because it's short and simple, is the Squaring 2-Digit Numbers Mentally tutorial.
Playing around with the methods in the math section of that tutorial, and doing a little research, I've run across an interesting pattern that make the calculations even simpler.
I'd noticed a pattern in some of the squares concerning their last two digits, but never really thought about the possibilities until I ran across this page about squares.
I've reproduced a slightly modified version of the number arrangement from that page below, but with the last two digits of each square highlighted. Although it's not shown here, the 2-digit pattern does continue on forever.
As I mention in the original tutorial, memorizing the squares of the numbers 1 through 25 is the most basic starting point, as is knowing how to square 2-digit numbers ending in 5. From here, though, we can take advantage of the pattern above in a different way.
x x2 x x2 1 01 49 2401 2 04 48 2304 3 09 47 2209 4 16 46 2116 5 25 45 2025 6 36 44 1936 7 49 43 1849 8 64 42 1764 9 81 41 1681 10 100 40 1600 11 121 39 1521 12 144 38 1444 13 169 37 1369 14 196 36 1296 15 225 35 1225 16 256 34 1156 17 289 33 1089 18 324 32 1024 19 361 31 961 20 400 30 900 21 441 29 841 22 484 28 784 23 529 27 729 24 576 26 676 25 625
SQUARING NUMBERS 26-50: When asked to square a number from 26-50, take the distance from 25 to that number, and multiply it by 100. Next, square the distance from the given to number to 50, and add it to the previous number you calculated. Those 2 steps give you the square.
For example, let's square 27. 27 - 25 = 2, so you multiply 2 × 100 = 200. The distance from 27 to 50 is 23, which you should know by heart as 529. Add 200 + 529 = 729, and you've got the answer to 27 squared!
How about, say, 38? How far is that from 25? Yes, it's 13, so we start with 1300. How far is 38 from 50? It's 12, and 12 squared is 144. 1300 + 144 = 1444, so we know 38 squared is 1444!
This approach makes numbers in the 40s almost ridiculously easy to square. You just say their distance from 25, then their distance from 50 squared. For 42, which is 25 + 17, you'd say, 17... then square 8 (the distance from 42 to 50) and say, "..64!" With a little practice, numbers in the 40s almost square themselves!
SQUARING NUMBERS 51-75: As I mentioned above, the squares of the numbers beyond 50 continue with this pattern. Here are the squares of the numbers from 51 to 75, and here are just the last digits of each of those numbers.
The method for squaring these numbers is slightly different. After you're given the number, work out its distance from 50, add 25 to that distance, then multiply by 100. The final step is to square the distance from 50 to the given number, and add that to your previous calculation.
Let's use 56 as our first example. 56 - 50 = 6, so we add 6 + 25 = 31, and multiply by 100 to get 3100. 56, as we've already worked out is 6 away from 50, so we square 6 to get 36. 3100 + 36 = 3136, which is 56 squared! Notice that the 50s almost multiply themselves, just like the 40s did above!
Can you handle, say, 67 squared? That's 17 away from 50, so we work out 17 + 25 = 42, and 42 × 100 = 4200. Next, 17 squared is 289 (you know that from memory, right?), so we add 4200 + 289 = 4489!
SQUARING NUMBERS BEYOND 75: Squaring the numbers from 76 up to 125 can be handled just as in the original tutorial. You'll probably understand why this approach works more completely after seeing the pattern above.
Perhaps you can work out a method for numbers beyond 125? It can be done, but it's more than a minor variation on the above patterns.