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!