Magical Maze - 35 Activities

The theme of the 1997 Christmas Lectures is "Mathematics and Nature". Here are a few activities and/or problems that are loosely related to the topics of the five lectures.

For a summary of the 5 lectures, see the Magical Maze Lecture Summary .

• Lecture 1: Sunflowers and Snowflakes
• Lecture 2: The Pattern of Tiny Feet
• Lecture 3: Outrageous Fortune
• Lecture 4: Chaos and Cauliflowers
• Lecture 5: Fearful Symmetry

Lecture 1: Sunflowers and Snowflakes

1. If its starts to snow, go outside with a dark coat and a magnifying glass, and look at the snowflakes before they can melt. See if they have hexagonal (six-sided) symmetry.
2. On a dark, clear night, go outside and look at the sky every half an hour or so for a couple of hours. Pay especial attention to some chosen region of sky (I use the bit just above my garage when seen from the front porch). Does the whole sky seem to be rotating slowly? Does one star seem not to move? How long do you think it takes the whole sky to rotate once, and why?
3. Borrow a guitar, and look at the spacing of the frets. Count 12 frets up from the end of the string, which is where you put your finger to sound the octave. Measure the distance from that point to the two ends of the string (at the guitar neck and at the bridge). What do you notice?
4. Tape a sheet of paper to a sloping board, and roll a marble that has been dipped in (washable!) paint upwards and sideways. See what shaped curve you get.
5. Get hold of a sunflower, daisy, or chrysanthemum. Look at the centre of the flower, perhaps using a magnifying glass. Notice the spiral patterns. Count how many clockwise spirals there are, and how many anti-clockwise ones there are.
6. The answer to 5 is in fact connected with the "Fibonacci series" of numbers, which goes
   

   1 1 2 3 5 8 13 21 34 55

   and so on, each number being the sum of the previous two. Continue this series and stop when you first come to a perfect square. Which square do you get?

7. 3 x 8 = 24 = 5 x 5 - 1 . Something very like this pattern continues throughout the Fibonacci series, but there's one tiny twist. What is it?

Lecture 2: The Pattern of Tiny Feet

1. Watch how your pet cat, dog, or gerbil moves its feet when it's walking. Can you see any patterns in the timing of feet hitting the ground?
2. Do the same for yourself when walking. Now the pattern is so familiar that it may take a few moments' thought to see that there is one!
3. Stand in front of a mirror, facing towards the side, and walk slowly sideways. How does the walking image in the mirror relate to the real you?
4. Why do geese fly in V-shaped patterns in the sky? Is the goose at the front of the V a special 'leader', or might it be there by accident?
5. Make a pendulum by tying a heavy object to a bit of string about a metre long, and hang it from the edge of a table or something similar. Make another one, exactly the same, and hang it about 5 cm to one side of the first. Join them by a short length of string, also 5cm long plus the knots at the ends, forming a kind of H shape relative to the two verticals, with the cross-bar of the H about 10 cm from the top of the vertical strings. Push one pendulum so that it starts to swing. What happens to the other one?
6. Start both pendulums swinging side by side, 'in step'. See what happens.
7. Start both pendulums swinging side by side, 'out of step' (That is, starting at opposite ends of their swings.) See what happens.
8. Try the same game with three or more pendulums in a row, all linked by strings near the top.

Lecture 3: Outrageous Fortune

1. a) Get a coin and keep tossing it until five heads turn up in a row.

   Do you think that on the next throw:

   1. Heads are more likely than tails?
   2. Heads are less likely than tails?
   3. Heads and tails are equally likely?
2. Test your answers to (a) by repeating the experiment twenty times. On each occasion keep tossing the coin until five heads turn up in a row; then record the NEXT toss.
3. On average, how many times should you toss a die (most people say 'dice' but really that's the plural) to get a six?
4. A drawer contains some red socks and some blue ones. When two socks are chosen at random, the probability that they are both red is 1/2.
   1. What is the smallest number of socks that there could be in the drawer?
   2. If the number of blue socks is known to be even, What is the smallest number of socks that there could be in the drawer?
5. In World War I, when soldiers were equipped with steel helmets to protect their heads from flying bullets, the number of head injuries INCREASED. Why?
6. Which is the less likely: being struck by lightning, or being hit by a falling meteorite?

Lecture 4: Chaos and Cauliflowers

1. Go to greengrocers until you find a 'broccoli romanesco'. What do you notice about this exceedingly curious plant?
2. How long is the coast of Britain?
3. Take a pocket calculator, choose some starting number between 0 and 1, such as 0.3541, and repeatedly square it (iterating the mapping x ---> x*x). What happens to the numbers?
4. Do the same, but now send x not to x*x but to x*x - 1 (and repeat starting from the new value). What happens to the numbers?
5. Do the same, but now send x not to x*x but to 2x*x - 1 (and repeat starting from the new value). What happens to the numbers?
6. Why do different things happen in the three cases?

Lecture 5: Fearful Symmetry

• Why does a mirror reverse left and right but not top and bottom?
• DOES a mirror actually reverse left and right?
• If not, what DOES it reverse?
• Get two small mirrors and place them on a sheet of plain paper so that they meet along one vertical edge at an angle of 60 degrees. Put a coin between them. How many images of the coin can you see? Why?
• Change the angle to 45 degrees, and repeat.
• Change the angl