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?
Length of the three safety curtains = 0.5πd (where d is the diameter of the work cell)
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
The diameter of the outer track is twice the diameter of the inner (in order to cause the wheels to turn at twice the rate), and the inner track is 10 feet less than the outer track (i.e. twice the axel width), so the outer track must be 20 feet in diameter. Thus the length of the outer track is 20π, or about 20.8 feet.
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
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
Can you explain the apparent paradox
in simple mathematical terms?
The parent is either not very good at math – or they wanted a small portion of the inheritance to go un-inherited (to charity? or perhaps ‘to the state’?). The portions stipulated (½ + 1/3rd +1/9th) only add up to 17/18ths of the total fortune. There is still 1/18th of the fortune that is not bequeathed (almost an entire factory).
When the friend artificially increases the inheritance to 18 factories, then all the fractions work out to even numbers, and there is no ‘unused portion’ in that solution – it is effectively divvied-up amongst the children.
And yes, each child gets more than was originally stipulated. And whether the parent’s wishes were honored is a very good question. In the first reckoning the children would only have received 8.5, 5.7, and 1.9 factories respectively.
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
Were the operators able to meet the
foreman's restrictions? Why or why not?
There are many ways to prove that the foreman’s challenge cannot be accomplished. Here’s one:
Imagine that the machine tools were laid out on a checker board, with alternating red and black squares. It’s easy to see that each operator would have to change his/her color during the move (i.e. each would have to move from either red to black, or black to red). Now count the occurrences of each color. Because 25 is an odd number, there must be more of one color that the other – so it follows that not all the operators can change their color. There is no way for all of them to move given the foreman’s restrictions – but he did solve their boredom problem for at least a few minutes while they argued about it ;-)
here to return to the Break Room
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Brainteaser Answers, Issue 25, 2013