Wednesday, September 17, 2014

modular tables

No, not a post about IKEA furniture. A while  ago I put up a post on colouring multiplication tables by assigning ranges of numbers a colour value. You end up with something that looks like a rainbow.

This image was made in Tinkerplots, so it was easy to go from a 10 x 10 table to a 50 x 50 table (removing the numbers and just keeping the colours, and shrinking each cell down a bit):

Inspired by the "Zn Multiplication visualizer" found here and mentioned here, and thinking about modular arithmetic from the last post, I decided to make a few more images.

If you take the values in this 50 x 50 table mod 9, you'll get this nice quilt - repeating in 9 x 9 blocks.

If you choose your mod value to line up with the width of the table (take the values mod 50) you get this:

The light blue along the bottom and the right side are all zeros - anything that is a multiple of 50, as the bottom and the far right side are, will have a remainder of 0 when you divide them by 50.

If you take the values mod 2500, then you get something that looks like the rainbow we started with - except for one square (maybe you can spot it, or figure out where it will be).

Here's a 75x75 table, mod 75, in shades of blue.

Monday, September 15, 2014

squashing multiples

An elementary school exercise leads to writing a simple program, a little proof by contradiction, and learning about some mostly-forgotten calculation tricks: just some of the fun that can be had when playing with simple math. Sound good? It all starts with squashing numbers...

No doubt you've noticed some patterns in the non-zero multiples of 9: 9, 18, 27, 36, 45,... One thing to notice is that if you (repeatedly) add up all the digits of a multiple of 9, you always get 9 as your answer.

This works immediately for many multiples of 9, like 9*14 = 126 (1 + 2 + 6 = 9), for others you need to keep squashing - if the first digit sum itself has more than one digit, sum those digits and repeat until you get a single digit answer. For example 9 * 42 = 378 (3 + 7 + 8 = 18, which has two digits, so keep squashing: 1 + 8 = 9). 

All multiples of 9 squash down to 9, which is neat. More importantly this also works in the other direction: any positive integer that squashes to 9 is a multiple of 9. This makes squashing an easy divisibility test for 9 and makes it easy to find multiples of 9 (is 359 a multiple of 9? No, but 369 is, and so is 459).  Also: Any number made by rearranging the digits of a multiple of 9 will also be a multiple of 9 (re-arranging the digits won't change the squash value). So, since I know that 882 is a multiple of 9, I also know that 288 and 828 are also multiples of 9.

Can you always squash a number? Could there ever be a number that gets bigger when you sum its digits?

One (wordy) argument goes like this: To obtain a contradiction, assume there are some positive integers greater than 9, whose digit sums are equal to or greater than the original number. Such a number would be a problem for us: we need to be sure that any number with two or more digits will always have a digit sum less than itself (so we can keep doing digit sums until the number is squashed down to a single digit).  Choose n to be the smallest such troublesome number. Suppose the digit sum of n is another number k such that n <= k. Now we are going to build a new number m by taking the digits of n and changing one of them: chose a non-zero digit in a position bigger than the ones place and decrease it by 1. For example, if our number n was 567 (which it isn't) our new number m could be 557. Now m is at least 10 less than n, but its digit sum is only one less than k (since only one digit was decreased by 1). Now m <= n -10 < - 1 <= k-1, and so m is also less than its digit sum (k - 1). But n was chosen to be the smallest number with this property, and m is definitely smaller than n. So we have a contradiction: it cannot happen that the digit sum of a number is greater than the number itself. This means that it is safe to squash: you will always get smaller and smaller numbers until you get down to the single digits.

Squashing multiples of 9 was interesting: What about patterns in other multiples? Consider positive multiples of 3: 3, 6, 9, 12, 15, 18, 21, 27, ... they don't squash down to the same value, but if you try it out you'll notice a pattern: 3, 6, 9, 3, 6, 9, 3, 6, 9, .... It's easy to see this if you write the multiples in a 3 column table.

In the chart above, the first column entries all squash to 3, the second to 6, and the third to 9. Some other multiples produces similar charts, for example 6 and 12. (I noticed these squashing patterns when looking at material from the JUMP math program for grades 3 and 4, where tables like these are used to explore patterns for learning multiplication facts.)

Not all integers will have their multiples fit into this pattern. For example, 4 needs a 9 column table to show a squash pattern.

You might want to write a little program to do your squashing for you - whether you do or squash by hand, you'll see a clear pattern.

Just a brief note on the Java utility below: the while(true) statement in the squash() method relies on our assumption that digit sums get smaller - otherwise we'd potentially have an infinite loop.  The digits() method is simple example of recursion.

package squash;
import java.util.ArrayList;
import java.util.List;

public class SquashCalculator {

public List digits(Integer start) {
if (start == 0) {
return new ArrayList();
int val = start % 10;
List recur = digits(start/10);
return recur;

public Integer sum(List list) {
Integer sum = 0;
for (Integer i: list) {
sum += i;
return sum;

public Integer digitSum(Integer i) {
return sum(digits(i));

public Integer squash(Integer n) {
if (n == 0) {
return 0;
Integer current = n;
while(true) {
current = digitSum(current);
if (digits(current).size() == 1) break;
return current;

Here's what you'll observe squashing the numbers 1 - 40:

Maybe you knew that this would happen, but I didn't. I was surprised at first that squashing numbers formed this regular pattern, but it really isn't surprising if you think about it. If you consider the squash of n  and then wonder what squash of n + 1 should be, it should just be one more, unless the squash of n was already 9, in which case adding one more will give you squash of 10, which is 1.

So, the squash function maps the integers onto a structure like the one on the right below, that is very close to taking the number "mod 9".

Putting the integers from 1 to 36 in a 9 column chart like the "multiples of 4" above shows this also.

This suggests that there is a direct formula for the squash of a number close to its mod 9 value. It turns out this can be expressed as:

One aspect of this is that n and squash (n) are congruent modulo 9 (which means that if you divide n by 9, or the squash of n by 9, you will get the same remainder).

This is the important relationship that makes squashing useful. It's kind of amazing that you can take a number and completely re-arrange its digits, or sum up all its digits, and still retain something about the original number (the remainder after dividing by 9). When you do something this violent to a number it's surprising that some information about the original number remains.

This is why every number that can be squashed to 9 is divisible by 9 (both are equal to 0 mod 9). This also helps to explain the number of columns in the tables above. For a number m whose multiples we are playing with, if m is divisible by 3 (like 3, 6, and 12), its multiples will have the same squash with a period of 3 (and will repeat in a 3 column table). If m is not divisible by 3 (the only factor 9 other than 1 and itself), then the squash of the multiples of m will have a period of 9 (and will repeat in a 9 column table). If m is a multiple of 9, then its multiples will be multiples of 9 also, and their squash will always be 9.

We also get a divisibility test for 3: If a number's squash value is 3, 6, or 9, then that number is divisible by 3. For example, suppose n squashes to 6. That means that n is congruent to 6 mod 9, which means that there is some positive integer k such that n = 9k + 6. Since the right hand side of that equation is divisible by 3 (dividing the right  by 3 gives 3k +2), so is the  left hand side.

But wait, there is more. Calculating squash values mod 9 has a short-cut: when you are adding up all the digits, you can throw out any multiples of 9, since they will always end up contributing zero to the final answer (because multiples of 9 have a remainder of 0 when divided by 9). This calculation is part of an error-checking technique called "casting out nines" which can be used to check arithmetic. When casting out nines, you essentially squash (ignoring 9s and multiples of 9s) the inputs and outputs of your calculation, and if they are different then you know you made a mistake.

If you want to learn more about this (and there is a lot more), you should Google "digital root" which is the standard name for what I've been calling "squash."

Friday, June 20, 2014

Wednesday, May 28, 2014

origami workshop

I mentioned in the previous post that I was considering doing some modular Sonobe origami in an upcoming workshop for middle school students. Wondering if this is a good idea, and thinking that I had better have some backup plans, I decided to make a list of origami models that I have used in school workshops in the past.

Simple Triangles
These simple models are nice for the very young and for beginners. Start with a triangle made from cutting standard origami paper along a diagonal.

A bit of playing around with the triangles should be enough for you to figure out how to fold examples like the ones above, and maybe even to come up with a few of your own. The instructions for the sailboat are on the OrigamiUSA diagrams page.

Hopping Frog
This is a very nice toy model from OrigamiUSA. See this post for some notes on it. Whenever I have used this with younger kids, I have always brought some pre-folded ones with me for them to play with and draw on.

Paper Cup
A traditional model using a square origami paper, the instructions are also available from OrigamiUSA. It makes a nice pocket to put your frog into. See this post for more on this model.

The origami t-shirt is a model that I use to start off workshops for older students - for younger children I usually bring some paper that has had the initial folds completed, so they can get to the finished product in a few folds - this motivates them to go back and do the whole model themselves. You can find the many origami shirts online, the one I use is one by Gay Merrill Gross that appeared in the first issue of Creased magazine.

Maybe its because we don't send letters much that this model does not grab everyone as overly exciting. I like it, and it is good to have some origami that you can do with standard letter paper (lots of these available in schools). See this post about the crease pattern. Also from the OrigamiUSA page.

Traditional Boat
This is the first and only origami model that I learned as child and I enjoy sharing it with children. It's another model (like the frog and the cup) that you can play with, and like the envelope it can be made from standard letter paper. The diagram is available on OrigamiUSA.  The boat becomes even more  playful when it is used as "storigami" in the tale of The Captain's Shirt. Happily destroying your boat while reciting the tragic story teaches an essential lesson about the impermanence of the origami art form.

Waterbomb Octahedron
I had success with this model in a workshop for grade 7 and 8 students. There is a very nice overview of it here, and the model was also featured in Creased magazine under the name "origami blowtop," but I have not found the issue.

Its modules are the first few folds of the waterbomb, another popular model for beginners. The waterbomb base has also made its way into origami tessellations (see here). If I were to do a tessellation in a workshop, it would be the waterbomb.

Picture Frame
All of the lesson plans for Creased magazine's "Teachers' Corner" are available online here. The picture frame is presented here in lesson 2. The blinz base used to create the picture frame is know to many children as the preliminary folds for the fortune teller or cootie catcher - another origami toy that is always fun to make.

Robinson's Butterfly
I've not yet used this in a workshop, but have enjoyed showing people how to fold it. You can make these from those pesky magazine inserts, or maybe from bus transfers. The model comes from Robinson's book "The Origami Bible." I have a post about it here.

When considering how to present these models, and how to conduct workshops for young people, I've looked into the "origametria" approach described in this paper by Miri Golan and Paul Jackson. Perhaps because I have only done origami with students in workshop settings where the emphasis is not on the mathematical value (which I hope is intrinsic anyway), rather than as part of a regular classroom program, I have not followed the origametria principles very closely. However, their advice on using positive language and respecting the students's work and efforts are essential in any setting - any feeling students have about what they experience will outlive anything mathematical they happen to notice about the creases.

The resources on OrigamiUSA and Creased are great, but keep in mind that very few beginners are good at reading origami diagrams, and most children are put off by them. I like to have the instructions available for the students, but most of them learn the pieces by following along with you, and having you (and others) repeat the folds in front of them. A document camera is a plus, but you have to be willing to walk around and demonstrate the folds, and recruit helpers who are ahead of the others to assist you.

Update: A handout used for the workshop, which did include a little modular origami, is here.

Monday, May 26, 2014

simple sonobe

I'm prepping for an origami workshop for middle school students, and am thinking about focusing on modular origami with sonobe units. In modular origami, many folded pieces of paper are assembled to make a model. Usually, all the pieces of paper (called units) are folded the same way. Like other forms of origami, modular origami is generally done without glue or scissors, so the pieces need to fit together so that they lock in place - putting the pieces together often feels like weaving or braiding. In past workshops for children at this age I have used a simple waterbomb octahedron (like this), but the sonobe presents a bit of a challenge - I might fall back on something with less frustration potential. On the other hand, that frustration is part of the lesson that sonobe teaches - it always seems like it just is not going to come together, and then it does.

There are some good instructions for modular origami projects using the sonobe unit out there - for folding the unit this one is good, and for assembly I like this one.

The version of the unit that I like to use is a simpler one than is presented in the instructions that I have found online, and is taken from Origami for the Connoisseur, by Kasahara and Takahama (not really being a connoisseur, it is one of the very few things I can put together from this book). I've attempted some diagrams for this below.

The unit has a few less folds than the standard version, and this makes a difference when you are making 30 or more of them.

The Simple Sonobe Unit

Sonobe Unit Assembly

Update: a handout based on these diagrams is here.