## Puzzle Vessels and The Knot

• ### Basin and bucket

The Fairness Cup is an example of a siphon. A siphon is a method of moving water uphill, but it requires special conditions for it to work. The water must receive some initial force to move it through the tube, and then it will continue flowing as long as there is no air in the line. In this case the initial force is provided by the water pressure in the cup. When the water reaches a level high enough above the straw, the water pressure at the bottom of the straw, where it is open inside the cup, is strong enough to push water up the straw. When the water reaches the curve at the top of the straw, it will go right over, pulling the rest of the water in the straw behind it. This water in turn pulls on water in the cup, draining it until it is empty of until air enters the straw. Then the suction is broken, and the siphon is no longer in action.

The Rising Head Cup is a simple demonstration of density. More dense things sink, less dense things rise. In this case, foam is much less dense than water, so the foam block at the bottom of the straw rises as the water level comes up. A good question to ask is, "would the straw still rise if the foam was not glued to the bottom of it?" Density can be described as "how much stuff there is in a certain space," or "how much air is in a material." The Styrofoam is a good example of something being less dense because it has big air pockets in it. Also a sponge.

The Towers of Hanoi is a mathematical game that can be predicted by an algorithm. An algorithm is a rule that predicts the outcome of some question as you change the inputs. In this case, we can predict the minimum number of moves necessary to "win" at Towers of Hanoi depending on how many disks we are using. When we use one disk, obviously one move finishes the game. It is a good idea to make a table to help you figure out the pattern:

# of disks (n).          Minimum # of moves to win

1                            1

2                            3

3

4

5

6

7

8

etc.

Warning! Here comes the rule. If you'd rather figure it out, don't look.

(2^n) - 1 = minimum number of moves to win

You can use this rule to figure out how many moves it would take to "win" with 100 disks, or 3,000, or 4 billion. Of course you could also cut out 3,000 disks and figure it out step by step if you want to...

The know is about topology, which is a very lovely and very abstract field of mathematics that is concerned with shapes in many dimensions, and how they change as you manipulate them. The knot is an example of a one-dimensional topology problem, since the string is a one-dimensional object. You could talk a little bit about 1, 2, and 3 dimensions with the kids if you'd like (string, paper, boxes are good examples). Then have them tell you examples of 1, 2, and 3 dimensional objects. It's an excellent math exercise, and one that any person can do without too much math training.

1. FC: A siphon will take liquid over a hill once it has started. The tube, or path, must be full of the liquid first, and the output must be lower than the intake.

2. Math is not just about numbers and counting. Shapes and patterns also play a large role.

3. Strings and knots and puzzles like the first two here are examples of the area of math called topology, which is about how geometric figures are different and similar.

1. What makes the Fairness cup work?

2. What would happen if the open end of the straw was sticking up without bending over and down?

3. Could you make a string puzzle that could not be solved?

4. If you had four holes in the wood of the "Two Rings" project, and two knots in the center holes, could you get the washers from one side to the other.