1. Circular References
A factory has a circular work cell, which they wish to divide into four equal sections, using three safety curtains, each the same length. How might this be done?
Extra Credit: What is the length of the three safety curtains?
2. The Circling Cart
That same factory has a four wheel cart that rides on a pair of circular tracks. The outside wheels of the cart turn twice as fast as the inner wheels. The cart’s axles are 5 feet wide.
What is the length of the outside track?
3. Ancient Riddle
This is a very old story – you may have heard it. It’s often told regarding 17 cows, but we couldn’t help taking a small liberty:
An eccentric automation tycoon passes away, leaving his empire (consisting of 17 factories) to his children in the following manner: his eldest is to receive half of his factories, his second child – one third of the factories, and his youngest gets one ninth of them. The children do not wish to share or ‘break up’ a factory, and thus they cannot figure out how to divide the empire.
They consult a wise friend (also an automation expert) who offers to loan them a factory, so that with a total of 18 factories; the eldest can take half (or 9 factories), the second child gets the one third share (6 factories), the youngest child receives the one ninth share (2 factories) AND they can give the ‘loaned’ factory back to the friend. The three children also realize that they are each better off in the end, than they would have been if they had subdivided one of the factories.
Can you explain the apparent paradox in simple mathematical terms?
4. Tooling Around
In one well-automated factory, the operators were getting bored, so the foreman offered them a challenge. The factory has 25 CNC machine tools arranged in a neat grid of 5 rows, and 5 columns. Each machine tool has an operator. The foreman offered to let each operator move to a new machine tool as long as they followed certain restrictions. Each operator could only move to a machine tool that was in the position directly in front, or directly behind, or directly to the right side, or directly to the left side – of that operator’s original position. All of the operators were required to move, and in the end there must be an operator at each station.
Were the operators able to meet the foreman’s restrictions? Why or why not?