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## Finding the Fun in Math at Home

March 30th, 2020

Editor’s note: Hi folks, meet Heather Moore, who I’ve been collaborating with on Pixidoku puzzles. Heather is a math teacher and has some great tips to offer for those of you who are looking for at-home learning activities for your kids. Take it away, Heather! — Jim

In the midst of a global crisis, I’ve been noticing some silver linings and surprising upsides.  This is not to downplay the seriousness of it all or dismiss any of the real loss and suffering in our world, but I feel it is important to take hold of the good things that can be found amidst all the uncertainty and struggle.

Being stuck at home can be seen as a chance to spend more time with family.  Being unable to do some kinds of work can free up time for projects that otherwise get neglected.  And a break from math class can be an opportunity to play with math together.

I know, I know. The words “play” and “math” in the same sentence?  Hear me out.  In my world of teaching math to homeschoolers I find that the more playful I can make math, the better.  So today I want to share with you a couple ways I’ve been playing with math with my students this week.

## The 11’s Game

Quick!  What’s 11 x 7?  If that was easy for you, it is probably because you know the rule for this pattern: 11, 22, 33, 44, 55,… Just repeat the digit you’re multiplying by.  This is cool by itself, especially for younger learners, but it gets better!

I introduce the 11’s game by showing off a little for my students. They have calculators in hand to check me.  I tell them, “Give me any two-digit number and I will multiply it by 11 in my head.” “43!” Someone calls out. “473” I respond immediately. “52”, “572!” and the inevitable “99” “1089!”

Once they are impressed, or at least intrigued, I inform them that with the use of a simple pattern they can learn to play the 11’s game just as fast as I can.

Check out the following multiples of 11. What do you notice?

The pattern is easier to see than it is to describe, but here’s one way to put it:  Take the digits of the multiplier and add them together.  Put the resulting digit in between the two original digits to create your three digit product.  So for 63 x 11 you add 6 + 3 to get 9 and nestle the 9 in between the 6 and the 3 to get 693.

Always start off practicing examples where the sum of the two digits of your multiplier is less than 10.  Then see if you can expand the rule to allow you to deal with cases where the sum is greater than 10.

All of my tutoring that is usually done in person has had to move online for now.  I find the change exhausting, but pulling this game out with one of my students had him literally shrieking with amazement and delight, giving our online tutoring session a much needed energy boost.

After enjoying the game in a purely playful spirit, I test the waters for some deeper mathematical exploration.  Some students will be intrigued enough by this trick that they will want to figure out why it works.  I try my best not to explain, but to instead encourage them to make sense of it for themselves.  It doesn’t need to happen all at once, and patience here is rewarded with the pure delight of genuine mathematical discovery.

## The Game of Nim, or “Last Pebble”

The week after the schools in my area shut down, my family went out camping on family property for a week to get away from populated areas.  Even though I had my small dry erase boards with me and a Life of Fred math book, one of my favorite math games didn’t require any of those things.

One of the girls and I set about preparing for the game: We collected about 20 smallish pebbles and found a flat log to use as a playing surface.  When I’m in the classroom I use base 10 block unit cubes, but you can use coins, poker chips, dried beans or any other kind of small counter or token.

The American mathematician Charles L. Bouton named the game ‘Nim’ in 1901, but it is an ancient game, no doubt played under many names, and I like to call it “Last Pebble”, “Last Coin”, or “Last Cube” depending on what we’re playing with.

To play, arrange your “pebbles” in three piles.  One player gets to set up the piles, choosing the number in each starting pile, and the other player gets to decide who goes first.  Players take turns removing some number of pebbles from one of the piles.  You are allowed to take an entire pile if you like.  You must take at least one pebble on your turn and you may not take pebbles from more than one pile on your turn. The player to take the last pebble wins.

When I’m playing with a new player I start off by trying to win about half the time.  If I’m able to win by taking the final pile, I always do so, but I often will make deliberate bad moves earlier in the game to give my new opponent opportunities to beat me.

Once a new player has caught on to some of the basic strategy, I offer “puzzle mode”.  We play once, with both of us trying our best to win.  Having worked out quite a bit of the ideal strategy, I usually win. When I do, I will have noticed an interesting position late in the game that could be used as an instructive puzzle. Usually it’s either a game state where I knew for sure I was going to win, or where I noticed they could have won with a different move than the one they made.  We return to this game state repeatedly, with the new player choosing whether to go first or second each time we replay it, until they can beat me at the puzzle.

A basic puzzle set up to try is three counters in the first pile, three in the second, and zero in the third.  Again, reset the puzzle each time the player loses and have them choose again whether to go first or second. Here are some other good puzzles I use: (2, 2, 4) (1, 2, 3) (1, 3, 5).

For a deeper challenge, my students and I will try to map out ideal strategies for each player and get to the point where we can look at any starting board and be able to tell if the first or second player will ultimately win, given perfect play from both players.

Thank you for for reading.  I hope you find these ideas useful and that they help bring a little mathematical silver lining into your world.

(More from Heather at monkeyflowermath.com)

## The Alexa Game

July 11th, 2019

The Alexa Game is a multiplayer game I made up in which the object is to correctly predict whether Alexa (or Siri or Google) can answer a question.

Rules:

Players take turns being the Interrogator.

The Interrogator comes up with a question to ask Alexa, and makes a prediction as to whether Alexa will answer it correctly or not. The question should be written down so it can be repeated exactly.

The prediction is recorded, but not revealed to the other players until they have made their predictions.

The Interrogator reads his Alexa question to the other players without saying “Alexa”, so they can also predict the outcome.

Note: Anyone who says “Alexa”, except when directly addressing her, is deducted a point or must take a drink, or whatever.

Everyone writes down their prediction. At this point, all predictions may be shared.

Then the Interrogator says “Alexa…” and asks the question exactly as originally phrased. The players should decide whether Alexa’s response is correct. In general incorrect answers are ones where Alexa clearly misunderstood the meaning of the question, or bailed on answering it. It’s okay if Alexa reads from Wikipedia, if the reading contains the answer.

If the Interrogator guessed correctly, he gets a point for everyone who guessed incorrectly.

If the Interrogator guessed incorrectly, everyone who guessed correctly gets a point.

Note: If everyone’s prediction is in agreement, nobody will score. In this case you may optionally ask Alexa and award a point to Alexa if she defies everyone’s expectations.

The first person to reach 42 points wins.

UPDATE:
My wife finds my scoring system too complex. She’s right. Her suggestion (which also helps prevents cheating) is to not have the Interrogator make a prediction. The Interrogator just forms the question and judges the predictions of everyone else. You score if you predict right, and the Interrogator doesn’t score that turn.

## Mailbag Monday

January 21st, 2019

I hope this message finds you well and that your new year is off to a wonderful start! I just made a small donation through PayPal and also wanted to write you a personal message. I was born and raised in the US and during my elementary school years in a “gifted” classroom, we started every day with logic puzzles. I will never forget the fun, challenge, or satisfaction of starting every day tackling a logic puzzle! I now live in India, where I started Kranti NGO, which empowers girls from Mumbai’s red-light areas to become agents of social change. We currently work with 20 girls, ages 13-23 who are daughters of sex workers, survivors of trafficking, or girls born and raised in Kamathipura, one of Asia’s most infamous red-light areas.

In the past five years, Kranti’s girls have: become the first girls from India’s red-light area to study abroad, received UN awards for their social justice work, given 25+ TEDx and other speeches around the world, and written, directed and performed their own play in front of 100K+ audience members. Their latest performance, at the world’s largest performing arts festival in 2017, was covered by BBC with over ten million views. In 2016 Kranti’s alternative school was named a Top 10 for the \$1 million Global Teacher Prize, and three Kranti girls are currently studying in universities in Rome, Chicago, and Los Angeles.

Please watch the “Global Teacher Prize” video if you have a few minutes….it will give you an idea of how our curriculum works and why you are such a vital part of it. I’ve been using your worksheets at least 2-3 times a week ever since we started Kranti School in 2014. There are some girls in our home who are completely illiterate and have never been to a mainstream school, but they LOVE your puzzles. The variety of puzzles has really given the girls a chance to see how intelligent they are and that they do love learning…quite a feat for girls who grew up their whole lives hearing from their teachers that “a whore’s daughter can only be a whore.”

As we are an NGO and I do all of the fundraising for Kranti as well as teach school full time, I can’t afford much more than \$20, but I will continue to make contributions every time we get an education grant. Your website saves me so much time every week and there’s no amount of money I could possible send you that would be adequate for the amount of energy, passion, and love for learning that you’ve single-handedly brought into our classroom. I hope you’ll take a minute today to appreciate and thank yourself for the thousands of lives you must be touching around the globe without even knowing it. Personally, I’ll just say you’ve made my life as a teacher significantly more fun and manageable!

Lots of love and gratitude,
Robin

## Pants-rank: Comparing Apples, Oranges with Google Trends

May 18th, 2018

I was recently talking to another puzzle maven, and the subject of how I measure the demand for specific logic puzzles came up. I’m interested in puzzle demand because it helps inform the choices I make when I add new puzzles to the website. It occurred to me that my methods may not be as widely employed as I had assumed, and it might be worth sharing them. This technique is not just useful for measuring demand for puzzles, but I’ll use puzzles as an example.

If you’re interested in measuring demand, you probably know that Google has a free service called Google Trends which allows you to measure comparative search volume for multiple search terms. One can debate how much Google search-engine demand correlates to “actual” demand (whatever that is), but it seems to me like a reasonable quantifier that is likely correlated (and is simple and easy to obtain). For the rest of the article, I will often use the word “demand” as a placeholder for “Google Search Traffic”.

For example, here’s a chart comparing apples (blue) and oranges (red). Notice there’s a big bump on searches for Apples in the Fall, when they are presumably in season (or maybe people are buying iPhones? I don’t know…).

If I extend the chart over a 5 year period, I can see that the demand for apples and oranges follows a cyclical seasonal pattern – a common phenomenon with many search terms. Apples are popular in the Fall, and oranges in the Winter.

As you can see from these charts, Google doesn’t reveal actual search volume, it only shows relative search volume, which it always maps to a scale from 0-100. Because the numbers are presented as rounded integers, it is not particularly helpful to compare two search terms in which one has 100 times more traffic, because the smaller item will get scaled down to the 0-1 range, which isn’t very informative. For example, if I compare Sudoku (blue) with Slitherink (red), I can’t really learn much about Slitherlink, other than that its search traffic is < 1% of Sudoku, which doesn’t tell me much.

It would be cool if there was a way to measure relative demand for a group of items where several magnitudes separate the most desired from the least desired items. As it happens, there is.

So my “one weird trick” is to use a small stable search term as a baseline (I call this kind of system “pants-rank” for reasons that will soon be clear). Recently, I’ve been using Futoshiki, a relatively unpopular Japanese logic puzzle, as a baseline for measuring search demand for other logic puzzles. So I measure puzzle demand in “F-Units” (or futoshikis, in which the demand for Futoshiki is normalized to 1. So for measuring puzzle demand, Futoshikis are my “inch”.

Here’s the demand for Futoshiki (blue) and Slitherlink (red), two puzzles with similar demand.

On the real chart, if you hover over the “average” bars on the left, you get two integers (again scaled to 0-100) which indicate that Futoshiki has an average relative demand of 61 and Slitherlink has 38.

So the F-units of Futoshiki are 1 (by definition) and the F-units of Slitherlink are 38/61 (or .62). I can then work my way up to successively more popular puzzles, and work out the F-units for Kakuro (5.38) and Kenken (8.77). Eventually the scaling for Futoshiki becomes too small, so instead I use a known puzzle whose F-Units I’ve already determined as a baseline for measuring more popular puzzles like Sudoku and Crosswords. Using this technique, I’ve worked out that Sudoku demand is at approximately 395 F-units and Crosswords are at a whopping 857 F-units. Here’s a table showing F-Units for a number of puzzles I’m interested in.

For what it’s worth, the demand indicated by Google Trends does not necessarily match the demand for puzzles that I see on my website. This is because other factors come into play, like how well I’m ranked on Search Engines, puzzle scarcity, and other intangibles. For example, Suguru is a very popular puzzle on my site, for some reason (in the KenKen/Kakuro range), but doesn’t seem to be hugely popular on Google.

In the past I’ve used this same trick to measure many other non-puzzly things. When I first did this, I used “pants” as my baseline (another fairly stable search term that has a slight bump in the late Fall preceding Christmas), and used them to measure celebrity popularity in “P-Units” (Pants-units). Hence “pants-rank”. “Chairs” is actually a more stable unit as it is less prone to seasonal fluctuations. Here, for example, is Lady Gaga (blue) compared to pants (red) and chairs (yellow) for the past 5 years. The spikes likely correspond to album releases and TV appearances.

I can see that Lady Gaga has been averaging 0.43 P-units and 1.11 C-units over this period (and I also happen to know she is about 435 Futoshikis – about Sudoku level, but not quite Crossword level popularity. But who is, really?

## Star Battle: Oh my gosh they’re full of stars!

May 12th, 2018

The latest addition to my puzzle menagerie is STAR BATTLE, a newish (well, 21st century, at least) puzzle that was originally created by Hans Eendebak for the 2003 World Puzzle Championship.

If you like Suguru you will probably find these fairly easy to get into, as they are somewhat similar. But instead of placing numbers, you are placing stars – a fixed number of stars in each row, column and bolded region. Like Suguru, the stars may not be adjacent to each other, not even diagonally.

I find these very aesthetically pleasing. I think it has something to do with the stars. If you were placing sturgeon or aardvarks into the puzzle, I don’t think I’d like it as much, but honestly, you can use any symbol you like.

I am currently carrying two versions of this puzzle: 8×8 puzzles that have 1-star in each container, and 10×10 puzzles that have 2-stars. The 10×10 2-star puzzles are significantly harder than the 8x8s.

If you’re new to the puzzles, and find them difficult to get started, then you might find this tutorial I wrote helpful. Or not. Let me know.

Enjoy the puzzles!

STAR BATTLE

## Binox Puzzles are here!

April 30th, 2018

I get a lot of requests for new puzzle varieties (all different kinds) but I started getting a lot of requests for one particular puzzle the past couple of years. I didn’t notice at first, because the different requesters were using different names: Binairo, Tic-tac-logic, Binoxxo, Binario, (I later discovered it is also called Takuzo, Binary Puzzle, Bineiro, Brain Snacks, Unruly, Zinero and Zernero). All names for the same curiously addictive puzzle, sometimes filled with Xs and Os, sometimes with 1s and 0s. Regardless of the symbols used, the rules are the same:

1. Fill the puzzle with two symbols (like Xs and Os).
2. Horizontally and vertically, there can be no more than 2 of the same symbol touching.
3. There are an equal number of Xs and Os in each row and column.
4. Each row must be unique. Each column must be unique.

Today I am introducing this fun simple puzzle to my collection. I am calling my version Binox — none of the other names seems to be used by more than one publisher, and who am I to break tradition? My version is available in 5 sizes ranging 6×6 to 14×14. I offer four different difficulty levels (Easy, Novice, Challenging and Toughest).

You’ll find both printable Binox puzzles and interactive Binox puzzles here at Krazydad. The interactive puzzles are available in both the X/O format and the 1/0 format (use the gear-icon to change it). For the moment, I’m offering the print puzzles only in X/O, however, if this omission is keeping you up at night, let me know, and I’ll add a duplicate set with 1s and 0s.

Enjoy the puzzles!

## Celebrity Prayer Candle Rankings, 2018

March 2nd, 2018

Celebrity rankings, based on the number of prayer candles for sale on Etsy, 2018 EDITION

1.
Tupac Shakur

2.
Beyoncé
Kanye West

3.
David Bowie
Princess Leia / Carrie Fisher

4.
Dolly Parton
Harry Styles

5.
Britney Spears
Notorious Biggy Smalls
Lana Del Ray

6.
Drake
Kim Kardashian

7.
Leo DiCaprio
Snoop Dogg
Kendrick Lamar
Bill Murray

8.
Bernie Sanders
Donald Trump
Rihanna
RuPaul
Eminem

9.
Hillary Clinton
Morrissey
James Franco
Chance the Rapper

10.
Cardi B
Kurt Cobain
Jimmy Fallon
Justin Trudeau

## Books, books and more books

February 22nd, 2018

I started publishing print-on-demand puzzle books a few months ago, and I’m now up to 22 books. Yay! The latest are four volumes of Killer Sudoku, available in Large Print and 4-per-page. The most popular, at the moment, are the “Tough Kakuro” and the “Stupendous Suguru” books.

The easiest way to find my books is to search for “Krazydad” on Amazon.

If you think I’m getting rich from my burgeoning publishing empire, you would be wrong! I make a few dollars per book, and I’m selling about a book a day, which is keeping me in coffee, but not exactly paying for my dog’s chiropractor. If you’d like to help me work my way up to TWO cups of coffee, you can help by reviewing my books (honestly) after you buy them. Thanks!

The print-on-demand company I’m using is CreateSpace which is owned by Amazon. There are rumors that CreateSpace might get merged into Amazon’s very similar KDP (Kindle Direct Publishing) service. If/when this happens, it may affect the availability of my books for a short time, but they should reappear. My reason for not initially publishing on KDP is that it requires that the book be made available in digital form for the Kindle. Since my books require pencils/pens, they need to be on paper, not Kindles. I wouldn’t want Kindle owners buying virtual versions of my books only to be disappointed, and leaving bad reviews.

If you’re curious how this whole self-publishing thing works, and want more gory details, I’m happy to share – just contact me via email.

## Topple puzzle magazine

December 19th, 2017

Looking for a little more variety in your daily puzzle consumption?

Gregory Gray wrote to tell me about his indie puzzle magazine, Topple. Each issue costs a buck to download and print yourself, and there is a very wide variety of puzzles, many with illustrations which remind me a bit of the Sam Loyd Cyclopedia, if you know what that is.

Greg is currently up to Issue #7, as of this writing. The latest issue contains 16 puzzles, of which four are Nikoli style puzzles (Masyu, Hashi, Nurikabe, Kakuro) and the others are from further afield. I very much like the anagram ring puzzle shown here.

Here’s a sampler that Greg sent me.

And here’s the website where you can get more!

## Where are the logic puzzles?

June 22nd, 2017

I sometimes get emails that ask where the logic puzzles are. “I thought your site had logic puzzles, where are the logic puzzles?” they ask.

My site does indeed have logic puzzles, by which I mean puzzles in which you employ logic to solve them. Pretty much every puzzle on my site meets this description. However, I’m being pedantic. I believe the logic puzzles being referred to by my correspondents are the kind that involve a group of girl scouts, sitting at a counter in a Boba shop, each with a different skill badge, a different pet, and a different beverage:

The 7 year old sits on a red stool.
The 8 year old has a schnauzer.
The 9 year old is drinking a green tea boba.
The green stool is just to the left of the white one.
The girl on the green stool is drinking a mocha boba.
The girl with the computer-expert badge has a one-legged parrot.
The girl on the yellow stool has a dancer badge.
The girl in the center is drinking a strawberry boba.
The 11 year old is on the first stool.
The girl with the hiking badge is next to the girl with an ocelot.
The girl with the pottery badge is drinking taro boba.
The girl with the magical pony is next to the dancer
The 12 year old has a home scientist badge.
The 11 year old is next to the girl on the blue stool.
The hiker is next to the girl drinking a plain tea boba.
Who’s got the Ferret?

Although these are commonly called logic puzzles, I prefer to call them “logic word puzzles”, since there are all kinds of puzzles that employ logic, not just these. I’ve also seen them called Einstein’s Puzzle, also a terrible name.

Whatever they are called, I don’t carry them. This is because of those pesky words. In order to produce them, I would have to come up with 4-6 lists of related things (girl scouts, boba beverages, sweaters, mood-rings, troll figurines, etc.) for each individual puzzle, and this would be extremely labor-intensive, especially since I like to put out many thousands of each puzzle variety, so you don’t run out.

This website currently has over a million puzzles on it, and the reason there are so many is because I’m able to use power tools (namely, computer programming) to produce them. I’ve thought about combining various word lists in combinatorial ways to produce these puzzles, but I don’t think the results would be very appealing (there are five girl scouts, each has a favorite muscle car, lives on a different planet, and has a different number of legs). Probably the best solution would be for me to crowd-source the word lists. The idea being that you, the visitors to this website, could suggest lists of things to use in puzzles), but setting this up would also be very time consuming, and would probably require moderation so as not to be abused by spammers and trolls.

For similar reasons, I don’t carry Fill-a-pix, Mosaik, Connect-the-dots, and other puzzles that employ unique pictures for the solutions, not to mention Word-Search and Crosswords. If you want to set up a moderated crowd-sourcing system for word-puzzle and picture-puzzle content that I can use, let me know when it’s running! I’d love to use it!