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Spring 2017

Week #9 - The ancient game of Nim

posted May 15, 2017, 8:24 AM by David Radnell   [ updated May 15, 2017, 8:24 AM ]

Being the last week of the club for spring I though it would be nice to end by using binary numbers that we learned in the first week. [Warning. There are some quite different variants of this game. The one we played is the traditional version]. 

 Nim is a simple and fair two player game. It has a winning strategy that involves some very nice mathematics. 

The game starts with any number of piles of objects. Each pile can have any number of objects (in class we used crayons). 

The players take alternate turns. During each turn a player can take one or more (even all) of the stones from one pile. Whoever makes the last move wins!
 Image result for nim game

What we did:
  • Let the kids play the game without any direction for about 15 minutes just to get used to the game. And to realize that its not so easy!
  • Mathematicians often approach difficult problems by looking at simple special cases. 
    • We played using just two piles and found the winning strategy to be to keep the piles equal.
    • We played using piles of just 1 or two objects. In this case the wining strategy to make even numbers of piles of each type. 
  • I explained the general strategy (explained fully in the links below). It involves thinking of the number in each pile in binary. The strategy is to make an even number of each power of two (1, 2, 4, 8, etc). For example if we have three piles of sizes A = 1, B = 3 and C = 6. We think of B as 1 +2 (or 11 in binary) and C as 2 + 4 (or 110 in binary). So we have two 1's, two 2's and one 4. We have an even number of 1's which is good, and even numbers of 2's. But an odd number of 4's. To fix this we take 4 stones away from pile C. 
Things to try at home: 
  • Play with your friends and family and try to understand the general strategy so you can beat them every time (unless by bad luck the starting position is a loosing one.). 
  • You can practice first with online games. Links below.

Explanations of the game and winning strategy
Online game

week #8

posted May 8, 2017, 11:02 PM by David Radnell   [ updated May 8, 2017, 11:09 PM ]

This week we worked on these puzzles. They come under the general category of combinatorics. They are mostly not too difficult and the kids managed to solve  them with some hints here and there. The last part of (1) is harder and we didn't discuss it. Something to work on at home :) 

Week #7 - Fractals I

posted Apr 24, 2017, 12:05 PM by David Radnell   [ updated Apr 24, 2017, 12:15 PM ]

 Fractal geometry will make you see everything differently. There is a danger in reading further. You risk the loss of your childhood vision of clouds, forests, flowers, galaxies, leaves, feathers, rocks, mountains, torrents of water, carpet, bricks, and much else besides. Never again will your interpretation of these things be quite the same. 
Fractals Everywhere (2000)
Sierpinski triangle  

 triangles GIF
 Why is geometry often described as “cold” and “dry?” One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line… Nature exhibits not simply a higher degree but an altogether different level of complexity. 
From The Fractal Geometry of Nature (1977)
Mandelbrot set 

What we did:
  • We continues from last week looking at the pattern of odd and even numbers in Pascals triangle. Surprisingly this pattern is like the Sierpinski triangle.
  • The we drew several steps of the Sierpinski triangle. Thes steps can be continued for ever the results shape is an example of a fractal
  • Perhaps the most famous fractal is the Mandelbrot set. I showed this on a tablet and the kids got to zoom in and get a hint at the incredible complexity and beauty. Links below for apps. 
  • Next week we'll talk in more detail about fractals and how they model many things in nature. 

Things to try at home: 
  • Carefully draw a Sierpinksi triangle. Experiment doing similar things with other shapes such as square or circles. Invent your own fractals and show me next time. 
  • Play around with the apps below. Zoom into different places in the Mandelbrot set to find. 
  • Family viewing! (just one simple equation make the most complex and beautiful object in existence)


There are many apps and webpages for exploring the Mandelbrot set and other fractals. Here are just some samples.

Web (make the number of iteration higher to get more detail. Draw a rectangle to zoom in)
  • Mandelbrot Explorer
  • Fractview (click on the gallery icon to select different fractals or reset)
  • Fractile plus (in options you can increase the iterations to get more detail when you zoom far)
  • Frax / FraxHD (many color and texture options for creating beautiful images.)

Week #6 - Pascal's Triangle

posted Apr 10, 2017, 4:43 AM by David Radnell   [ updated Apr 10, 2017, 4:52 AM ]

 As usual, the name of something does not refer to its discoverer. Pascal's triangle was known in many parts of the world centuries before Pascal. 

There are many interesting number patterns to explore as well as the relation to counting combinations of objects. 
[Both images from ]

What we did:
  • From 4 coloured pens how many ways can you choose: 0 pens, 1 pen, 2 pens, 3 pens, or 4 pens. We repeated this for 1 coloured pen up to 5 coloured pens. We saw these numbers for Pascal's triangled and we discussed how the addition pattern related to the previous counting question. 
  • Looked for different patterns in Pascal's triangle.
  • Using the hexagon patter we started colouring the even and odd numbers different colours. We then realized that we don't need the actual numbers to figure out which are odd and even. Just that odd+odd = even, odd+even= odd and even + even = even. We continued the colouring enough to see the triangle patterns emerge. 

Things to try at home: 


Week #4, #5

posted Apr 10, 2017, 2:44 AM by David Radnell   [ updated Apr 10, 2017, 2:48 AM ]

Due to scheduled swimming and ice skating the grade 3 and 4 kids were away these two weeks which meant only 2 or 3 kids attended. We just played some games and did some puzzles from the sheet below. After trying to do the cube problem just by visualizing we built a cube by folding paper. 


Week #4 - Mobius Strips

posted Mar 20, 2017, 7:12 AM by David Radnell   [ updated Sep 17, 2017, 10:53 AM ]

We continued this week on the general theme of shape and space (topology). 

                                                                                                    [photo by Steve Krave]
The second photo is a sculpture at Fermilab (the high energy particle physics lab in the USA).

What we did:
  • Made  Mobius strips out of paper and observed it only has one side by starting at a point and drawing down the center until you eventually return to the same point. Strangely, the line you draw is on "both" side of the paper 
  • We then cut along this center line for a surprising result. Cutting the paper in half again leads to more surprises.
  • Next we made a new Mobius strip and cut it into thirds. Again the result is surprising. 
  • Briefly, we figured out how to draw a diagram to understand why these odd things happens. 
  • Looked at the tri-flexagon we made last week. It is actually a Mobius band!!! 
Things to try at home: 
  • Experiment by cutting Mobius strips in different ways
  • Make different loops with a different number of twists and see what happens when you cut them in half.

Week #3 - flexagons

posted Mar 13, 2017, 6:00 AM by David Radnell   [ updated Mar 13, 2017, 6:00 AM ]

This is one of my all time favourite things!

Here are a sequence of entertaining and instructive videos by Vi Hart. 
Please watch!!! :) 

She has lots of other good youtube videos too. Her father, George Hart, is a mathematician/sculptor. He has visited Aalto University and did some cool things as part of a math/art/architecture course. 

What we did:
  • Made tri-flexagons. [image from] 
Things to try at home: 
  • Make more triflexagons. Instructions below. Color or write different non-symmetrical patterns on each triangle so you can see how they kind of rotate when you do the flex. 
  • Make a hexaflexagon. Instructions below. This requires some patience as you need to make the folds quite accurate. You might need a parent or friend to help the first time you do it. Its somehow not as easy as it looks in the videos above :) 

Week #2 - powers of 2 and 3

posted Mar 6, 2017, 12:20 PM by David Radnell   [ updated Mar 13, 2017, 5:23 AM ]

What we did:
  • Figured out the solution to problem 1 from week #1. The weights to use are 1, 3, 9, 27. We thought about why these are going up in powers of 3.  In the case the weights can only be placed on one side of the scale we saw in week#1 that the weights were powers of 2. This is a binary system as there are only two possibilities - the weight is either off the scale or on the scale. Now that the weight can be placed on either side there are three possibilities -  the weight is off the scale, on the left (subtracted) or on the right (added). We then talked quickly about base three (ternary), as compared to the usual base 10 number system and the base two (binary) number system we discussed last week. 
  • While we were talking about powers of 2, we talked about the famous wheat and chess board problem. I was hoping we could estimate (without a calculator) the number of grains of wheat but there was not enough time. Maybe some other week. The idea is that 2^10 is 1024 which approximately 1000. So 2^20 is approximately 1,000,000. And so on.
  • We ended by playing this two person game. The secret will be revealed next week! It is a familiar game in disguise :).  Write down the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9. Player X chooses a number and that number is crossed off the list. Then player Y chooses a number and that number is crossed off the list. And so on. The aim is to be able to make exactly 3 of your numbers add to 15. For example if after thee turns player X has 1, 5, 8. And Y has 3, 9, 2. Then X chooses 6 and wins as 1+8+6 = 15.   

Things to try at home: 
  • Force your friends and parents of to play the game described above. Write down all the ways you can make 15. Try to organize the information in a useful way. Maybe you will discover the secret! 


Meeting #1 - Binary Numbers

posted Feb 27, 2017, 12:04 PM by David Radnell   [ updated Feb 27, 2017, 11:05 PM ]

What we did: Discussed binary numbers and in particular how to count to 1023 on your fingers.
  • We motivated this by solving the problem of how to weigh items of wright  1kg, 2kg, 3kg, ..., 15kg on a balance scale using the minimum number of weights. The weights are only allowed to be placed on one site of the scale. The answer is to use weights of 1kg, 2kg, 4kg and 8kg. 
  • Labeling the fingers on one hand with values 1, 2, 4, 8, 16, we found we can count to 31 on one hand. On two hands we can count to 1023.
  • We played briefly with the magic trick here. The kids all got copies of the cards to take home. 
Things to try at home: 
  1. Modify the scale problem above where you can now put the weights on either side of the scales. What is the minimum number of weights needed so you can weigh items from 1kg to 40kg. This is really the famous weighing problem of the French mathematician Claude Gaspard Bachet de Meziriac (1581-1638). "A merchant had a forty-pound measuring weight that broke into four pieces as the result of a fall. When the pieces were subsequently weighed, it was found that the weight of each piece was a whole number of pounds and that the four pieces could be used to weigh every integral weight between 1 and 40 pounds. What were the weights of the pieces?"
  2. What is the highest number you can count to using your fingers and toes?
  3. Understand how the magic trick works. 
  4. Binary numbers game. A good mental arithmetic workout!

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